List of polygons, polyhedra and polytopes
Encyclopedia
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.
Quadrilateral
Star polygons
Uniform tilings in hyperbolic plane
Archimedean tiling
Tetrahedron
Pentahedron
Hexahedron
Heptahedron
Octahedron
Enneahedron
Decahedron
Dodecahedron
Antiprism
Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space
and Uniform polyhedron compound
Polychoron
- general term for a four dimensional polytope
Regular convex polychoron
Abstract regular polytope
Schläfli–Hess polychoron (Regular star polychoron)
Uniform polychoron
Prismatic uniform polychoron
Uniform antiprismatic prism
and uniform 5-polytope
Prismatic uniform 5-polytope
and uniform 6-polytope
, uniform 7-polytope
, uniform 8-polytope
and List of regular polytopes
Uniform polytope
Honeycombs
Polygon (2-polytope)
- VertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
- EdgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
the facet or (n-1)-face of the polygon
Polyhedron (3-polytope)
- VertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
the peak or (n-3)-face of the polyhedron - EdgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
the ridge or (n-2)-face of the polyhedron - FaceFace (geometry)In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
the facet or (n-1)-face of the polyhedron
Polychoron (4-polytope)
- VertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
- EdgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
the peak or (n-3)-face of the polychoron - FaceFace (geometry)In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
the ridge or (n-2)-face of the polychoron - CellCell (geometry)In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
the facet or (n-1)-face of the polychoron
Polyteron (5-polytope)
- VertexVertex (geometry)In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
the (n-5)-face of the polyteron - EdgeEdge (geometry)In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
the (n-4)-face of the Polyteron - FaceFace (geometry)In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
the peak or (n-3)-face of the polyteron - CellCell (geometry)In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb For example, a cubic honeycomb is made...
the ridge or (n-2)-face of the polyteron - HypercellHypercellIn geometry, a hypercell is a descriptive term for an element of a polytope or tessellation, usually representing an element one dimension higher than a cell. The most generally accepted term is 4-face because it contains a 4-dimensional interior...
the facet or (n-1)-face of the polyteron
Other
- PointPoint (geometry)In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
- Line segmentLine segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
- Vertex figureVertex figureIn geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...
- PeakPeak (geometry)In geometry, a peak is an -face of an n-dimensional polytope. A peak attaches at least three facets ....
- (n-3)-face - RidgeRidge (geometry)In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a facet.By dimension, this corresponds to:*a vertex of a polygon;...
- (n-2)-face - Facet - (n-1)-face
Two dimensional
TriangleTriangle
A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....
- Equilateral triangle
- Isosceles triangle
Quadrilateral
Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
- SquareSquare (geometry)In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
, QuadrilateralQuadrilateralIn Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
, RectangleRectangleIn Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. The term "oblong" is occasionally used to refer to a non-square rectangle...
, RhombusRhombusIn Euclidean geometry, a rhombus or rhomb is a convex quadrilateral whose four sides all have the same length. The rhombus is often called a diamond, after the diamonds suit in playing cards, or a lozenge, though the latter sometimes refers specifically to a rhombus with a 45° angle.Every...
, ParallelogramParallelogramIn Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure...
, TrapezoidTrapezoidIn Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
, Isosceles trapezoidIsosceles trapezoidIn Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides, making it automatically a trapezoid...
, KiteKite (geometry)In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...
, RhomboidRhomboidTraditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are oblique.A parallelogram with sides of equal length is a rhombus but not a rhomboid....
- PentagonPentagonIn geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
- Hexagon
- Heptagon
- Octagon
- EnneagonEnneagonIn geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century...
- DecagonDecagonIn geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and each internal angle equal to 144°...
- HendecagonHendecagonIn geometry, a hendecagon is an 11-sided polygon....
- DodecagonDodecagonIn geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...
- Triskaidecagon
- Tetradecagon
- PentadecagonPentadecagonIn geometry, a pentadecagon is any 15-sided, 15-angled, polygon.- Regular pentadecagon:A regular pentadecagon has interior angles of 156°, and with a side length a, has an area given by...
- HexadecagonHexadecagonIn mathematics, a hexadecagon is a polygon with 16 sides and 16 vertices.- Regular hexadecagon :A regular hexadecagon is constructible with a compass and straightedge....
- HeptadecagonHeptadecagonIn geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
- OctadecagonOctadecagonAn octadecagon is a polygon with 18 sides and 18 vertices. Another name for an octadecagon is octakaidecagon.- Construction :A regular octadecagon cannot be constructed using compass and straightedge.- Petrie polygons :...
- EnneadecagonEnneadecagonIn geometry, an enneadecagon is a polygon with 19 sides and angles. It is also known as an enneakaidecagon or a nonadecagon.The radius of the circumcircle of the regular enneadecagon with side length t is...
- IcosagonIcosagonIn geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.One interior angle in a regular icosagon is 162° meaning that one exterior angle would be 18°...
- TriacontagonTriacontagonIn geometry, an triacontagon is a thirty-sided polygon. The sum of any triacontagon's interior angles is 5040 degrees.One interior angle in a regular triacontagon is 168° meaning that one exterior angle would be 12°...
- GnomonGnomon (figure)In geometry, a gnomon is a plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram. More generically, the term gnomon denotes the form that is to be added to a figure to produce a larger figure of the same shape....
- GolygonGolygonA golygon is any polygon with all right angles, whose sides are consecutive integer lengths. Golygons were invented and named by Lee Sallows, and popularized by A. K. Dewdney in a 1990 Scientific American column...
Star polygons
- PentagramPentagramA pentagram is the shape of a five-pointed star drawn with five straight strokes...
, Hexagram, HeptagramHeptagramA heptagram or septegram is a seven-pointed star drawn with seven straight strokes.- Geometry :In general, a heptagram is any self-intersecting heptagon ....
, OctagramOctagramIn geometry, an octagram is an eight-sided star polygon.- Geometry :In general, an octagram is any self-intersecting octagon ....
, Enneagram (geometry)Enneagram (geometry)In geometry, an enneagram is a nine-pointed geometric figure. It is sometimes called a nonagram.-Regular enneagram:A regular enneagram is constructed using the same points as the regular enneagon but connected in fixed steps...
, DecagramDecagramDecagram may refer to:* 10 grams, unit of mass* Decagram , geometric figure...
, Hendecagram, Dodecagram
Families
- PolygonPolygonIn geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
- Concave polygon
- Cyclic polygon
- Regular polygonRegular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
- Star polygon
- PolyformPolyformIn recreational mathematics, a polyform is a plane figure constructed by joining together identical basic polygons. The basic polygon is often a convex plane-filling polygon, such as a square or a triangle. More specific names have been given to polyforms resulting from specific basic polygons, as...
- Johnson solidJohnson solidIn geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
- Prism (geometry)Prism (geometry)In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
- AntiprismAntiprismIn geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
- Pyramid (geometry)Pyramid (geometry)In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
- BipyramidBipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
- TrapezohedronTrapezohedronThe n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...
- CupolaCupolaIn architecture, a cupola is a small, most-often dome-like, structure on top of a building. Often used to provide a lookout or to admit light and air, it usually crowns a larger roof or dome....
- Bicupola
- Truncated trapezohedronTruncated trapezohedronAn n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....
- Snub polyhedronSnub polyhedronA snub polyhedron is a polyhedron obtained by adding extra triangles around each vertex.Chiral snub polyhedra do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other...
- Archimedean solidArchimedean solidIn geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
- Catalan solidCatalan solidIn mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
- DeltahedronDeltahedronA deltahedron is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek majuscule delta , which has the shape of an equilateral triangle. There are infinitely many deltahedra, but of these only eight are convex, having 4, 6, 8, 10, 12, 14, 16 and 20 faces...
- ZonohedronZonohedronA zonohedron is a convex polyhedron where every face is a polygon with point symmetry or, equivalently, symmetry under rotations through 180°. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional...
- PrismatoidPrismatoidIn geometry, a prismatoid is a polyhedron where all vertices lie in two parallel planes....
- Semiregular polyhedronSemiregular polyhedronThe term semiregular polyhedron is used variously by different authors.In its original definition, it is a polyhedron with regular faces and a symmetry group which is transitive on its vertices, which is more commonly referred to today as a uniform polyhedron...
Tilings
List of uniform tilingsUniform tilings in hyperbolic plane
Uniform tilings in hyperbolic plane
There are an infinite number of uniform tilings on the hyperbolic plane based on the where 1/p + 1/q + 1/r ...
Archimedean tiling
Uniform tiling
In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-uniform.Uniform tilings can exist in both the Euclidean plane and hyperbolic plane...
- Square tiling
- Triangular tiling
- Hexagonal tiling
- Truncated square tilingTruncated square tilingIn geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. There is one square and two octagons on each vertex. This is the only edge-to-edge tiling by regular convex polygons which contains an octagon...
- Snub square tiling
- Trihexagonal tilingTrihexagonal tilingIn geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex...
- Truncated hexagonal tilingTruncated hexagonal tilingIn geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons and one triangle on each vertex....
- Rhombitrihexagonal tiling
- Truncated trihexagonal tiling
- Snub hexagonal tilingSnub hexagonal tilingIn geometry, the Snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex...
- Elongated triangular tilingElongated triangular tilingIn geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex.Conway calls it a isosnub quadrille....
Regular
Regular polyhedronRegular polyhedron
A regular polyhedron is a polyhedron whose faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive - i.e. it is transitive on its flags...
- Platonic solidPlatonic solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
:- TetrahedronTetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
, CubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, OctahedronOctahedronIn geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
, Dodecahedron, IcosahedronIcosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
- Tetrahedron
- Regular spherical polyhedronSpherical polyhedronIn mathematics, a spherical polyhedron is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons...
- DihedronDihedronA dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is...
, Hosohedron
- Dihedron
- Kepler–Poinsot polyhedron (Regular star polyhedra)
- Abstract regular polyhedra (Projective polyhedronProjective polyhedronIn geometry, a projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids....
)- Hemicube (geometry), hemi-octahedronHemi-octahedronA hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.It has 4 triangular faces, 6 edges, and 3 vertices...
, hemi-dodecahedronHemi-dodecahedronA hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and...
, hemi-icosahedronHemi-icosahedronA hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
- Hemicube (geometry), hemi-octahedron
Tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
- DisphenoidDisphenoidA disphenoid is a polyhedron whose four faces are identical triangles. The faces of a tetragonal disphenoid are isosceles; the faces of a rhombic disphenoid are scalene. If the faces are equilateral triangles, one obtains a regular tetrahedron, which is not normally considered a disphenoid.All the...
Pentahedron
Pentahedron
In geometry, a pentahedron is a polyhedron with five faces. Since there are no face-transitive polyhedra with five sides and there are two distinct topological types, this term is less frequently used than tetrahedron or octahedron....
- Square pyramidSquare pyramidIn geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
, Triangular prismTriangular prismIn geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
Hexahedron
Hexahedron
A hexahedron is any polyhedron with six faces, although usually implies the cube as a regular hexahedron with all its faces square, and three squares around each vertex....
- ParallelepipedParallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
, CuboidCuboidIn geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing definitions of a cuboid in mathematical literature...
, RhombohedronRhombohedronIn geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length....
, Trigonal trapezohedronTrigonal trapezohedronIn geometry, the trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has six faces which are congruent rhombi....
, CubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, Pentagonal pyramidPentagonal pyramidIn geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point . Like any pyramid, it is self-dual....
, Triangular dipyramidTriangular dipyramidIn geometry, the triangular bipyramid is the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces....
, quadrilateral frustumFrustumIn geometry, a frustum is the portion of a solid that lies between two parallel planes cutting it....
Heptahedron
Heptahedron
A heptahedron is a polyhedron having seven sides, or faces.A heptahedron can take a surprising number of different basic forms, or topologies. Probably most familiar are the hexagonal pyramid and the pentagonal prism. Also notable is the tetrahemihexahedron, whose seven equilateral triangle faces...
- hexagonal pyramidHexagonal pyramidIn geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point...
, pentagonal prismPentagonal prismIn geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...
, tetrahemihexahedronTetrahemihexahedronIn geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
Octahedron
Octahedron
In geometry, an octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex....
- Hexagonal prismHexagonal prismIn geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
, Truncated tetrahedronTruncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
, Tetragonal trapezohedronTetragonal trapezohedronThe tetragonal trapezohedron or deltohedron is the second in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has eight faces which are congruent kites.- External links :*...
Enneahedron
Enneahedron
In geometry, a enneahedron is a polyhedron with 9 faces. There are 2606 topologically distinct enneahedra and none are regular, so this name is ambiguous.-Examples:...
- Octagonal pyramid, Heptagonal prismHeptagonal prismIn geometry, the heptagonal prism is the fifth of the infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two heptagons....
Decahedron
Decahedron
In geometry, a decahedron is a polyhedron with 10 faces. There are 32300 topologically distinct decahedra and none are regular, so this name is ambiguous.With regular faces:* Octagonal prism * Square antiprism...
- Octagonal prismOctagonal prismIn geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :...
, Square antiprismSquare antiprismIn geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps...
, Square cupolaSquare cupolaIn geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids . It can be obtained as a slice of the rhombicuboctahedron...
, Pentagonal dipyramidPentagonal dipyramidIn geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids.Each bipyramid is the dual of a uniform prism.If the faces are equilateral triangles, it is a deltahedron and a Johnson solid...
, Augmented pentagonal prismAugmented pentagonal prismIn geometry, the augmented pentagonal prism is one of the Johnson solids . As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces....
Dodecahedron
- Pentagonal antiprismPentagonal antiprismIn geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...
, Decagonal prismDecagonal prismIn geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....
, Pentagonal cupolaPentagonal cupolaIn geometry, the pentagonal cupola is one of the Johnson solids . It can be obtained as a slice of the rhombicosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
, Snub disphenoidSnub disphenoidIn geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
, Elongated square dipyramidElongated square dipyramidIn geometry, the elongated square bipyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating an octahedron by inserting a cube between its congruent halves....
, Metabidiminished icosahedronMetabidiminished icosahedronIn geometry, the metabidiminished icosahedron is one of the Johnson solids . The name refers to one way of constructing it, by removing two pentagonal pyramids from a regular icosahedron, replacing two sets of five triangular faces of the icosahedron with two adjacent pentagonal faces...
, Hexagonal bipyramidHexagonal bipyramidA hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.It is one of an infinite set of bipyramids...
, Hexagonal trapezohedronHexagonal trapezohedronThe hexagonal trapezohedron or deltohedron is the fourth in an infinite series of face-uniform polyhedra which are dual polyhedron to the antiprisms. It has twelve faces which are congruent kites.- External links :* The Encyclopedia of Polyhedra...
, Triakis tetrahedronTriakis tetrahedronIn geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron...
, Rhombic dodecahedronRhombic dodecahedronIn geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
, Hendecagonal pyramid, Trapezo-rhombic dodecahedronTrapezo-rhombic dodecahedronThe trapezo-rhombic dodecahedron is a convex polyhedron with 6 rhombic and 6 trapezoidal faces.This shape could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom...
, Rhombo-hexagonal dodecahedronRhombo-hexagonal dodecahedronThe rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces.It is also called an elongated dodecahedron and extended rhombic dodecahedron because it is related to the rhombic dodecahedron by expanding four rhombic faces of the rhombic dodecahedron into...
