List of Wenninger polyhedron models
Encyclopedia
This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger
.
The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes.
It contains the 75 nonprismatic uniform polyhedra
, as well as 44 stellated forms
of the convex regular and semiregular polyhedra.
This list was written to honor this early polyhedral work from Wenninger, and to provide a detailed reference to the 119 numbered models in his book.
Models listed here can be cited as "Wenninger Model Number N", or WN for brevity.
The polyhedra are grouped below in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.
Kepler–Poinsot polyhedra (Regular star polyhedra
Magnus Wenninger
Father Magnus J. Wenninger OSB is a mathematician who works on constructing polyhedron models, and wrote the first book on their construction.-Early life and education:...
.
The book was written as a guide book to building polyhedra as physical models. It includes templates of face elements for construction and helpful hints in building, and also brief descriptions on the theory behind these shapes.
It contains the 75 nonprismatic uniform polyhedra
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
, as well as 44 stellated forms
Stellation
Stellation 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...
of the convex regular and semiregular polyhedra.
This list was written to honor this early polyhedral work from Wenninger, and to provide a detailed reference to the 119 numbered models in his book.
Models listed here can be cited as "Wenninger Model Number N", or WN for brevity.
The polyhedra are grouped below in 5 tables: Regular (1–5), Semiregular (6–18), regular star polyhedra (20–22,41), Stellations and compounds (19–66), and uniform star polyhedra (67–119). The four regular star polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.
Platonic solids (regular) W1 to W5
Index | Name | Picture | Dual name | Dual picture | Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... and Schläfli symbol |
Symmetry group | U# | K# | V | E | F | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 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... |
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... |
3|2 3 | {3,3} |
Td | U01 | K06 | 4 | 6 | 4 | 4{3} | ||
2 | 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.... |
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.... |
4|2 3 | {3,4} |
Oh | U05 | K10 | 6 | 12 | 8 | 8{3} | ||
3 | 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.... (Cube) |
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.... |
3|2 4 | {4,3} |
Oh | U06 | K11 | 8 | 12 | 6 | 6{4} | ||
4 | Icosahedron Icosahedron In 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.... |
Dodecahedron | 5|2 3 | {3,5} |
Ih | U22 | K27 | 12 | 30 | 20 | 20{3} | ||
5 | Dodecahedron | Icosahedron Icosahedron In 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.... |
3|2 5 | {5,3} |
Ih | U23 | K28 | 20 | 30 | 12 | 12{5} |
Archimedean solids (Semiregular) W6 to W18
Index | Name | Picture | Dual name | Dual picture | Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... |
Symmetry group | U# | K# | V | E | F | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
6 | Truncated tetrahedron Truncated tetrahedron In 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 :... |
triakis tetrahedron Triakis tetrahedron In 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... |
2 3|3 | 3.6.6 |
Td | U02 | K07 | 12 | 18 | 8 | 4{3} + 4{6} | ||
7 | Truncated octahedron Truncated octahedron In 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.... |
tetrakis hexahedron Tetrakis hexahedron In 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.... |
2 4|3 | 4.6.6 |
Oh | U08 | K13 | 24 | 36 | 14 | 6{4} + 8{6} | ||
8 | Truncated hexahedron Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... |
triakis octahedron Triakis octahedron In 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... |
2 3|4 | 3.8.8 |
Oh | U09 | K14 | 24 | 36 | 14 | 8{3} + 6{8} | ||
9 | Truncated icosahedron Truncated icosahedron In 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.... |
pentakis dodecahedron Pentakis dodecahedron In 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... |
2 5|3 | 5.6.6 |
Ih | U25 | K30 | 60 | 90 | 32 | 12{5} + 20{6} | ||
10 | Truncated dodecahedron Truncated dodecahedron In 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 :... |
triakis icosahedron Triakis icosahedron In 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... |
2 3|5 | 3.10.10 |
Ih | U26 | K31 | 60 | 90 | 32 | 20{3} + 12{10} | ||
11 | Cuboctahedron Cuboctahedron In 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,... |
rhombic dodecahedron Rhombic dodecahedron In 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:... |
2|3 4 | 3.4.3.4 |
Oh | U07 | K12 | 12 | 24 | 14 | 8{3} + 6{4} | ||
12 | Icosidodecahedron Icosidodecahedron In 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... |
rhombic triacontahedron Rhombic triacontahedron In 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.... |
2|3 5 | 3.5.3.5 |
Ih | U24 | K29 | 30 | 60 | 32 | 20{3} + 12{5} | ||
13 | Small rhombicuboctahedron | deltoidal icositetrahedron Deltoidal icositetrahedron In geometry, a deltoidal icositetrahedron is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.... |
3 4|2 | 3.4.4.4 |
Oh | U10 | K15 | 24 | 48 | 26 | 8{3}+(6+12){4} | ||
14 | Small rhombicosidodecahedron | deltoidal hexecontahedron Deltoidal hexecontahedron In 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... |
3 5|2 | 3.4.5.4 |
Ih | U27 | K32 | 60 | 120 | 62 | 20{3} + 30{4} + 12{5} | ||
15 | Great rhombicuboctahedron (Rhombitruncated cuboctahedron) (Truncated cuboctahedron) |
disdyakis dodecahedron Disdyakis dodecahedron In 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... |
2 3 4| | 4.6.8 |
Oh | U11 | K16 | 48 | 72 | 26 | 12{4} + 8{6} + 6{8} | ||
16 | Great rhombicosidodecahedron (Rhombitruncated icosidodecahedron) (Truncated icosidodecahedron) |
disdyakis triacontahedron Disdyakis triacontahedron In 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... |
2 3 5| | 4.6.10 |
Ih | U28 | K33 | 120 | 180 | 62 | 30{4} + 20{6} + 12{10} | ||
17 | Snub cube Snub cube In 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... |
pentagonal icositetrahedron Pentagonal icositetrahedron In 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.... |
|2 3 4 | 3.3.3.3.4 |
O | U12 | K17 | 24 | 60 | 38 | (8 + 24){3} + 6{4} | ||
18 | Snub dodecahedron Snub dodecahedron In 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.... |
pentagonal hexecontahedron Pentagonal hexecontahedron