Archimedean solids
Archimedean solidArchimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
- Truncated tetrahedronTruncated tetrahedronIn geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.- Area and volume :...
, CuboctahedronCuboctahedronIn geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
, Truncated cubeTruncated cubeIn geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices....
, Truncated octahedronTruncated octahedronIn geometry, the truncated octahedron is an Archimedean solid. It has 14 faces , 36 edges, and 24 vertices. Since each of its faces has point symmetry the truncated octahedron is a zonohedron....
, RhombicuboctahedronRhombicuboctahedronIn geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...
, Truncated cuboctahedronTruncated cuboctahedronIn geometry, the truncated cuboctahedron is an Archimedean solid. It has 12 square faces, 8 regular hexagonal faces, 6 regular octagonal faces, 48 vertices and 72 edges...
, Snub cubeSnub cubeIn geometry, the snub cube, or snub cuboctahedron, is an Archimedean solid.The snub cube has 38 faces, 6 of which are squares and the other 32 are equilateral triangles. It has 60 edges and 24 vertices. It is a chiral polyhedron, that is, it has two distinct forms, which are mirror images of each...
, IcosidodecahedronIcosidodecahedronIn geometry, an icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon...
, Truncated dodecahedronTruncated dodecahedronIn geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...
, Truncated icosahedronTruncated icosahedronIn geometry, the truncated icosahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids whose faces are two or more types of regular polygons.It has 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges....
, RhombicosidodecahedronRhombicosidodecahedronIn geometry, the rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces....
, Truncated icosidodecahedronTruncated icosidodecahedronIn geometry, the truncated icosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....
, Snub dodecahedronSnub dodecahedronIn geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces....
Prisms and antiprisms
PrismPrism (geometry)
In geometry, a prism is a polyhedron with an n-sided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All cross-sections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
- Triangular prismTriangular prismIn geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides....
, Pentagonal prismPentagonal prismIn geometry, the pentagonal prism is a prism with a pentagonal base. It is a type of heptahedron with 7 faces, 15 edges, and 10 vertices.- As a semiregular polyhedron :...
, Hexagonal prismHexagonal prismIn geometry, the hexagonal prism is a prism with hexagonal base. The shape has 8 faces, 18 edges, and 12 vertices.Since it has eight faces, it is an octahedron. However, the term octahedron is primarily used to refer to the regular octahedron, which has eight triangular faces...
, Heptagonal prismHeptagonal prismIn geometry, the heptagonal prism is the fifth of the infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two heptagons....
, Octagonal prismOctagonal prismIn geometry, the octagonal prism is the sixth in an infinite set of prisms, formed by square sides and two regular octagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :...
, Enneagonal prism, Decagonal prismDecagonal prismIn geometry, the decagonal prism is the eighth in an infinite set of prisms, formed by ten square side faces and two regular decagon caps. With twelve faces, it is one of many nonregular dodecahedra.If faces are all regular, it is a semiregular polyhedron....
, Hendecagonal prismHendecagonal prismIn geometry, the hendecagonal prism is one in an infinite set of convex prisms formed by square sides and two regular polygon caps, in this case two hendecagons....
, Dodecagonal prismDodecagonal prismIn geometry, the dodecagonal prism is the tenth in an infinite set of prisms, formed by square sides and two regular dodecagon caps.If faces are all regular, it is a semiregular polyhedron.- Use :...
Antiprism
Antiprism
In geometry, an n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles...
- Square antiprismSquare antiprismIn geometry, the square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps...
, Pentagonal antiprismPentagonal antiprismIn geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. It consists of two pentagons joined to each other by a ring of 10 triangles for a total of 12 faces...
, Hexagonal antiprismHexagonal antiprismIn geometry, the hexagonal antiprism is the 4th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
, Heptagonal antiprismHeptagonal antiprismIn geometry, the heptagonal antiprism is the fourth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
, Octagonal antiprismOctagonal antiprismIn geometry, the octagonal antiprism is the 6th in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- See also :* Set of antiprisms...
, Enneagonal antiprismEnneagonal antiprismIn geometry, the enneagonal antiprism is one in an infinite set of convex antiprisms formed by triangle sides and two regular polygon caps, in this case two enneagons....
, Decagonal antiprismDecagonal antiprismIn geometry, the decagonal antiprism is the eighth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron.- External links :*...
, Dodecagonal antiprismDodecagonal antiprismIn geometry, the dodecagonal antiprism is the tenth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps.If faces are all regular, it is a semiregular polyhedron....
Catalan solids
Catalan solidCatalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
- Triakis tetrahedronTriakis tetrahedronIn geometry, a triakis tetrahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated tetrahedron.It can be seen as a tetrahedron with triangular pyramids added to each face; that is, it is the Kleetope of the tetrahedron...
, Rhombic dodecahedronRhombic dodecahedronIn geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.-Properties:...
, Triakis octahedronTriakis octahedronIn geometry, a triakis octahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated cube.It can be seen as an octahedron with triangular pyramids added to each face; that is, it is the Kleetope of the octahedron. It is also sometimes called a trisoctahedron, or, more...
, Tetrakis hexahedronTetrakis hexahedronIn geometry, a tetrakis hexahedron is a Catalan solid. Its dual is the truncated octahedron, an Archimedean solid. It can be seen as a cube with square pyramids covering each square face; that is, it is the Kleetope of the cube....
, Deltoidal icositetrahedronDeltoidal icositetrahedronIn geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron....
, Disdyakis dodecahedronDisdyakis dodecahedronIn geometry, a disdyakis dodecahedron, or hexakis octahedron, is a Catalan solid and the dual to the Archimedean truncated cuboctahedron. As such it is face-transitive but with irregular face polygons...
, Pentagonal icositetrahedronPentagonal icositetrahedronIn geometry, a pentagonal icositetrahedron is a Catalan solid which is the dual of the snub cube. It has two distinct forms, which are mirror images of each other....
, Rhombic triacontahedronRhombic triacontahedronIn geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron....
, Triakis icosahedronTriakis icosahedronIn geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron...
, Pentakis dodecahedronPentakis dodecahedronIn geometry, a pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron...
, Deltoidal hexecontahedronDeltoidal hexecontahedronIn geometry, a deltoidal hexecontahedron is a catalan solid which looks a bit like an overinflated dodecahedron. It is sometimes also called the trapezoidal hexecontahedron or strombic hexecontahedron...
, Disdyakis triacontahedronDisdyakis triacontahedronIn geometry, a disdyakis triacontahedron, or hexakis icosahedron is a Catalan solid and the dual to the Archimedean truncated icosidodecahedron. As such it is face uniform but with irregular face polygons...
, Pentagonal hexecontahedronPentagonal hexecontahedronIn geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images of each other. It is also well-known to be the Catalan Solid with the most vertices...
Bipyramids and Trapezohedron
- BipyramidBipyramidAn n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base.The referenced n-gon in the name of the bipyramids is not an external face but an internal one, existing on the primary symmetry plane which connects the two pyramid halves.The...
- Triangular dipyramidTriangular dipyramidIn geometry, the triangular bipyramid is the first in the infinite set of face-transitive bipyramids. It is the dual of the triangular prism with 6 isosceles triangle faces....
, Pentagonal dipyramidPentagonal dipyramidIn geometry, the pentagonal bipyramid is third of the infinite set of face-transitive bipyramids.Each bipyramid is the dual of a uniform prism.If the faces are equilateral triangles, it is a deltahedron and a Johnson solid...
, Hexagonal bipyramidHexagonal bipyramidA hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.It is one of an infinite set of bipyramids...
, Octagonal bipyramidOctagonal bipyramidThe octagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If an octagonal bipyramid is to be face-transitive, all faces must be isosceles triangles...
, Decagonal bipyramidDecagonal bipyramidIn geometry, a decagonal bipyramid is one of the infinite set of bipyramids, dual to the infinite prisms. If a decagonal bipyramid is to be face-transitive, all faces must be isosceles triangles...
- Triangular dipyramid
- TrapezohedronTrapezohedronThe n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...
Uniform star polyhedra
Uniform star polyhedron- Cubitruncated cuboctahedronCubitruncated cuboctahedronIn geometry, the cubitruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16.- Convex hull :Its convex hull is a nonuniform truncated cuboctahedron.- Cartesian coordinates :...
- CubohemioctahedronCubohemioctahedronIn geometry, the cubohemioctahedron is a nonconvex uniform polyhedron, indexed as U15. Its vertex figure is a crossed quadrilateral.A nonconvex polyhedron has intersecting faces which do not represent new edges or faces...
- Ditrigonal dodecadodecahedronDitrigonal dodecadodecahedronIn geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.- Related polyhedra :Its convex hull is a regular dodecahedron...
- Dodecadodecahedron
- Great cubicuboctahedronGreat cubicuboctahedronIn geometry, the great cubicuboctahedron is a nonconvex uniform polyhedron, indexed as U14.- Related polyhedra :It shares the vertex arrangement with the convex truncated cube and two other nonconvex uniform polyhedra...
- Great dirhombicosidodecahedronGreat dirhombicosidodecahedronIn geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...
- Great disnub dirhombidodecahedronGreat disnub dirhombidodecahedronIn geometry, the great disnub dirhombidodecahedron, also called Skilling's figure, is a uniform star polyhedron.John Skilling discovered this one further uniform polyhedron, by relaxing the condition that only two faces may meet at an edge...
- Great ditrigonal dodecicosidodecahedronGreat ditrigonal dodecicosidodecahedronIn geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...
- Great ditrigonal icosidodecahedron
- Great dodecahemicosahedronGreat dodecahemicosahedronIn geometry, the great dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U65. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :...
- Great dodecahemidodecahedronGreat dodecahemidodecahedronIn geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. Its vertex figure is a crossed quadrilateral....
- Great dodecicosahedronGreat dodecicosahedronIn geometry, the great dodecicosahedron is a nonconvex uniform polyhedron, indexed as U63. Its vertex figure is a crossed quadrilateral.It has a composite Wythoff symbol, 3 5/3 |, requiring two different Schwarz triangles to generate it: and .Its vertex figure 6.10/3.6/5.10/7 is also ambiguous,...
- Great dodecicosidodecahedronGreat dodecicosidodecahedronIn geometry, the great dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U61.- Related polyhedra :It shares its vertex arrangement with the truncated great dodecahedron and the uniform compounds of 6 or 12 pentagonal prisms...
- Great icosicosidodecahedronGreat icosicosidodecahedronIn geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...
- Great icosidodecahedron
- Great icosihemidodecahedronGreat icosihemidodecahedronIn geometry, the great icosihemidodecahedron is a nonconvex uniform polyhedron, indexed as U71. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with 6 decagrammic faces passing through the model center.- Related polyhedra :...
- Great inverted snub icosidodecahedron
- Great retrosnub icosidodecahedron
- Great rhombidodecahedronGreat rhombidodecahedronIn geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...
- Great rhombihexahedronGreat rhombihexahedronIn geometry, the great rhombihexahedron is a nonconvex uniform polyhedron, indexed as U21. Its dual is the great rhombihexacron. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...
- Great snub dodecicosidodecahedronGreat snub dodecicosidodecahedronIn geometry, the great snub dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U64.- Related polyhedra :It shares its vertices and edges, as well as 20 of its triangular faces and all its pentagrammic faces, with the great dirhombicosidodecahedron,...
- Great snub icosidodecahedron
- Great stellated truncated dodecahedron
- Great truncated cuboctahedron
- Great truncated icosidodecahedron
- IcosidodecadodecahedronIcosidodecadodecahedronIn geometry, the icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U44. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the uniform compounds of 10 or 20 triangular prisms...
- Icositruncated dodecadodecahedronIcositruncated dodecadodecahedronIn geometry, the icositruncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U45.- Convex hull :Its convex hull is a nonuniform great rhombicosidodecahedron.- Cartesian coordinates :...
- Inverted snub dodecadodecahedron
- Nonconvex great rhombicosidodecahedron
- Nonconvex great rhombicuboctahedron
- OctahemioctahedronOctahemioctahedronIn geometry, the octahemioctahedron is a nonconvex uniform polyhedron, indexed as U3. Its vertex figure is a crossed quadrilateral.It is one of nine hemipolyhedra with 4 hexagonal faces passing through the model center.- Related polyhedra :...
- RhombicosahedronRhombicosahedronIn geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :...
- Rhombidodecadodecahedron
- Small cubicuboctahedronSmall cubicuboctahedronIn geometry, the small cubicuboctahedron is a uniform star polyhedron, indexed as U13. It has 20 faces , 48 edges, and 24 vertices. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...
- Small ditrigonal dodecicosidodecahedronSmall ditrigonal dodecicosidodecahedronIn geometry, the small ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U43. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron...
- Small ditrigonal icosidodecahedronSmall ditrigonal icosidodecahedronIn geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron...
- Small dodecahemicosahedronSmall dodecahemicosahedronIn geometry, the small dodecahemicosahedron is a nonconvex uniform polyhedron, indexed as U62. Its vertex figure is a crossed quadrilateral.It is a hemipolyhedron with ten hexagonal faces passing through the model center.- Related polyhedra :...
- Small dodecahemidodecahedronSmall dodecahemidodecahedronIn geometry, the small dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U51. Its vertex figure alternates two regular pentagons and decagons as a crossed quadrilateral....
- Small dodecicosahedronSmall dodecicosahedronIn geometry, the small dodecicosahedron is a nonconvex uniform polyhedron, indexed as U50. Its vertex figure is a crossed quadrilateral.-Related polyhedra:It shares its vertex arrangement with the great stellated truncated dodecahedron...
- Small dodecicosidodecahedronSmall dodecicosidodecahedronIn geometry, the small dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U33. Its vertex figure is a crossed quadrilateral.-Related polyhedra:...
- Small icosicosidodecahedronSmall icosicosidodecahedronIn geometry, the small icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U31.- Related polyhedra :It shares its vertex arrangement with the great stellated truncated dodecahedron...
- Small icosihemidodecahedronSmall icosihemidodecahedronIn geometry, the small icosihemidodecahedron is a uniform star polyhedron, indexed as U49. Its vertex figure alternates two regular triangles and decagons as a crossed quadrilateral....
- Small retrosnub icosicosidodecahedronSmall retrosnub icosicosidodecahedronIn geometry, the small retrosnub icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U72.- Convex hull :Its convex hull is a nonuniform truncated dodecahedron.- Cartesian coordinates :...
- Small rhombidodecahedronSmall rhombidodecahedronIn geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. Its vertex figure is a crossed quadrilateral.- Related polyhedra :...
- Small rhombihexahedronSmall rhombihexahedronIn geometry, the small rhombihexahedron is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces , 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram.-Related polyhedra:...
- Small snub icosicosidodecahedronSmall snub icosicosidodecahedronIn geometry, the small snub icosicosidodecahedron is a uniform star polyhedron, indexed as U32. It has 112 faces , 180 edges, and 60 vertices.- Convex hull :Its convex hull is a nonuniform truncated icosahedron....
- Small stellated truncated dodecahedron
- Snub dodecadodecahedron
- Snub icosidodecadodecahedronSnub icosidodecadodecahedronIn geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46.- Cartesian coordinates :Cartesian coordinates for the vertices of a snub icosidodecadodecahedron are all the even permutations of...
- Stellated truncated hexahedron
- TetrahemihexahedronTetrahemihexahedronIn geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 6 vertices and 12 edges, and 7 faces: 4 triangular and 3 square. Its vertex figure is a crossed quadrilateral. It has Coxeter-Dynkin diagram of ....
- Truncated dodecadodecahedron
- Truncated great dodecahedron
- Truncated great icosahedron
Uniform prismatic star polyhedra
Prismatic uniform polyhedronPrismatic uniform polyhedron
In geometry, a prismatic uniform polyhedron is a uniform polyhedron with dihedral symmetry. They exist in two infinite families, the uniform prisms and the uniform antiprisms...
- Pentagrammic prismPentagrammic prismIn geometry, the pentagrammic prism is one in an infinite set of nonconvex prisms formed by square sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U78 as a uniform polyhedron....
, Pentagrammic antiprismPentagrammic antiprismIn geometry, the pentagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.This polyhedron is identified with the indexed name U79 as a uniform polyhedron....
, Pentagrammic crossed-antiprismPentagrammic crossed-antiprismIn geometry, the pentagrammic crossed-antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two pentagrams.... - Heptagrammic antiprism (7/2), Heptagrammic antiprism (7/3)
- Enneagrammic antiprism (9/2). Enneagrammic antiprism (9/4)
- Enneagrammic crossed-antiprism, Enneagrammic prism (9/2), Enneagrammic prism (9/4)
- Decagrammic prismDecagrammic prismIn geometry, the decagrammic prism is one of an infinite set of nonconvex prisms formed by squares sides and two regular star polygon caps, in this case two decagrams....
, Decagrammic antiprismDecagrammic antiprismIn geometry, the decagrammic antiprism is one in an infinite set of nonconvex antiprisms formed by triangle sides and two regular star polygon caps, in this case two decagrams....
Johnson solids
Johnson solidJohnson solid
In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i.e., not a Platonic solid, Archimedean solid, prism or antiprism. There is no requirement that each face must be the same polygon, or that the same polygons join around...
- Augmented dodecahedronAugmented dodecahedronIn geometry, the augmented dodecahedron is one of the Johnson solids , consisting of a dodecahedron with a pentagonal pyramid attached to one of the faces...
- Augmented hexagonal prismAugmented hexagonal prismIn geometry, the augmented hexagonal prism is one of the Johnson solids . As the name suggests, it can be constructed by augmenting a hexagonal prism by attaching a square pyramid to one of its equatorial faces....
- Augmented pentagonal prismAugmented pentagonal prismIn geometry, the augmented pentagonal prism is one of the Johnson solids . As the name suggests, it can be constructed by augmenting a pentagonal prism by attaching a square pyramid to one of its equatorial faces....
- Augmented sphenocoronaAugmented sphenocoronaIn geometry, the augmented sphenocorona is one of theJohnson solids , and is obtained by addinga square pyramid to one of the square faces of the sphenocorona....
- Augmented triangular prismAugmented triangular prismIn geometry, the augmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by augmenting a triangular prism by attaching a square pyramid to one of its equatorial faces...
- Augmented tridiminished icosahedronAugmented tridiminished icosahedronIn geometry, the augmented tridiminished icosahedron is one of theJohnson solids .It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron....
- Augmented truncated cubeAugmented truncated cubeIn geometry, the augmented truncated cube is one of theJohnson solids . As its name suggests, it is created by attaching square cupola onto one octagonal face of a truncated cube.-References:...
- Augmented truncated dodecahedronAugmented truncated dodecahedronIn geometry, the augmented truncated dodecahedron is one of the Johnson solids .-External links:**...
- Augmented truncated tetrahedronAugmented truncated tetrahedronIn geometry, the augmented truncated tetrahedron is one of the Johnson solids . It is created by attaching a triangular cupola to one hexagonal face of an truncated tetrahedron.-External links:**...
- Biaugmented pentagonal prismBiaugmented pentagonal prismIn geometry, the biaugmented pentagonal prism is one of the Johnson solids . As the name suggests, it can be constructed by doubly augmenting a pentagonal prism by attaching square pyramids to two of its nonadjacent equatorial faces...
- Biaugmented triangular prismBiaugmented triangular prismIn geometry, the biaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by augmenting a triangular prism by attaching square pyramids to two of its equatorial faces....
- Biaugmented truncated cubeBiaugmented truncated cubeIn geometry, the biaugmented truncated cube is one of the Johnson solids .-External links:**...
- Bigyrate diminished rhombicosidodecahedronBigyrate diminished rhombicosidodecahedronIn geometry, the bigyrate diminished rhombicosidodecahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- BilunabirotundaBilunabirotundaIn geometry, the bilunabirotunda is one of the Johnson solids . It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
- Diminished rhombicosidodecahedronDiminished rhombicosidodecahedronIn geometry, the diminished rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with one pentagonal cupola removed.The 92 Johnson solids were named and described by Norman Johnson in 1966....
- DisphenocingulumDisphenocingulumIn geometry, the disphenocingulum is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
- Elongated pentagonal bipyramid
- Elongated pentagonal cupolaElongated pentagonal cupolaIn geometry, the elongated pentagonal cupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal cupola by attaching a decagonal prism to its base...
- Elongated pentagonal gyrobicupolaElongated pentagonal gyrobicupolaIn geometry, the elongated pentagonal gyrobicupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal gyrobicupola by inserting a decagonal prism between its congruent halves...
- Elongated pentagonal gyrobirotundaElongated pentagonal gyrobirotundaIn geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron , by inserting a decagonal prism between its congruent halves...
- Elongated pentagonal gyrocupolarotundaElongated pentagonal gyrocupolarotundaIn geometry, the elongated pentagonal gyrocupolarotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal gyrocupolarotunda by inserting a decagonal prism between its halves...
- Elongated pentagonal orthobicupolaElongated pentagonal orthobicupolaIn geometry, the elongated pentagonal orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal orthobicupola by inserting a decagonal prism between its two congruent halves...
- Elongated pentagonal orthobirotundaElongated pentagonal orthobirotundaIn geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda by inserting a decagonal prism between its congruent halves...
- Elongated pentagonal orthocupolarotundaElongated pentagonal orthocupolarotundaIn geometry, the elongated pentagonal orthocupolarotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal orthocupolarotunda by inserting a decagonal prism between its halves...
- Elongated pentagonal pyramidElongated pentagonal pyramidIn geometry, the elongated pentagonal pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal pyramid by attaching a pentagonal prism to its base....
- Elongated pentagonal rotundaElongated pentagonal rotundaIn geometry, the elongated pentagonal rotunda is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal rotunda by attaching a decagonal prism to its base...
- Elongated square bipyramid
- Elongated square cupolaElongated square cupolaIn geometry, the elongated square cupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a square cupola by attaching an octagonal prism to its base...
- Elongated square gyrobicupolaElongated square gyrobicupolaIn geometry, the elongated square gyrobicupola or pseudorhombicuboctahedron is one of the Johnson solids . The 92 Johnson solids were named and described by Norman Johnson in 1966.- Relation to Rhombicuboctahedron :...
- Elongated square pyramid Elongated square pyramidIn geometry, the elongated square pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base...
- Elongated triangular bipyramid
- Elongated triangular cupolaElongated triangular cupolaIn geometry, the elongated triangular cupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a triangular cupola by attaching a hexagonal prism to its base....
- Elongated triangular gyrobicupolaElongated triangular gyrobicupolaIn geometry, the elongated triangular gyrobicupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a "triangular gyrobicupola," or cuboctahedron, by inserting a hexagonal prism between its two halves, which are congruent triangular cupolae...
- Elongated triangular orthobicupolaElongated triangular orthobicupolaIn geometry, the elongated triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by elongating a triangular orthobicupola by inserting a hexagonal prism between its two halves...
- Elongated triangular pyramid Elongated triangular pyramidIn geometry, the elongated triangular pyramid is one of the Johnson solids . Norman Johnson discovered elongated triangular pyramids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base...
- Gyrate bidiminished rhombicosidodecahedronGyrate bidiminished rhombicosidodecahedronIn geometry, the gyrate bidiminished rhombicosidodecahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Gyrate rhombicosidodecahedronGyrate rhombicosidodecahedronIn geometry, the gyrate rhombicosidodecahedron is one of theJohnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.- Related polyhedron :...
- GyrobifastigiumGyrobifastigiumIn geometry, the gyrobifastigium is the 26th Johnson solid . It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a half-turn to one prism....
- Gyroelongated pentagonal bicupolaGyroelongated pentagonal bicupolaIn geometry, the gyroelongated pentagonal bicupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal bicupola by inserting a decagonal antiprism between its congruent halves....
- Gyroelongated pentagonal birotundaGyroelongated pentagonal birotundaIn geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda by inserting a decagonal antiprism between its two halves.The gyroelongated pentagonal birotunda is one of five Johnson solids...
- Gyroelongated pentagonal cupolaGyroelongated pentagonal cupolaIn geometry, the gyroelongated pentagonal cupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal cupola by attaching a decagonal antiprism to its base...
- Gyroelongated pentagonal cupolarotundaGyroelongated pentagonal cupolarotundaIn geometry, the gyroelongated pentagonal cupolarotunda is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal cupolarotunda by inserting a decagonal antiprism between its two halves.The gyroelongated pentagonal cupolarotunda is one of five...
- Gyroelongated pentagonal pyramid Gyroelongated pentagonal pyramidIn geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids . As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base....
- Gyroelongated pentagonal rotundaGyroelongated pentagonal rotundaIn geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda by attaching a decagonal antiprism to its base...
- Gyroelongated square bicupolaGyroelongated square bicupolaIn geometry, the gyroelongated square bicupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a square bicupola by inserting an octagonal antiprism between its congruent halves....
- Gyroelongated square bipyramid
- Gyroelongated square cupolaGyroelongated square cupolaIn geometry, the gyroelongated square cupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a square cupola by attaching an octagonal antiprism to its base...
- Gyroelongated square pyramidGyroelongated square pyramidIn geometry, the gyroelongated square pyramid is one of the Johnson solids . As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base....
- Gyroelongated triangular bicupolaGyroelongated triangular bicupolaIn geometry, the gyroelongated triangular bicupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a triangular bicupola by inserting a hexagonal antiprism between its congruent halves.The gyroelongated triangular bicupola is one of five Johnson solids...
- Gyroelongated triangular cupolaGyroelongated triangular cupolaIn geometry, the gyroelongated triangular cupola is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating a triangular cupola by attaching a hexagonal antiprism to its base. It can also be seen as a gyroelongated triangular bicupola with one triangular cupola...
- HebesphenomegacoronaHebesphenomegacoronaIn geometry, the hebesphenomegacorona is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. It has 21 faces, 18 triangles and 3 squares, 33 edges, and 14 vertices.The 92 Johnson solids...
- Metabiaugmented dodecahedronMetabiaugmented dodecahedronIn geometry, the metabiaugmented dodecahedron is one of the Johnson solids . It can be viewed as a dodecahedron with two pentagonal pyramids attached to two faces that are separated by one face...
- Metabiaugmented hexagonal prismMetabiaugmented hexagonal prismIn geometry, the metabiaugmented hexagonal prism is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Metabiaugmented truncated dodecahedronMetabiaugmented truncated dodecahedronIn geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids .-External links:**...
- Metabidiminished icosahedronMetabidiminished icosahedronIn geometry, the metabidiminished icosahedron is one of the Johnson solids . The name refers to one way of constructing it, by removing two pentagonal pyramids from a regular icosahedron, replacing two sets of five triangular faces of the icosahedron with two adjacent pentagonal faces...
- Metabidiminished rhombicosidodecahedronMetabidiminished rhombicosidodecahedronIn geometry, the metabidiminished rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae removed.Related Johnson solids are...
- Metabigyrate rhombicosidodecahedronMetabigyrate rhombicosidodecahedronIn geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids . It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees....
- Metagyrate diminished rhombicosidodecahedronMetagyrate diminished rhombicosidodecahedronIn geometry, the metagyrate diminished rhombicosidodecahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Parabiaugmented dodecahedronParabiaugmented dodecahedronIn geometry, the parabiaugmented dodecahedron is one of the Johnson solids . It can be seen as a dodecahedron with two pentagonal pyramids attached to opposite faces...
- Parabiaugmented hexagonal prismParabiaugmented hexagonal prismIn geometry, the parabiaugmented hexagonal prism is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Parabiaugmented truncated dodecahedronParabiaugmented truncated dodecahedronIn geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids .-External links:**...
- Parabidiminished rhombicosidodecahedronParabidiminished rhombicosidodecahedronIn geometry, the parabidiminished rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupola removed.Related Johnson solids are...
- Parabigyrate rhombicosidodecahedronParabigyrate rhombicosidodecahedronIn geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids . It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees....
- Paragyrate diminished rhombicosidodecahedronParagyrate diminished rhombicosidodecahedronIn geometry, the paragyrate diminished rhombicosidodecahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Pentagonal bipyramid
- Pentagonal cupolaPentagonal cupolaIn geometry, the pentagonal cupola is one of the Johnson solids . It can be obtained as a slice of the rhombicosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966....
- Pentagonal gyrobicupolaPentagonal gyrobicupolaIn geometry, the pentagonal gyrobicupola is one of the Johnson solids . Like the pentagonal orthobicupola , it can be obtained by joining two pentagonal cupolae along their bases...
- Pentagonal gyrocupolarotundaPentagonal gyrocupolarotundaIn geometry, the pentagonal gyrocupolarotunda is one of the Johnson solids . Like the pentagonal orthocupolarotunda , it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases...
- Pentagonal orthobicupolaPentagonal orthobicupolaIn geometry, the pentagonal orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal cupolae along their decagonal bases, matching like faces...
- Pentagonal orthobirotundaPentagonal orthobirotundaIn geometry, the pentagonal orthobirotunda is one of the Johnson solids . As the name suggests, it can be constructed by joining two pentagonal rotundae along their decagonal faces, matching like faces...
- Pentagonal orthocupolarotundaPentagonal orthocupolarotundaIn geometry, the pentagonal orthocupolarotunda is one of the Johnson solids . As the name suggests, it can be constructed by joining a pentagonal cupola and a pentagonal rotunda along their decagonal bases, matching the pentagonal faces...
- Pentagonal pyramidPentagonal pyramidIn geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point . Like any pyramid, it is self-dual....
- Pentagonal rotundaPentagonal rotundaIn geometry, the pentagonal rotunda is one of the Johnson solids . It can be seen as half an icosidodecahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966.-Formulae:...
- Snub disphenoidSnub disphenoidIn geometry, the snub disphenoid is one of the Johnson solids . It is a three-dimensional solid that has only equilateral triangles as faces, and is therefore a deltahedron. It is not a regular polyhedron because some vertices have four faces and others have five...
- Snub square antiprismSnub square antiprismIn geometry, the snub square antiprism is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the Icosahedron that has fourfold symmetry instead of...
- SphenocoronaSphenocoronaIn geometry, the sphenocorona is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
- SphenomegacoronaSphenomegacoronaIn geometry, the sphenomegacorona is one of theJohnson solids .It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids....
- Square cupolaSquare cupolaIn geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids . It can be obtained as a slice of the rhombicuboctahedron...
- Square gyrobicupolaSquare gyrobicupolaIn geometry, the square gyrobicupola is one of the Johnson solids . Like the square orthobicupola , it can be obtained by joining two square cupolae along their bases...
- Square orthobicupolaSquare orthobicupolaIn geometry, the square orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by joining two square cupolae along their octagonal bases, matching like faces...
- Square pyramidSquare pyramidIn geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
- Triangular bipyramid
- Triangular cupolaTriangular cupolaIn geometry, the triangular cupola is one of the Johnson solids . It can be seen as half a cuboctahedron.The 92 Johnson solids were named and described by Norman Johnson in 1966.-Formulae:...
- Triangular hebesphenorotundaTriangular hebesphenorotundaIn geometry, the triangular hebesphenorotunda is one of the Johnson solids . It is one of the elementary Johnson solids, which do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. However, it does have a strong relationship to the icosidodecahedron, an Archimedean...
- Triangular orthobicupolaTriangular orthobicupolaIn geometry, the triangular orthobicupola is one of the Johnson solids . As the name suggests, it can be constructed by attaching two triangular cupolas along their bases...
- Triaugmented dodecahedronTriaugmented dodecahedronIn geometry, the triaugmented dodecahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Triaugmented hexagonal prismTriaugmented hexagonal prismIn geometry, the triaugmented hexagonal prism is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Triaugmented triangular prismTriaugmented triangular prismIn geometry, the triaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by attaching square pyramids to each of the three equatorial faces of the triangular prism...
- Triaugmented truncated dodecahedronTriaugmented truncated dodecahedronIn geometry, the triaugmented truncated dodecahedron is one of the Johnson solids ; of them, it has the greatest volume in proportion to side length.The 92 Johnson solids were named and described by Norman Johnson in 1966.-External links:**...
- Tridiminished icosahedronTridiminished icosahedronIn geometry, the tridiminished icosahedron is one of the Johnson solids .The 92 Johnson solids were named and described by Norman Johnson in 1966.- Related polytopes :...
- Tridiminished rhombicosidodecahedronTridiminished rhombicosidodecahedronIn geometry, the tridiminished rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with three pentagonal cupola removed.Related Johnson solids are:...
- Trigyrate rhombicosidodecahedronTrigyrate rhombicosidodecahedronIn geometry, the trigyrate rhombicosidodecahedron is one of theJohnson solids .It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees.Related Johnson solids are...
Dual uniform star polyhedra
- Great complex icosidodecahedronGreat complex icosidodecahedronIn geometry, the great complex icosidodecahedron is a degenerate uniform star polyhedron. It has 12 vertices, and 60 edges, and 32 faces, 12 pentagrams and 20 pentagons...
- Great deltoidal hexecontahedronGreat deltoidal hexecontahedronIn geometry, the great deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the nonconvex great rhombicosidodecahedron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices....
- Great deltoidal icositetrahedronGreat deltoidal icositetrahedronIn geometry, the great deltoidal icositetrahedron is the dual of the uniform great rhombicuboctahedron.-External links:...
- Great dirhombicosidodecacronGreat dirhombicosidodecacronIn geometry, the great dirhombicosidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great dirhombicosidodecahedron.In Magnus Wenninger's Dual Models, it is represented with intersecting infinite prisms passing through the model center, cut off at a certain point that is...
- Great dirhombicosidodecahedronGreat dirhombicosidodecahedronIn geometry, the great dirhombicosidodecahedron is a nonconvex uniform polyhedron, indexed last as U75.This is the only uniform polyhedron with more than six faces meeting at a vertex...
- Great disdyakis dodecahedronGreat disdyakis dodecahedronIn geometry, the great disdyakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great truncated cuboctahedron....
- Great disdyakis triacontahedronGreat disdyakis triacontahedronIn geometry, the great disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron....
- Great disnub dirhombidodecacron
- Great ditrigonal dodecacronic hexecontahedronGreat ditrigonal dodecacronic hexecontahedronIn geometry, the great ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron....
- Great dodecacronic hexecontahedronGreat dodecacronic hexecontahedronIn geometry, the great dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great dodecicosidodecahedron....
- Great dodecahemicosacronGreat dodecahemicosacronIn geometry, the great dodecahemicosacron is the dual of the great dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemicosacron....
- Great dodecicosacronGreat dodecicosacronIn geometry, the great dodecicosacron is the dual of the great dodecicosahedron . It has 60 intersecting bow-tie-shaped faces.- External links :*...
- Great hexacronic icositetrahedronGreat hexacronic icositetrahedronIn geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron....
- Great hexagonal hexecontahedronGreat hexagonal hexecontahedronIn geometry, the great hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great snub dodecicosidodecahedron....
- Great icosacronic hexecontahedronGreat icosacronic hexecontahedronIn geometry, the great icosacronic hexecontahedron is the dual of the great icosicosidodecahedron.- External links :...
- Great icosihemidodecacronGreat icosihemidodecacronIn geometry, the great icosihemidodecacron is the dual of the great icosihemidodecahedron, and is one of nine dual hemipolyhedra. It appears indistinct from the great dodecahemidodecacron....
- Great inverted pentagonal hexecontahedronGreat inverted pentagonal hexecontahedronIn geometry, the great inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It's composed of 60 self-intersecting pentagonal faces, 150 edges and 92 vertices.It is the dual of the uniform great inverted snub icosidodecahedron....
- Great pentagonal hexecontahedronGreat pentagonal hexecontahedronIn geometry, the great pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the great snub icosidodecahedron. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.- External links :*...
- Great pentagrammic hexecontahedronGreat pentagrammic hexecontahedronIn geometry, the great pentagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron....
- Great pentakis dodecahedronGreat pentakis dodecahedronIn geometry, the great pentakis dodecahedron is a nonconvex isohedral polyhedron.It is the dual of the uniform small stellated truncated dodecahedron. The decagrammic faces pass close to the origin in the uniform polyhedron, causing this dual to be very spikey....
- Great rhombic triacontahedronGreat rhombic triacontahedronIn geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron...
- Great rhombidodecacronGreat rhombidodecacronIn geometry, the great rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. Its faces are antiparallelograms....
- Great rhombihexacronGreat rhombihexacronIn geometry, the great rhombihexacron is a nonconvex isohedral polyhedron. It is the dual of the uniform great rhombihexahedron . It has 24 identical bow-tie-shaped faces, 18 vertices, and 48 edges....
- Great stellapentakis dodecahedronGreat stellapentakis dodecahedronIn geometry, the great stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great icosahedron. It has 60 intersecting triangular faces.- External links :*...
- Great triakis icosahedronGreat triakis icosahedronIn geometry, the great triakis icosahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great stellated truncated dodecahedron....
- Great triakis octahedronGreat triakis octahedronIn geometry, the great triakis octahedron is the dual of the stellated truncated hexahedron . It has 24 intersecting isosceles triangle faces....
- Great triambic icosahedronGreat triambic icosahedronIn geometry, the great triambic icosahedron and medial triambic icosahedron are visually identical dual uniform polyhedra. The exterior surface also represents the De1f1 stellation of the icosahedron. The only way to differentiate these two polyhedra is to mark which intersections between edges are...
- Medial deltoidal hexecontahedronMedial deltoidal hexecontahedronIn geometry, the medial deltoidal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the rhombidodecadodecahedron. It has 60 intersecting quadrilateral faces.- External links :*...
- Medial disdyakis triacontahedronMedial disdyakis triacontahedronIn geometry, the medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron....
- Medial hexagonal hexecontahedronMedial hexagonal hexecontahedronIn geometry, the small ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron....
- Medial icosacronic hexecontahedronMedial icosacronic hexecontahedronIn geometry, the medial icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform icosidodecadodecahedron....
- Medial inverted pentagonal hexecontahedronMedial inverted pentagonal hexecontahedronIn geometry, the medial inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron....
- Medial pentagonal hexecontahedronMedial pentagonal hexecontahedronIn geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.- External links :*...
- Medial rhombic triacontahedronMedial rhombic triacontahedronIn geometry, the medial rhombic triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the dodecadodecahedron. It has 30 intersecting rhombic faces.It can also be called the small stellated triacontahedron....
- HexahemioctacronHexahemioctacronIn geometry, the hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the octahemioctacron....
- HemipolyhedronHemipolyhedronIn geometry, a hemipolyhedron is a uniform star polyhedron some of whose faces pass through its center. These "hemi" faces lie parallel to the faces of some other symmetrical polyhedron, and their count is half the number of faces of that other polyhedron - hence the "hemi" prefix.The prefix "hemi"...
- OctahemioctacronOctahemioctacronIn geometry, the octahemioctacron is the dual of the octahemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the hexahemioctacron....
- RhombicosacronRhombicosacronIn geometry, the rhombicosacron is a nonconvex isohedral polyhedron. It is the dual of the uniform rhombicosahedron, U56. It has 50 vertices, 120 edges, and 60 crossed-quadrilateral faces.- External links :*...
- Small complex icosidodecahedronSmall complex icosidodecahedronIn geometry, the small complex icosidodecahedron is a degenerate uniform star polyhedron. It has 32 faces , 60 edges and 12 vertices...
- Small ditrigonal dodecacronic hexecontahedronSmall ditrigonal dodecacronic hexecontahedronIn geometry, the small ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron....
- Small dodecacronic hexecontahedronSmall dodecacronic hexecontahedronIn geometry, the small dodecacronic hexecontahedron is the dual of the small dodecicosidodecahedron.-External links:...
- Small dodecahemicosacronSmall dodecahemicosacronIn geometry, the small dodecahemicosacron is the dual of the small dodecahemicosahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the great dodecahemicosacron....
- Small dodecahemidodecacronSmall dodecahemidodecacronIn geometry, the small dodecahemidodecacron is the dual of the small dodecahemidodecahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small icosihemidodecacron....
- Small dodecicosacronSmall dodecicosacronIn geometry, the small dodecicosacron is the dual of the small dodecicosahedron . It has 60 intersecting bow-tie-shaped faces.-External links:*...
- Small hexacronic icositetrahedronSmall hexacronic icositetrahedronIn geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron....
- Small hexagonal hexecontahedronSmall hexagonal hexecontahedronIn geometry, the small hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small snub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces....
- Small hexagrammic hexecontahedronSmall hexagrammic hexecontahedronIn geometry, the small hexagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the small retrosnub icosicosidodecahedron. It is partially degenerate, having coincident vertices, as its dual has coplanar triangular faces....
- Small icosacronic hexecontahedronSmall icosacronic hexecontahedronIn geometry, the small icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small icosicosidodecahedron. A Hexacontahedron has 60 faces....
- Small icosihemidodecacronSmall icosihemidodecacronIn geometry, the small icosihemidodecacron is the dual of the small icosihemidodecahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the small dodecahemidodecacron....
- Small rhombidodecacronSmall rhombidodecacronIn geometry, the small rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the small rhombidodecahedron. It has 60 intersecting antiparallelogram faces.- External links :*...
- Small rhombihexacronSmall rhombihexacronIn geometry, the small rhombihexacron is the dual of the small rhombihexahedron. Its faces are antiparallelograms formed by pairs of coplanar triangles....
- Small stellapentakis dodecahedronSmall stellapentakis dodecahedronIn geometry, the small stellapentakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the truncated great dodecahedron. It has 60 intersecting triangular faces.- External links :*...
- Small triambic icosahedronSmall triambic icosahedronIn geometry, the small triambic icosahedron is the dual to the uniform small ditrigonal icosidodecahedron. It is composed of 20 intersecting isogonal hexagon faces. It has 60 edges and 32 vertices, and Euler characteristic of −8....
- TetrahemihexacronTetrahemihexacronIn geometry, the tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra.Since the tetrahemihexahedron has three square faces passing through the model centre, the tetrahemihexacron has three vertices at infinity...
Honeycombs
Convex uniform honeycombConvex uniform honeycomb
In geometry, a convex uniform honeycomb is a uniform tessellation which fills three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells.Twenty-eight such honeycombs exist:* the familiar cubic honeycomb and 7 truncations thereof;...
- Cubic honeycombCubic honeycombThe cubic honeycomb is the only regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron....
- Truncated cubic honeycombTruncated cubic honeycombThe truncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cubes and octahedra in a ratio of 1:1.- Symmetry :...
- Bitruncated cubic honeycombBitruncated cubic honeycombThe bitruncated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra.It is one of 28 uniform honeycombs. It has 4 truncated octahedra around each vertex....
- Cantellated cubic honeycombCantellated cubic honeycombThe cantellated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cuboctahedra, and cubes in a ratio of 1:1:3.- References :...
- Cantitruncated cubic honeycombCantitruncated cubic honeycombThe cantitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space, made up of truncated cuboctahedra, truncated octahedra, and cubes in a ratio of 1:1:3.- Uniform colorings :...
- Rectified cubic honeycombRectified cubic honeycombThe rectified cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of octahedra and cuboctahedra in a ratio of 1:1.- Symmetry :...
- Runcitruncated cubic honeycombRuncitruncated cubic honeycombThe runcitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, truncated cubes, octagonal prisms, and cubes in a ratio of 1:1:3:3....
- Omnitruncated cubic honeycombOmnitruncated cubic honeycombThe omnitruncated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cuboctahedra and octagonal prisms in a ratio of 1:3.A wireframe rendering- References :...
- Tetrahedral-octahedral honeycombTetrahedral-octahedral honeycombThe tetrahedral-octahedral honeycomb or alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of alternating octahedra and tetrahedra in a ratio of 1:2....
- Truncated alternated cubic honeycombTruncated alternated cubic honeycombThe truncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated octahedra, cuboctahedra and truncated tetrahedra in a ratio of 1:1:2. Its vertex figure is a rectangular pyramid....
- Cantitruncated alternated cubic honeycombCantitruncated alternated cubic honeycombThe cantitruncated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of truncated cuboctahedra, truncated cubes and truncated tetrahedra in a ratio of 1:1:2.- References :...
- Runcinated alternated cubic honeycombRuncinated alternated cubic honeycombThe runcinated alternated cubic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of small rhombicuboctahedra, cubes, and tetrahedra in a ratio of 1:1:2.- References :...
- Quarter cubic honeycombQuarter cubic honeycombThe quarter cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1...
- Gyrated tetrahedral-octahedral honeycombGyrated tetrahedral-octahedral honeycombThe gyrated tetrahedral-octahedral honeycomb or gyrated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space made up of octahedra and tetrahedra in a ratio of 1:2....
- Gyrated triangular prismatic honeycombGyrated triangular prismatic honeycombThe gyrated triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space made up of triangular prisms. It is vertex-uniform with 12 triangular prisms per vertex....
- Gyroelongated alternated cubic honeycombGyroelongated alternated cubic honeycombThe gyroelongated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2....
- Gyroelongated triangular prismatic honeycombGyroelongated triangular prismatic honeycombThe gyroelongated triangular prismatic honeycomb is a uniform space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2....
- Elongated triangular prismatic honeycombElongated triangular prismatic honeycombThe elongated triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from an elongated triangular tiling extruded into prisms....
- Elongated alternated cubic honeycombElongated alternated cubic honeycombThe elongated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octahedra, triangular prisms, and tetrahedra in a ratio of 1:2:2....
- Hexagonal prismatic honeycombHexagonal prismatic honeycombThe hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space made up of hexagonal prisms.It is constructed from a hexagonal tiling extruded into prisms.It is one of 28 convex uniform honeycombs.- References :...
- Triangular prismatic honeycombTriangular prismatic honeycombThe triangular prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.It is constructed from a triangular tiling extruded into prisms.It is one of 28 convex uniform honeycombs....
- Triangular-hexagonal prismatic honeycombTriangular-hexagonal prismatic honeycombThe triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:2.It is constructed from a trihexagonal tiling extruded into prisms....
- Truncated hexagonal prismatic honeycombTruncated hexagonal prismatic honeycombThe truncated hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, and triangular prisms in a ratio of 1:2....
- Truncated square prismatic honeycombTruncated square prismatic honeycombThe truncated square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of octagonal prisms and cubes in a ratio of 1:1.It is constructed from a truncated square tiling extruded into prisms....
- Rhombitriangular-hexagonal prismatic honeycombRhombitriangular-hexagonal prismatic honeycombThe rhombitriangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms, cubes, and triangular prisms in a ratio of 1:3:2....
- Omnitruncated triangular-hexagonal prismatic honeycombOmnitruncated triangular-hexagonal prismatic honeycombThe omnitruncated triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of dodecagonal prisms, hexagonal prisms, and cubes in a ratio of 1:2:3....
- Snub triangular-hexagonal prismatic honeycombSnub triangular-hexagonal prismatic honeycombThe Snub triangular-hexagonal prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of hexagonal prisms and triangular prisms in a ratio of 1:8....
- Snub square prismatic honeycombSnub square prismatic honeycombThe snub square prismatic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of cubes and triangular prisms in a ratio of 1:2.It is constructed from a Snub square tiling extruded into prisms....
Dual uniform honeycomb
- Disphenoid tetrahedral honeycombDisphenoid tetrahedral honeycombThe disphenoid tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of identical nonregular tetrahedral cells....
- Rhombic dodecahedral honeycombRhombic dodecahedral honeycombThe rhombic dodecahedra honeycomb is a space-filling tessellation in Euclidean 3-space. It is the Voronoi diagram of the face-centered cubic sphere-packing, which is believed to be the densest possible packing of equal spheres in ordinary space .It consists of copies of a single cell, the rhombic...
Others
- Trapezo-rhombic dodecahedral honeycombTrapezo-rhombic dodecahedral honeycombThe trapezo-rhombic dodecahedral honeycomb is a space-filling tessellation in Euclidean 3-space.It consists of copies of a single cell, the trapezo-rhombic dodecahedron....
- Weaire–Phelan structure
Convex uniform honeycombs in hyperbolic space
Convex uniform honeycombs in hyperbolic space
In geometry, a convex uniform honeycomb is a tessellation of convex uniform polyhedron cells. In 3-dimensional hyperbolic space there are nine Coxeter group families of compact convex uniform honeycombs, generated as Wythoff constructions, and represented by ring permutations of the Coxeter–Dynkin...
- Order-4 dodecahedral honeycomb
- Order-5 cubic honeycomb
- Order-5 dodecahedral honeycomb
- Icosahedral honeycomb
Other
- Apeirogonal prismApeirogonal prismIn geometry, an apeirogonal prism or infinite prism is the arithmetic limit of the family of prisms; it can be considered an infinite polyhedron or a tiling of the plane....
- ApeirohedronApeirohedronAn apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded.Two main...
- BicupolaBicupola (geometry)In geometry, a bicupola is a solid formed by connecting two cupolae on their bases.There are two classes of bicupola because each cupola half is bordered by alternating triangles and squares...
- CupolaCupola (geometry)In geometry, a cupola is a solid formed by joining two polygons, one with twice as many edges as the other, by an alternating band of triangles and rectangles...
- BifrustumBifrustumAn n-agonal bifrustum is a polyhedron composed of three parallel planes of n-agons, with the middle plane largest and usually the top and bottom congruent....
- Boerdijk–Coxeter helixBoerdijk–Coxeter helixThe Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra. There are two chiral forms, with either clockwise or counterclockwise windings. Contrary to any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally...
- Császár polyhedronCsászár polyhedronIn geometry, the Császár polyhedron is a nonconvex polyhedron, topologically a toroid, with 14 triangular faces.This polyhedron has no diagonals; every pair of vertices is connected by an edge. The seven vertices and 21 edges of the Császár polyhedron form an embedding of the complete graph K_7...
- Flexible polyhedronFlexible polyhedronIn geometry, a flexible polyhedron is a polyhedral surface that allows continuous non-rigid deformations such that all faces remain rigid. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex .The first examples of flexible polyhedra, now called Bricard's...
- Gyroelongated square dipyramidGyroelongated square dipyramidIn geometry, the gyroelongated square bipyramid is one of the Johnson solids . As the name suggests, it can be constructed by gyroelongating an octahedron by inserting a square antiprism between its congruent halves...
- Heronian tetrahedronHeronian tetrahedronA Heronian tetrahedron is a tetrahedron whose side lengths, face areas and volume are all rational numbers. The faces must therefore all be Heronian triangles....
- Hexagonal bifrustumHexagonal bifrustumThe hexagonal bifrustum or truncated bipyramid is the fourth in an infinite series of bifrustum polyhedra. It has 12 trapezoid and 2 hexagonal faces....
- Hexagonal truncated trapezohedronHexagonal truncated trapezohedronThe hexagonal truncated trapezohedron is the fourth in an infinite series of truncated trapezohedron polyhedra. It has 12 pentagon and 2 hexagon faces....
- Hill tetrahedronHill tetrahedronIn geometry, the Hill tetrahedra are a family of space-filling tetrahedra. They were discovered in 1896 by M.J.M. Hill, a professor of mathematics at the University College London, who showed that they are scissor-congruent to a cube.- Construction :...
- HolyhedronHolyhedronIn mathematics, a holyhedron is a type of 3-dimensional geometric body: a polyhedron each of whose faces contains at least one polygon-shaped hole, and whose holes' boundaries share no point with each other or the face's boundary....
- Infinite skew polyhedronInfinite skew polyhedronIn geometry, an infinite skew polyhedron is an extension of the idea of a polyhedron, consisting of regular polygon faces with nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface....
- Jessen's icosahedronJessen's icosahedronJessen's icosahedron, sometimes called Jessen's orthogonal icosahedron is a non-convex polyhedron with the same number of vertices, edges and faces as the regular icosahedron...
- Near-miss Johnson solidNear-miss Johnson solidIn geometry, a near-miss Johnson solid is a strictly convex polyhedron, where every face is a regular or nearly regular polygon, and excluding the 5 Platonic solids, the 13 Archimedean solids, the infinite set of prisms, the infinite set of antiprisms, and the 92 Johnson solids.The set of...
- ParallelepipedParallelepipedIn geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms. By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean geometry, its definition encompasses all four concepts...
- Pentagonal bifrustumPentagonal bifrustumThe pentagonal bifrustum or pentagonal truncated bipyramid is the third in an infinite series of bifrustum polyhedra. It has 10 trapezoid and 2 pentagonal faces. It is the dual of the Johnson solid, elongated pentagonal dipyramid....
- PolytetrahedronPolytetrahedronPolytetrahedron is a term used for three distinct types of objects, all basedon the tetrahedron:*A uniform convex polychoron made up of 600 tetrahedral cells. It is more commonly known as a 600-cell or hexacosichoron...
- PyritohedronPyritohedronIn geometry, a pyritohedron is an irregular dodecahedron with pyritohedral symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes...
- Rhombic enneacontahedronRhombic enneacontahedronA rhombic enneacontahedron is a polyhedron composed of 90 rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim...
- Rhombic icosahedronRhombic icosahedronA rhombic icosahedron is a polyhedron shaped like an oblate sphere.It is composed of 20 rhombic faces, of which three, four, or five meet at each vertex...
- Rhombo-hexagonal dodecahedronRhombo-hexagonal dodecahedronThe rhombo-hexagonal dodecahedron is a convex polyhedron with 8 rhombic and 4 equilateral hexagonal faces.It is also called an elongated dodecahedron and extended rhombic dodecahedron because it is related to the rhombic dodecahedron by expanding four rhombic faces of the rhombic dodecahedron into...
- RhombohedronRhombohedronIn geometry, a rhombohedron is a three-dimensional figure like a cube, except that its faces are not squares but rhombi. It is a special case of a parallelepiped where all edges are the same length....
- ScalenohedronScalenohedronIn geometry, a scalenohedron is a polyhedron containing three or more pairs of mutually congruent scalene triangles as faces....
- Schönhardt polyhedronSchönhardt polyhedronIn geometry, the Schönhardt polyhedron is the simplest non-convex polyhedron that cannot be triangulated into tetrahedra without adding new vertices...
- Square bifrustumSquare bifrustumThe square bifrustum or square truncated bipyramid is the second in an infinite series of bifrustum polyhedra. It has 4 trapezoid and 2 square faces....
- Square truncated trapezohedronSquare truncated trapezohedronThe square truncated trapezohedron is the second in an infinite series of truncated trapezohedron polyhedra. It has 8 pentagon and 2 square faces.This polyhedron can be constructed by taking a tetragonal trapezohedron and truncating the polar axis vertices....
- Szilassi polyhedronSzilassi polyhedronThe Szilassi polyhedron is a nonconvex polyhedron, topologically a torus, with seven hexagonal faces.Each face of this polyhedron shares an edge with each other face. As a result, it requires seven colours to colour each adjacent face, providing the lower bound for the seven colour theorem...
- TetradecahedronTetradecahedronA tetradecahedron is a polyhedron with 14 faces. There are numerous topologically distinct forms of a tetradecahedron, with many constructible entirely with regular polygon faces....
- Tetradyakis hexahedronTetradyakis hexahedronIn geometry, the tetradyakis hexahedron is a nonconvex isohedral polyhedron. It has 48 intersecting scalene triangle faces, 72 edges, and 20 vertices.It is the dual of the uniform cubitruncated cuboctahedron.-External links:*...
- Tetrated dodecahedronTetrated dodecahedronThe tetrated dodecahedron is a near-miss Johnson solid. It was first discovered in 2002 by Alex Doskey. It was then independently rediscovered in 2003 and named by Robert Austin....
- Triangular bifrustumTriangular bifrustumThe triangular bifrustum or triangular truncated bipyramid is the second in an infinite series of bifrustum polyhedra. It has 6 trapezoid and 2 triangle faces....
- Triaugmented triangular prismTriaugmented triangular prismIn geometry, the triaugmented triangular prism is one of the Johnson solids . As the name suggests, it can be constructed by attaching square pyramids to each of the three equatorial faces of the triangular prism...
- Truncated rhombic dodecahedronTruncated rhombic dodecahedronThe truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 vertices.The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become flattened hexagons, and the truncated vertices become squares.The...
- Truncated trapezohedronTruncated trapezohedronAn n-agonal truncated trapezohedron is a polyhedron formed by a n-agonal trapezohedron with n-agonal pyramids truncated from its two polar axis vertices....
- Truncated triakis tetrahedronTruncated triakis tetrahedronThe truncated triakis tetrahedron is a convex polyhedron with 16 faces: 4 sets of 3 pentagons arranged in a tetrahedral arrangement, with 4 hexagons in the gaps. It is constructed from taking a triakis tetrahedron by truncating the order-6 vertices...
- Tridyakis icosahedronTridyakis icosahedronIn geometry, the tridyakis icosahedron is the dual polyhedron of the nonconvex uniform polyhedron, icositruncated dodecadodecahedron. It has 44 vertices, 180 edges, and 120 scalene triangular faces.-See also:* Catalan solid Duals to convex uniform polyhedra...
- Trigonal trapezohedronTrigonal trapezohedronIn geometry, the trigonal trapezohedron or deltohedron is the first in an infinite series of face-uniform polyhedra which are dual to the antiprisms. It has six faces which are congruent rhombi....
- Regular skew polyhedronRegular skew polyhedronIn geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures....
- Waterman polyhedronWaterman polyhedronWaterman polyhedra were invented around 1990 by the mathematician Steve Waterman.Waterman polyhedra are created by packing spheres according to the cubic close packing , then sweeping away the spheres that are farther from the center than a defined radius...
- WedgeWedge (geometry)In solid geometry, a wedge is a polyhedron defined by two triangles and three trapezoid faces. A wedge has five faces, nine edges, and six vertices.A wedge is a subclass of the prismatoids with the base and opposite ridge in two parallel planes....
Regular and uniform compound polyhedra
Polyhedral compoundPolyhedral compound
A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram....
and Uniform polyhedron compound
Uniform polyhedron compound
A uniform polyhedron compound is a polyhedral compound whose constituents are identical uniform polyhedra, in an arrangement that is also uniform: the symmetry group of the compound acts transitively on the compound's vertices.The uniform polyhedron compounds were first enumerated by John Skilling...
- Compound of cube and octahedronCompound of cube and octahedronThis polyhedron can be seen as either a polyhedral stellation or a compound.- As a compound :It can be seen as the compound of an octahedron and a cube...
- Compound of dodecahedron and icosahedronCompound of dodecahedron and icosahedronIn geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.- As a compound :It can be seen as the compound of an icosahedron and dodecahedron...
- Compound of eight octahedra with rotational freedomCompound of eight octahedra with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 8 octahedra, considered as triangular antiprisms. It can be constructed by superimposing eight identical octahedra, and then rotating them in pairs about the four axes that pass through the centres of two opposite octahedral faces...
- Compound of eight triangular prismsCompound of eight triangular prismsThis uniform polyhedron compound is a symmetric arrangement of 8 triangular prisms, aligned in pairs with the axes of three-fold rotational symmetry of an octahedron. It results from composing the two enantiomorphs of the compound of 4 triangular prisms....
- Compound of five cubesCompound of five cubesThis polyhedral compound is a symmetric arrangement of five cubes. This compound was first described by Edmund Hess in 1876.It is one of five regular compounds, and dual to the compound of five octahedra....
- Compound of five cuboctahedraCompound of five cuboctahedraIn geometry, this uniform polyhedron compound is a composition of 5 cuboctahedra. It has icosahedral symmetry Ih.- Cartesian coordinates :Cartesian coordinates for the vertices of this compound are all the cyclic permutations of...
- Compound of five cubohemioctahedraCompound of five cubohemioctahedraThis uniform polyhedron compound is a composition of 5 cubohemioctahedra, in the same arrangement as in the compound of 5 cuboctahedra....
- Compound of five great cubicuboctahedraCompound of five great cubicuboctahedraThis uniform polyhedron compound is a composition of 5 great cubicuboctahedra, in the same arrangement as the compound of 5 uniform great rhombicuboctahedra....
- Compound of five great dodecahedraCompound of five great dodecahedraThis uniform polyhedron compound is a composition of 5 great dodecahedra, in the same arrangement as in the compound of 5 icosahedra....
- Compound of five great icosahedraCompound of five great icosahedraThis uniform polyhedron compound is a composition of 5 great icosahedra, in the same arrangement as in the compound of 5 icosahedra.The triangles in this compound decompose into two orbits under action of the symmetry group: 40 of the triangles lie in coplanar pairs in icosahedral planes, while the...
- Compound of five great rhombihexahedraCompound of five great rhombihexahedraThis uniform polyhedron compound is a composition of 5 great rhombihexahedra, in the same arrangement as the compound of 5 uniform great rhombicuboctahedra....
- Compound of five icosahedraCompound of five icosahedraThis uniform polyhedron compound is a composition of 5 icosahedra. It has icosahedral symmetry Ih.The triangles in this compound decompose into two orbits under action of the symmetry group: 40 of the triangles lie in coplanar pairs in icosahedral planes, while the other 60 lie in unique planes.-...
- Compound of five octahedraCompound of five octahedraThis polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.- As a stellation :It is the second stellation of the icosahedron, and given as Wenninger model index 23....
- Compound of five octahemioctahedraCompound of five octahemioctahedraIn geometry, this uniform polyhedron compound is a composition of 5 octahemioctahedra, in the same vertex arrangement as in the compound of 5 cuboctahedra....
- Compound of five small cubicuboctahedraCompound of five small cubicuboctahedraThis uniform polyhedron compound is a composition of 5 small cubicuboctahedra, in the same vertex arrangement as the compound of 5 small rhombicuboctahedra....
- Compound of five small rhombicuboctahedraCompound of five small rhombicuboctahedraThis uniform polyhedron compound is a composition of 5 small rhombicuboctahedra, in the same vertex arrangement the compound of 5 stellated truncated cubes....
- Compound of five small rhombihexahedraCompound of five small rhombihexahedraThis uniform polyhedron compound is a composition of 5 small rhombihexahedra, in the same arrangement as the compound of 5 small rhombicuboctahedra....
- Compound of five small stellated dodecahedraCompound of five small stellated dodecahedraThis uniform polyhedron compound is a composition of 5 small stellated dodecahedra, in the same arrangement as in the compound of 5 icosahedra....
- Compound of five stellated truncated cubesCompound of five stellated truncated cubesThis uniform polyhedron compound is a composition of 5 stellated truncated cubes, formed by star-truncating each of the cubes in the compound of 5 cubes.- Cartesian coordinates :...
- Compound of five tetrahedraCompound of five tetrahedraThis compound polyhedron is also a stellation of the regular icosahedron. It was first described by Edmund Hess in 1876.-As a compound:It can be constructed by arranging five tetrahedra in rotational icosahedral symmetry , as colored in the upper right model...
- Compound of five tetrahemihexahedraCompound of five tetrahemihexahedraA compound of five tetrahemihexahedra is a uniform polyhedron compound and a symmetric arrangement of five tetrahemihexahedra. It is chiral with icosahedral symmetry .- Related polyhedra :...
- Compound of five truncated cubesCompound of five truncated cubesThis uniform polyhedron compound is a composition of 5 truncated cubes, formed by truncating each of the cubes in the compound of 5 cubes.- Cartesian coordinates :Cartesian coordinates for the vertices of this compound are all the cyclic permutations of...
- Compound of five truncated tetrahedraCompound of five truncated tetrahedraThis uniform polyhedron compound is a composition of 5 truncated tetrahedra, formed by truncating each of the tetrahedra in the compound of 5 tetrahedra. A far-enough truncation creates the Compound of five octahedra...
- Compound of five uniform great rhombicuboctahedraCompound of five uniform great rhombicuboctahedraThis uniform polyhedron compound is a composition of 5 uniform great rhombicuboctahedra, in the same arrangement the compound of 5 truncated cubes....
- Compound of four hexagonal prismsCompound of four hexagonal prismsThis uniform polyhedron compound is a symmetric arrangement of 4 hexagonal prisms, aligned with the axes of threefold rotational symmetry of an octahedron.- Cartesian coordinates :Cartesian coordinates for the vertices of this compound are all the permutations of...
- Compound of four octahedraCompound of four octahedraThis uniform polyhedron compound is a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by 60 degrees about a separate axis .- Cartesian coordinates :Cartesian coordinates for the...
- Compound of four octahedra with rotational freedomCompound of four octahedra with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 4 octahedra, considered as triangular antiprisms. It can be constructed by superimposing four identical octahedra, and then rotating each by an equal angle θ about a separate axis passing through the centres of two opposite octahedral...
- Compound of four tetrahedraCompound of four tetrahedraIn geometry, a compound of three tetrahedra can be constructed by three tetrahedra in a number of different symmetry positions.One compound can be constructed rotating a tetrahedron by 45 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D4...
- Compound of four triangular prismsCompound of four triangular prismsThis uniform polyhedron compound is a chiral symmetric arrangement of 4 triangular prisms, aligned with the axes of three-fold rotational symmetry of an octahedron.- Cartesian coordinates :...
- Compound of great icosahedron and great stellated dodecahedronCompound of great icosahedron and great stellated dodecahedronThis polyhedron can be seen as either a polyhedral stellation or a compound.- As a compound :It can be seen as a polyhedron compound of a great icosahedron and great stellated dodecahedron...
- Compound of six cubes with rotational freedomCompound of six cubes with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 6 cubes, considered as square prisms. It can be constructed by superimposing six identical cubes, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each cube is rotated by an...
- Compound of six decagonal prismsCompound of six decagonal prismsThis uniform polyhedron compound is a symmetric arrangement of 6 decagonal prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.- Cartesian coordinates :...
- Compound of six decagrammic prismsCompound of six decagrammic prismsThis uniform polyhedron compound is a symmetric arrangement of 6 decagrammic prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.- Cartesian coordinates :...
- Compound of six pentagonal antiprismsCompound of six pentagonal antiprismsThis uniform polyhedron compound is a symmetric arrangement of 6 pentagonal antiprisms. It can be constructed by inscribing within an icosahedron one pentagonal antiprism in each of the six possible ways, and then rotating each by 36 degrees about its axis .It shares its vertex arrangement with the...
- Compound of six pentagonal prismsCompound of six pentagonal prismsThis uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagonal prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.- Related polyhedra :...
- Compound of six pentagrammic antiprismsCompound of six pentagrammic antiprismsThis uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic antiprisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.- Cartesian coordinates :...
- Compound of six pentagrammic crossed antiprismsCompound of six pentagrammic crossed antiprismsThis uniform polyhedron compound is a symmetric arrangement of 6 pentagrammic crossed antiprisms. It can be constructed by inscribing within a great icosahedron one pentagrammic crossed antiprism in each of the six possible ways, and then rotating each by 36 degrees about its axis...
- Compound of six pentagrammic prismsCompound of six pentagrammic prismsThis uniform polyhedron compound is a chiral symmetric arrangement of 6 pentagrammic prisms, aligned with the axes of fivefold rotational symmetry of a dodecahedron.- Related polyhedra :...
- Compound of six square antiprismsCompound of six square antiprismsThis uniform polyhedron compound is a symmetric arrangement of 6 square antiprisms, aligned in pairs with the three axes of 4-fold rotational symmetry of a cube. It results from composing the two enantiomers of the compound of 3 square antiprisms....
- Compound of six tetrahedraCompound of six tetrahedraThis uniform polyhedron compound is a symmetric arrangement of 6 tetrahedra. It can be constructed by inscribing a stella octangula within each cube in the compound of three cubes, or by stellating each octahedron in the compound of three octahedra....
- Compound of six tetrahedra with rotational freedomCompound of six tetrahedra with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 6 tetrahedra, considered as antiprisms. It can be constructed by superimposing six tetrahedra within a cube, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic faces. Each...
- Compound of small stellated dodecahedron and great dodecahedronCompound of small stellated dodecahedron and great dodecahedronThe compound of small stellated dodecahedron and great dodecahedron is a polyhedron compound where the great dodecahedron is interior to its dual, the small stellated dodecahedron.-See also:*Great dodecahedron*Small stellated dodecahedron...
- Compound of ten hexagonal prismsCompound of ten hexagonal prismsThis uniform polyhedron compound is a symmetric arrangement of 10 hexagonal prisms, aligned with the axes of three-fold rotational symmetry of an icosahedron.- Cartesian coordinates :...
- Compound of ten octahedraCompound of ten octahedraThese uniform polyhedron compounds are symmetric arrangements of 10 octahedra, considered as triangular antiprisms, aligned with the axes of three-fold rotational symmetry of an icosahedron...
- Compound of ten tetrahedraCompound of ten tetrahedraThis polyhedron can be seen as either a polyhedral stellation or a compound. This compound was first described by Edmund Hess in 1876.- As a compound :It can also be seen as the compound of ten tetrahedra with full icosahedral symmetry...
- Compound of ten triangular prismsCompound of ten triangular prismsThis uniform polyhedron compound is a chiral symmetric arrangement of 10 triangular prisms, aligned with the axes of three-fold rotational symmetry of an icosahedron.- Relatde polyhedra :...
- Compound of ten truncated tetrahedraCompound of ten truncated tetrahedraThis uniform polyhedron compound is a composition of 10 truncated tetrahedra, formed by truncating each of the tetrahedra in the compound of 10 tetrahedra. It also results from composing the two enantiomers of the compound of 5 truncated tetrahedra....
- Compound of three cubesCompound of three cubesThis uniform polyhedron compound is a symmetric arrangement of 3 cubes, considered as square prisms. It can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis .This compound famously appears in the lithograph print Waterfall by M.C....
- Compound of three square antiprismsCompound of three square antiprismsThis uniform polyhedron compound is a symmetric arrangement of 3 square antiprisms, aligned with the three axes of 4-fold rotational symmetry of a cube.- Cartesian coordinates :...
- Compound of three tetrahedraCompound of three tetrahedraIn geometry, a compound of three tetrahedra can be constructed by three tetrahedra rotated by 60 degree turns along an axis of the middle of an edge. It has dihedral symmetry, D3It is similar to the compound of two tetrahedra with 90 degree turns....
- Compound of twelve pentagonal antiprisms with rotational freedomCompound of twelve pentagonal antiprisms with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 12 pentagonal antiprisms. It can be constructed by inscribing one pair of pentagonal antiprisms within an icosahedron, in each of the six possible ways, and then rotating each by an equal and opposite angle θ.When θ is 36 degrees, the...
- Compound of twelve pentagonal prismsCompound of twelve pentagonal prismsThis uniform polyhedron compound is a symmetric arrangement of 12 pentagonal prisms, aligned in pairs with the axes of fivefold rotational symmetry of a dodecahedron.It results from composing the two enantiomorphs of the compound of six pentagonal prisms...
- Compound of twelve pentagrammic antiprismsCompound of twelve pentagrammic antiprismsThis uniform polyhedron compound is a symmetric arrangement of 12 pentagrammic antiprisms, aligned in pairs with the axes of fivefold rotational symmetry of a dodecahedron. It results from composing the two enantiomorphs of the compound of 6 pentagrammic antiprisms....
- Compound of twelve pentagrammic crossed antiprisms with rotational freedomCompound of twelve pentagrammic crossed antiprisms with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 12 pentagrammic crossed antiprisms. It can be constructed by inscribing one pair of pentagrammic crossed antiprisms within a great icosahedron, in each of the six possible ways, and then rotating each by an equal and opposite angle...
- Compound of twelve pentagrammic prismsCompound of twelve pentagrammic prismsThis uniform polyhedron compound is a symmetric arrangement of 12 pentagrammic prisms, aligned in pairs with the axes of fivefold rotational symmetry of a dodecahedron.It results from composing the two enantiomorphs of the compound of six pentagrammic prisms...
- Compound of twelve tetrahedra with rotational freedomCompound of twelve tetrahedra with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 12 tetrahedra, considered as antiprisms. It can be constructed by superimposing six identical copies of the stella octangula, and then rotating them in pairs about the three axes that pass through the centres of two opposite cubic...
- Compound of twenty octahedraCompound of twenty octahedraThis uniform polyhedron compound is a symmetric arrangement of 20 octahedra . It is a special case of the compound of 20 octahedra with rotational freedom, in which pairs of octahedral vertices coincide.- Related polyhedra :...
- Compound of twenty octahedra with rotational freedomCompound of twenty octahedra with rotational freedomThis uniform polyhedron compound is a symmetric arrangement of 20 octahedra, considered as triangular antiprisms. It can be constructed by superimposing two copies of the compound of 10 octahedra UC16, and for each resulting pair of octahedra, rotating each octahedron in the pair by an equal and...
- Compound of twenty tetrahemihexahedraCompound of twenty tetrahemihexahedraThis uniform polyhedron compound is a symmetric arrangement of 20 tetrahemihexahedra. It is chiral with icosahedral symmetry .John Skilling notes, in his enumeration of uniform compounds of uniform polyhedra, that this compound of 20 tetrahemihexahedra is unique in that it cannot be obtained by...
- Compound of twenty triangular prismsCompound of twenty triangular prismsThis uniform polyhedron compound is a symmetric arrangement of 20 triangular prisms, aligned in pairs with the axes of three-fold rotational symmetry of an icosahedron.It results from composing the two enantiomorphs of the compound of 10 triangular prisms...
- Compound of two great dodecahedraCompound of two great dodecahedraThis uniform polyhedron compound is a composition of 2 great dodecahedra, in the same arrangement as in the compound of 2 icosahedra.- External links :* VRML model:...
- Compound of two great icosahedraCompound of two great icosahedraThis uniform polyhedron compound is a composition of 2 great icosahedra, in the same arrangement as in the compound of 2 icosahedra.The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral planes, while the...
- Compound of two great inverted snub icosidodecahedraCompound of two great inverted snub icosidodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the great inverted snub icosidodecahedron....
- Compound of two great retrosnub icosidodecahedraCompound of two great retrosnub icosidodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the great retrosnub icosidodecahedron....
- Compound of two great snub icosidodecahedraCompound of two great snub icosidodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the great snub icosidodecahedron....
- Compound of two icosahedraCompound of two icosahedraThis uniform polyhedron compound is a composition of 2 icosahedra. It has octahedral symmetry Oh.The triangles in this compound decompose into two orbits under action of the symmetry group: 16 of the triangles lie in coplanar pairs in octahedral planes, while the other 24 lie in unique planes.It...
- Compound of two inverted snub dodecadodecahedraCompound of two inverted snub dodecadodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the inverted snub dodecadodecahedron....
- Compound of two small stellated dodecahedraCompound of two small stellated dodecahedraThis uniform polyhedron compound is a composition of 2 small stellated dodecahedra, in the same arrangement as in the compound of 2 icosahedra.- External links :* VRML model:...
- Compound of two snub cubesCompound of two snub cubesThis uniform polyhedron compound is a composition of the 2 enantiomers of the snub cube.The vertex arrangement of this compound is shared by a convex nonuniform truncated cuboctahedron, having rectangular faces, alongside irregular hexagons and octagons, each alternating with two edge lengths....
- Compound of two snub dodecadodecahedraCompound of two snub dodecadodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecadodecahedron....
- Compound of two snub dodecahedraCompound of two snub dodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the snub dodecahedron.The vertex arrangement of this compound is shared by a convex nonuniform truncated icosidodecahedron, with rectangular faces, alongside irregular hexagons and decagons, each alternating two different...
- Compound of two snub icosidodecadodecahedraCompound of two snub icosidodecadodecahedraThis uniform polyhedron compound is a composition of the 2 enantiomers of the snub icosidodecadodecahedron....
- Compound of two truncated tetrahedraCompound of two truncated tetrahedraThis uniform polyhedron compound is a composition of two truncated tetrahedra, formed by truncating each of the tetrahedra in the stella octangula.The vertex arrangement is the same as a convex, but nonuniform rhombicuboctahedron having 12 rectangular faces....
- Prismatic compound of antiprismsPrismatic compound of antiprismsEach member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry.This infinite family can be enumerated as follows:...
- Prismatic compound of antiprisms with rotational freedomPrismatic compound of antiprisms with rotational freedomEach member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of antiprisms sharing a common axis of rotational symmetry...
- Prismatic compound of prismsPrismatic compound of prismsEach member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of prisms sharing a common axis of rotational symmetry.This infinite family can be enumerated as follows:...
- Prismatic compound of prisms with rotational freedomPrismatic compound of prisms with rotational freedomEach member of this infinite family of uniform polyhedron compounds is a symmetric arrangement of prisms sharing a common axis of rotational symmetry...
Four dimensions
Four-dimensional spacePolychoron
Polychoron
In geometry, a polychoron or 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
- general term for a four dimensional polytope
Regular convex polychoron
- 5-cell, TesseractTesseractIn geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analog of the cube. The tesseract is to the cube as the cube is to the square. Just as the surface of the cube consists of 6 square faces, the hypersurface of the tesseract consists of 8...
, 16-cell16-cellIn four dimensional geometry, a 16-cell or hexadecachoron is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century....
, 24-cell, 120-cell , 600-cell
Abstract regular polytope
- 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
, 57-cell
Schläfli–Hess polychoron (Regular star polychoron)
- Icosahedral 120-cell, Small stellated 120-cell, Great 120-cell, Grand 120-cell, Great stellated 120-cell, Grand stellated 120-cell, Great grand 120-cell, Great icosahedral 120-cell, Grand 600-cell, Great grand stellated 120-cell
Uniform polychoron
Uniform polychoron
In geometry, a uniform polychoron is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra....
- Rectified 5-cellRectified 5-cellIn four dimensional geometry, the rectified 5-cell is a uniform polychoron composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10...
, Truncated 5-cellTruncated 5-cellIn geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...
, Cantellated 5-cellCantellated 5-cellIn four-dimensional geometry, a cantellated 5-cell is a convex uniform polychoron, being a cantellation of the regular 5-cell.There are 2 unique degrees of runcinations of the 5-cell including with permutations truncations....
, Runcinated 5-cell - Rectified tesseractRectified tesseractIn geometry, the rectified tesseract, or rectified 8-cell is a uniform polychoron bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra....
, Truncated tesseractTruncated tesseractIn geometry, a truncated tesseract is a uniform polychoron formed as the truncation of the regular tesseract.There are three trunctions, including a bitruncation, and a tritruncation, which creates the truncated 16-cell....
, Cantellated tesseractCantellated tesseractIn four-dimensional geometry, a cantellated tesseract is a convex uniform polychoron, being a cantellation of the regular tesseract.There are four degrees of cantellations of the tesseract including with permutations truncations...
, Runcinated tesseractRuncinated tesseractIn four-dimensional geometry, a runcinated tesseract is a convex uniform polychoron, being a runcination of the regular tesseract.... - Rectified 16-cell, Truncated 16-cell
- Rectified 24-cellRectified 24-cellIn geometry, the rectified 24-cell is a uniform 4-dimensional polytope , which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. It can be obtained by reducing the icositetrachoron's cells to cubes or cuboctahedra....
, Truncated 24-cellTruncated 24-cellIn geometry, a truncated 5-cell is a uniform polychoron formed as the truncation of the regular 5-cell.There are two degrees of trunctions, including a bitruncation.- Truncated 5-cell:...
, Cantellated 24-cellCantellated 24-cellIn four-dimensional geometry, a cantellated 24-cell is a convex uniform polychoron, being a cantellation of the regular 24-cell.There are 2 unique degrees of runcinations of the 24-cell including with permutations truncations....
, Runcinated 24-cellRuncinated 24-cellIn four-dimensional geometry, a runcinated 24-cell is a convex uniform polychoron, being a runcination of the regular 24-cell....
, Snub 24-cellSnub 24-cellIn geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices.... - Rectified 120-cellRectified 120-cellIn geometry, a rectified 120-cell is a uniform polychoron formed as the rectification of the regular 120-cell.There are four rectifications of the 120-cell, including the zeroth, the 120-cell itself...
, Truncated 120-cellTruncated 120-cellIn geometry, a truncated 120-cell is a uniform polychoron formed as the truncation of the regular 120-cell.There are three trunctions, including a bitruncation, and a tritruncation, which creates the truncated 600-cell....
, Cantellated 120-cellCantellated 120-cellIn four-dimensional geometry, a cantellated 120-cell is a convex uniform polychoron, being a cantellation of the regular 120-cell....
, Runcinated 120-cellRuncinated 120-cellIn four-dimensional geometry, a runcinated 120-cell is a convex uniform polychoron, being a runcination of the regular 120-cell.... - Rectified 600-cellRectified 600-cellIn geometry, a rectified 600-cell is a uniform polychoron formed as the rectification of the regular 600-cell.There are four rectifications of the 600-cell, including the zeroth, the 600-cell itself...
, Truncated 600-cell, Cantellated 600-cell
Prismatic uniform polychoron
Prismatic uniform polychoron
In four-dimensional geometry, a prismatic uniform polytope is a uniform polychoron with a nonconnected Coxeter-Dynkin diagram symmetry group. These figures are analogous to the set of prisms and antiprism uniform polyhedra, but add a third category called duoprisms, constructed as a product of two...
- Grand antiprismGrand antiprismIn geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform polychoron, discovered in 1965 by Conway and Guy.- Alternate names :* Pentagonal double...
- DuoprismDuoprismIn geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher...
- Tetrahedral prism, Truncated tetrahedral prismTruncated tetrahedral prismIn geometry, a truncated tetrahedral prism is a convex uniform polychoron . This polychoron has 10 polyhedral cells: 2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms. It has 24 faces: 8 triangular, 18 square, and 8 hexagons. It has 48 edges and 24 vertices.Alternative...
- Truncated cubic prismTruncated cubic prismIn geometry, a truncated cubic prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Truncated-cubic hyperprism* Truncated-cubic...
, Truncated octahedral prismTruncated octahedral prismIn 4-dimensional geometry, a truncated octahedral prism is a convex uniform polychoron . This polychoron has 16 cells It has 64 faces , and 96 edges and 48 vertices.It has two symmetry constructions, one from the truncated octahedron, and one as an omnitruncation of...
, Cuboctahedral prismCuboctahedral prismIn geometry, a cuboctahedral prism is a convex uniform polychoron . This polychoron has 16 polyhedral cells: 2 cuboctahedra connected by 8 triangular prisms, and 6 cubes....
, Rhombicuboctahedral prismRhombicuboctahedral prismIn geometry, a rhombicuboctahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* rhombicuboctahedral dyadic prism In...
, Truncated cuboctahedral prismTruncated cuboctahedral prismIn geometry, a truncated cuboctahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Truncated-cuboctahedral dyadic prism...
, Snub cubic prismSnub cubic prismIn geometry, a truncated cuboctahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Snub-cuboctahedral dyadic prism In... - Truncated dodecahedral prismTruncated dodecahedral prismIn geometry, a truncated dodecahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Truncated-dodecahedral dyadic prism...
, Truncated icosahedral prismTruncated icosahedral prismIn geometry, a truncated icosahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Truncated-icosahedral dyadic prism In...
, Icosidodecahedral prismIcosidodecahedral prismIn geometry, an icosidodecahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids or Archimedean solids....
, Rhombicosidodecahedral prismRhombicosidodecahedral prismIn geometry, a rhombicosidodecahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* rhombicosidodecahedral dyadic prism...
, Truncated icosidodecahedral prismTruncated icosidodecahedral prismIn geometry, a truncated icosidodecahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Truncated-icosidodecahedral...
, Snub dodecahedral prismSnub dodecahedral prismIn geometry, a snub dodecahedral prism is a convex uniform polychoron .It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of Platonic solids or Archimedean solids in parallel hyperplanes.- Alternative names :* Snub-icosidodecahedral dyadic prism In...
Uniform antiprismatic prism
Uniform antiprismatic prism
In 4-dimensional geometry, a uniform antiprismatic prism is a uniform polychoron with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces....
- Triangular antiprismatic prism, Square antiprismatic prismSquare antiprismatic prismIn geometry, a square antiprismatic prism is a convex uniform polychoron . It is formed as two parallel square antiprisms connected by cubes and triangular prisms.- Related polytopes :...
, Pentagonal antiprismatic prism, Hexagonal antiprismatic prism, Heptagonal antiprismatic prism, Octagonal antiprismatic prism, Enneagonal antiprismatic prism, Decagonal antiprismatic prism - Pentagrammic antiprismatic prism, Hexagrammic antiprismatic prism, Heptagrammic antiprismatic prism, Octagrammic antiprismatic prism, Enneagrammic antiprismatic prism, Decagrammic antiprismatic prism
- Pentagrammic crossed antiprismatic prism, Hexagrammic crossed antiprismatic prism, Heptagrammic crossed antiprismatic prism, Octagrammic crossed antiprismatic prism, Enneagrammic crossed antiprismatic prism, Decagrammic crossed antiprismatic prism
Honeycombs
- Tesseractic honeycomb
- 24-cell honeycomb
- Snub 24-cell honeycombSnub 24-cell honeycombIn four-dimensional Euclidean geometry, the snub 24-cell honeycomb, or snub icositetrachoric honeycomb is a uniform space-filling tessellation by snub 24-cells, 16-cells, and 5-cells. It was discovered by Thorold Gosset with his 1900 paper of semiregular polytopes...
- Rectified 24-cell honeycombRectified 24-cell honeycombIn four-dimensional Euclidean geometry, the rectified 24-cell honeycomb is a uniform space-filling honeycomb. It is construcated rectification of the regular 24-cell honeycomb, containing tesseract and rectified 24-cell cells.- Symmetry constructions :...
- Truncated 24-cell honeycombTruncated 24-cell honeycombIn four-dimensional Euclidean geometry, the truncated 24-cell honeycomb is a uniform space-filling honeycomb. It can be seen as a truncation of the regular 24-cell honeycomb, containing tesseract and truncated 24-cell cells....
- 16-cell honeycomb
- 5-cell honeycomb5-cell honeycombIn four-dimensional Euclidean geometry, the 4-simplex honeycomb, 5-cell honeycomb or pentachoric-dispentachoric honeycomb is a space-filling tessellation honeycomb...
- Omnitruncated 5-cell honeycomb
- Truncated 5-cell honeycomb
- Omnitruncated 5-simplex honeycombOmnitruncated 5-simplex honeycombIn five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation . It is composed entirely of omnitruncated 5-simplex facets....
Five dimensions
Five-dimensional space, 5-polytope5-polytope
In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:...
and uniform 5-polytope
- 5-simplex, Rectified 5-simplex, Truncated 5-simplex, Cantellated 5-simplex, Runcinated 5-simplexRuncinated 5-simplexIn six-dimensional geometry, a runcinated 5-simplex is a convex uniform 5-polytope with 3rd order truncations of the regular 5-simplex.There are 4 unique runcinations of the 5-simplex with permutations of truncations, and cantellations....
, Stericated 5-simplex - 5-demicube, Truncated 5-demicube, Cantellated 5-demicubeCantellated 5-demicubeIn six-dimensional geometry, a cantellated 5-demicube is a convex uniform 5-polytope, being a cantellation of the uniform 5-demicube.There are 2 unique cantellation for the 5-demicube including a truncation.- Cantellated 5-demicube:...
, Runcinated 5-demicubeRuncinated 5-demicubeIn five-dimensional geometry, a runcinated 5-demicube is a convex uniform 5-polytope with a runcination operation, a 3rd order truncations the uniform 5-demicube.... - 5-cube, Rectified 5-cubeRectified 5-cubeIn give-dimensional geometry, a rectified 5-cube is a convex uniform 5-polytope, being a rectification of the regular 5-cube.There are 5 degrees of rectifications of a 5-polytope, the zeroth here being the 5-cube, and the 4th and last being the 5-orthoplex. Vertices of the rectified 5-cube are...
, 5-cube, Truncated 5-cubeTruncated 5-cubeIn five-dimensional geometry, a truncated 5-cube is a convex uniform 5-polytope, being a truncation of the regular 5-cube.There are four unique truncations of the 5-cube. Vertices of the truncated 5-cube are located as pairs on the edge of the 5-cube. Vertices of the bitruncated 5-cube are located...
, Cantellated 5-cubeCantellated 5-cubeIn six-dimensional geometry, a cantellated 5-cube is a convex uniform 5-polytope, being a cantellation of the regular 5-cube.There are 6 unique cantellation for the 5-cube, including truncations...
, Runcinated 5-cubeRuncinated 5-cubeIn five-dimensional geometry, a runcinated 5-cube is a convex uniform 5-polytope that is a runcination of the regular 5-cube....
, Stericated 5-cubeStericated 5-cubeIn five-dimensional geometry, a stericated 5-cube is a convex uniform 5-polytope with fourth-order truncations of the regular 5-cube.... - 5-orthoplex, Rectified 5-orthoplex, Truncated 5-orthoplexTruncated 5-orthoplexIn six-dimensional geometry, a truncated 5-orthoplex is a convex uniform 5-polytope, being a truncation of the regular 5-orthoplex.There are 4 unique truncations of the 5-orthoplex. Vertices of the truncation 5-orthoplex are located as pairs on the edge of the 5-orthoplex. Vertices of the...
, Cantellated 5-orthoplexCantellated 5-orthoplexIn six-dimensional geometry, a cantellated 5-orthoplex is a convex uniform 5-polytope, being a cantellation of the regular 5-orthoplex.There are 6 cantellation for the 5-orthoplex, including truncations...
, Runcinated 5-orthoplexRuncinated 5-orthoplexIn six-dimensional geometry, a runcinated 5-orthoplex is a convex uniform 5-polytope with 3rd order truncation of the regular 5-orthoplex.There are 8 runcinations of the 5-orthoplex with permutations of truncations, and cantellations...
Prismatic uniform 5-polytope
- 5-cell prism, Rectified 5-cell prism, Truncated 5-cell prism, Cantellated 5-cell prism, Runcinated 5-cell prism, Bitruncated 5-cell prism, Cantitruncated 5-cell prism, Runcitruncated 5-cell prism, Omnitruncated 5-cell prism
- Tesseractic prism, Rectified tesseractic prism, Truncated tesseractic prism, Cantellated tesseractic prism, Runcinated tesseractic prism, Bitruncated tesseractic prism, Cantitruncated tesseractic prism, Runcitruncated tesseractic prism, Omnitruncated tesseractic prism
- 16-cell prism, Truncated 16-cell prism, Runcitruncated 16-cell prism
- 24-cell prism, rectified 24-cell prism, truncated 24-cell prism, cantellated 24-cell prism, runcinated 24-cell prism, bitruncated 24-cell prism, cantitruncated 24-cell prism, runcitruncated 24-cell prism, omnitruncated 24-cell prism, snub 24-cell prism
- 120-cell prism, Rectified 120-cell prism, Truncated 120-cell prism, Cantellated 120-cell prism, Runcinated 120-cell prism, Bitruncated 120-cell prism, Cantitruncated 120-cell prism, Runcitruncated 120-cell prism, Omnitruncated 120-cell prism
- 600-cell prism, Rectified 600-cell prism, Truncated 600-cell prism, Cantellated 600-cell prism, Cantitruncated 600-cell prism, Runcitruncated 600-cell prism
- Grand antiprism prism
Honeycombs
- 5-cubic honeycomb
- 5-simplex honeycomb5-simplex honeycombIn five-dimensional Euclidean geometry, the 5-simplex honeycomb or hexateric honeycomb is a space-filling tessellation . Each vertex is shared by 12 5-simplexes, 30 rectified 5-simplexes, and 20 birectified 5-simplexes...
- Truncated 5-simplex honeycombTruncated 5-simplex honeycombIn five-dimensional Euclidean geometry, the truncated 5-simplex honeycomb or truncated hexateric honeycomb is a space-filling tessellation...
- 5-demicubic honeycomb
Six dimensions
Six-dimensional space, 6-polytope6-polytope
In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....
and uniform 6-polytope
- 6-simplex, Rectified 6-simplexRectified 6-simplexIn six-dimensional geometry, a rectified 6-simplex is a convex uniform 6-polytope, being a rectification of the regular 6-simplex.There are three unique degrees of rectifications, including the zeroth, the 6-simplex itself. Vertices of the rectified 6-simplex are located at the edge-centers of the...
, Truncated 6-simplexTruncated 6-simplexIn six-dimensional geometry, a truncated 6-simplex is a convex uniform 6-polytope, being a truncation of the regular 6-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 6-simplex are located as pairs on the edge of the 6-simplex. Vertices of the bitruncated 6-simplex are...
, Cantellated 6-simplexCantellated 6-simplexIn six-dimensional geometry, a cantellated 6-simplex is a convex uniform 6-polytope, being a cantellation of the regular 6-simplex.There are unique 4 degrees of cantellation for the 6-simplex, including truncations.- Cantellated 6-simplex:...
, Runcinated 6-simplexRuncinated 6-simplexIn six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination of the regular 6-simplex....
, Stericated 6-simplexStericated 6-simplexIn six-dimensional geometry, a stericated 6-simplex is a convex uniform 6-polytope with 4th order truncations of the regular 6-simplex....
, Pentellated 6-simplexPentellated 6-simplexIn six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.There are unique 10 degrees of pentellations of the 6-simplex with permutations of truncations, cantellations, runcinations, and sterications... - 6-demicube, Truncated 6-demicube, Cantellated 6-demicubeCantellated 6-demicubeIn six-dimensional geometry, a cantellated 6-demicube is a convex uniform 6-polytope, being a cantellation of the uniform 6-demicube. There are 2 unique cantellation for the 6-demicube including a truncation.- Cantellated 6-demicube:...
, Runcinated 6-demicubeRuncinated 6-demicubeIn six-dimensional geometry, a runcinated 6-demicube is a convex uniform 6-polytope with 3rd order truncations of the uniform 6-demicube.There are unique 4 runcinations of the 6-demicube, including permutations of truncations, and cantellations....
, Stericated 6-demicubeStericated 6-demicubeIn six-dimensional geometry, a stericated 6-demicube is a convex uniform 6-polytope, constructed as a sterication of the 6-demicube.... - 6-cube, Rectified 6-cubeRectified 6-cubeIn six-dimensional geometry, a rectified 6-cube is a convex uniform 6-polytope, being a rectification of the regular 6-cube.There are unique 6 degrees of rectifications, the zeroth being the 6-cube, and the 6th and last being the 6-orthoplex. Vertices of the rectified 6-cube are located at the...
, 6-cube, Truncated 6-cube, Cantellated 6-cubeCantellated 6-cubeIn six-dimensional geometry, a cantellated 6-cube is a convex uniform 6-polytope, being a cantellation of the regular 6-cube.There are 8 cantellations for the 6-cube, including truncations...
, Runcinated 6-cubeRuncinated 6-cubeIn six-dimensional geometry, a runcinated 6-cube is a convex uniform 6-polytope with 3rd order truncations of the regular 6-cube.There are 12 unique runcinations of the 6-cube with permutations of truncations, and cantellations...
, Stericated 6-cubeStericated 6-cubeIn six-dimensional geometry, a stericated 6-cube is a convex uniform 6-polytope, constructed as a sterication of the regular 6-cube....
, Pentellated 6-cubePentellated 6-cubeIn six-dimensional geometry, a pentellated 6-cube is a convex uniform 6-polytope with 5th order truncations of the regular 6-cube.There are unique 16 degrees of pentellations of the 6-cube with permutations of truncations, cantellations, runcinations, and sterications... - 6-orthoplex, Rectified 6-orthoplex, Truncated 6-orthoplexTruncated 6-orthoplexIn six-dimensional geometry, a truncated 6-orthoplex is a convex uniform 6-polytope, being a truncation of the regular 6-orthoplex.There are 5 degrees of truncation for the 6-orthoplex. Vertices of the truncated 6-orthoplex are located as pairs on the edge of the 6-orthoplex. Vertices of the...
, Cantellated 6-orthoplexCantellated 6-orthoplexIn six-dimensional geometry, a cantellated 6-orthoplex is a convex uniform 6-polytope, being a cantellation of the regular 6-orthoplex.There are 8 cantellation for the 6-orthoplex including truncations...
, Runcinated 6-orthoplexRuncinated 6-orthoplexIn six-dimensional geometry, a runcinated 6-orthplex is a convex uniform 6-polytope with 3rd order truncations of the regular 6-orthoplex.There are 12 unique runcinations of the 6-orthoplex with permutations of truncations, and cantellations...
, Stericated 6-orthoplexStericated 6-orthoplexIn six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication of the regular 6-orthoplex.... - 1 22 polytope, 2 21 polytope
Honeycombs
- 6-cubic honeycomb
- 6-simplex honeycomb6-simplex honeycombIn six-dimensional Euclidean geometry, the 6-simplex honeycomb is a space-filling tessellation . The tessellation fills space by 6-simplex, rectified 6-simplex, and birectified 6-simplex facets. These facet types occur in proportions of 1:1:1 respectively in the whole honeycomb.This vertex...
- 6-demicubic honeycomb
- 2 22 honeycomb
Seven dimensions
Seven-dimensional spaceSeven-dimensional space
In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space...
, uniform 7-polytope
- 7-simplex, Rectified 7-simplexRectified 7-simplexIn seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...
, Truncated 7-simplexTruncated 7-simplexIn seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex...
, Cantellated 7-simplexCantellated 7-simplexIn seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.There are unique 6 degrees of cantellation for the 7-simplex, including truncations.- Cantellated 7-simplex:...
, Runcinated 7-simplexRuncinated 7-simplexIn seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-simplex.There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations....
, Stericated 7-simplexStericated 7-simplexIn seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-simplex....
, Pentellated 7-simplexPentellated 7-simplexIn seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-simplex....
, Hexicated 7-simplexHexicated 7-simplexIn seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-simplex.... - 7-demicube, Truncated 7-demicubeTruncated 7-demicubeIn seven-dimensional geometry, a truncated 7-demicube is a uniform 7-polytope, being a truncation of the 7-demicube.- Cartesian coordinates :The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd...
, Cantellated 7-demicubeCantellated 7-demicubeIn seven-dimensional geometry, a cantellated 7-demicube is a convex uniform 7-polytope, being a cantellation of the uniform 7-demicube. There are 2 unique cantellation for the 7-demicube including a truncation.- Cantellated 7-demicube:...
, Runcinated 7-demicube, Stericated 7-demicube, Pentellated 7-demicube - 7-cube, Rectified 7-cubeRectified 7-cubeIn seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the...
, 7-cube, Truncated 7-cubeTruncated 7-cubeIn seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the...
, Cantellated 7-cubeCantellated 7-cubeIn seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.There are 10 degrees of cantellation for the 7-cube, including truncations...
, Runcinated 7-cubeRuncinated 7-cubeIn seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations of the regular 7-cube.There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations...
, Stericated 7-cubeStericated 7-cubeIn seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations of the regular 7-cube.There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations...
, Pentellated 7-cubePentellated 7-cubeIn seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications...
, Hexicated 7-cubeHexicated 7-cubeIn seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-cube.... - 7-orthoplex, Rectified 7-orthoplex, Truncated 7-orthoplexTruncated 7-orthoplexIn seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated...
, Cantellated 7-orthoplexCantellated 7-orthoplexIn seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.There are ten degrees of cantellation for the 7-orthoplex, including truncations...
, Runcinated 7-orthoplexRuncinated 7-orthoplexIn seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-orthoplex.There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations...
, Stericated 7-orthoplexStericated 7-orthoplexIn seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-orthoplex....
, Pentellated 7-orthoplexPentellated 7-orthoplexIn seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-orthoplex.... - 1 32 polytope, 2 31 polytope, 3 21 polytope
Honeycombs
- 7-cubic honeycomb
- 7-demicubic honeycomb
- 3 31 honeycomb, 1 33 honeycomb
Eight dimension
Eight-dimensional spaceEight-dimensional space
In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 8, the set of all such locations is called 8-dimensional Euclidean space...
, uniform 8-polytope
- 8-simplex, Rectified 8-simplexRectified 8-simplexIn eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex.There are unique 3 degrees of rectifications. Vertices of the rectified 8-simplex are located at the edge-centers of the 8-simplex. Vertices of the birectified...
, Truncated 8-simplexTruncated 8-simplexIn eight-dimensional geometry, a truncated 8-simplex is a convex uniform 8-polytope, being a truncation of the regular 8-simplex.There are a four unique degrees of truncation. Vertices of the truncation 8-simplex are located as pairs on the edge of the 8-simplex. Vertices of the bitruncated...
, Cantellated 8-simplexCantellated 8-simplexIn eight-dimensional geometry, a cantellated 8-simplex is a convex uniform 8-polytope, being a cantellation of the regular 8-simplex.There are six unique cantellations for the 8-simplex, including permutations of truncation.- Cantellated 8-simplex:...
, Runcinated 8-simplexRuncinated 8-simplexIn eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations of the regular 8-simplex....
, Stericated 8-simplexStericated 8-simplexIn eight-dimensional geometry, a stericated 8-simplex is a convex uniform 8-polytope with 4th order truncations of the regular 8-simplex...
, Pentellated 8-simplexPentellated 8-simplexIn eight-dimensional geometry, a pentellated 8-simplex is a convex uniform 8-polytope with 5th order truncations of the regular 8-simplex.There are two unique pentellations of the 8-simplex. Including truncations, canetellations, runcinations, and sterications, there are 32 more pentallations....
, Hexicated 8-simplexHexicated 8-simplexIn eight-dimensional geometry, a hexicated 8-simplex is a uniform 8-polytope, being a hexication of the regular 8-simplex.- Coordinates :...
, Heptellated 8-simplexHeptellated 8-simplexIn eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations from the regular 8-simplex.... - 8-orthoplex, Rectified 8-orthoplex, Truncated 8-orthoplexTruncated 8-orthoplexIn eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated...
, Cantellated 8-orthoplex, Runcinated 8-orthoplex, Stericated 8-orthoplex, Pentellated 8-orthoplex, Hexicated 8-orthoplex, - 8-cube, Rectified 8-cubeRectified 8-cubeIn eight-dimensional geometry, a rectified 8-cube is a convex uniform 8-polytope, being a rectification of the regular 8-cube.There are unique 8 degrees of rectifications, the zeroth being the 8-cube, and the 7th and last being the 8-orthoplex. Vertices of the rectified 8-cube are located at the...
, Truncated 8-cubeTruncated 8-cubeIn eight-dimensional geometry, a truncated 8-cube is a convex uniform 8-polytope, being a truncation of the regular 8-cube.There are unique 7 degrees of truncation for the 8-cube. Vertices of the truncation 8-cube are located as pairs on the edge of the 8-cube. Vertices of the bitruncated 8-cube...
, Cantellated 8-cube, Runcinated 8-cube, Stericated 8-cube, Pentellated 8-cube, Hexicated 8-cube, Heptellated 8-cube - 8-demicube, Truncated 8-demicubeTruncated 8-demicubeIn eight-dimensional geometry, a truncated 8-demicube is a uniform 8-polytope, being a truncation of the 8-demicube.- Cartesian coordinates :The Cartesian coordinates for the vertices of a truncated 8-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd number...
, Cantellated 8-demicube, Runcinated 8-demicube, Stericated 8-demicube, Pentellated 8-demicube, Hexicated 8-demicube - 1 42 polytope, 2 41 polytope, 4 21 polytope, Truncated 4 21 polytope, Truncated 2 41 polytope, Truncated 1 42 polytope, Cantellated 4 21 polytope, Cantellated 2 41 polytope, Runcinated 4 21 polytope
Honeycombs
- 8-cubic honeycomb
- 8-demicubic honeycomb
- 5 21 honeycomb5 21 honeycombIn geometry, the 521 honeycomb is a uniform tessellation of 8-dimensional Euclidean space.This honeycomb was first studied by Gosset who called it a 9-ic semi-regular figure...
, 2 51 honeycomb, 1 52 honeycomb
Dimensional families
Regular polytopeRegular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
and List of regular polytopes
- SimplexSimplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
- HypercubeHypercubeIn geometry, a hypercube is an n-dimensional analogue of a square and a cube . It is a closed, compact, convex figure whose 1-skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length.An...
- Cross-polytopeCross-polytopeIn geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in any number of dimensions. The vertices of a cross-polytope are all the permutations of . The cross-polytope is the convex hull of its vertices...
Uniform polytope
Uniform polytope
A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
- Demihypercube
- Uniform 1 k2 polytopeUniform 1 k2 polytopeIn geometry, 1k2 polytope is a uniform polytope in n-dimensions constructed from the En Coxeter group. The family was named by Coxeter as 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence...
- Uniform 2 k1 polytopeUniform 2 k1 polytopeIn geometry, 2k1 polytope is a uniform polytope in n dimensions constructed from the En Coxeter group. The family was named by Coxeter as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence...
- Uniform k 21 polytope
Honeycombs
Geometric operators
- Rectification (geometry)Rectification (geometry)In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...
- Truncation (geometry)Truncation (geometry)In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
- BitruncationBitruncationIn geometry, a bitruncation is an operation on regular polytopes. It represents a truncation beyond rectification. The original edges are lost completely and the original faces remain as smaller copies of themselves....
- Cantellation
- RuncinationRuncinationIn geometry, runcination is an operation that cuts a regular polytope simultaneously along the faces, edges and vertices, creating new facets in place of the original face, edge, and vertex centers....
- Sterication
- Omnitruncation
- Expansion (geometry)Expansion (geometry)In geometry, expansion is a polytope operation where facets are separated and moved radially apart, and new facets are formed at separated elements...
- Snub (geometry)Snub (geometry)In geometry, an alternation is an operation on a polyhedron or tiling that removes alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided...
- Alternation (geometry)
- Dual polyhedronDual polyhedronIn geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
- Gyration (geometry)
- Elongation (geometry)
- Augmentation (geometry)
- Diminishment (geometry)
- Greatening (geometry)
- Aggrandizement (geometry)
- StellationStellationStellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...
- KleetopeKleetopeIn geometry and polyhedral combinatorics, the Kleetope of a polyhedron or higher-dimensional convex polytope is another polyhedron or polytope formed by replacing each facet of with a shallow pyramid...
- Conway polyhedron notationConway polyhedron notationConway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...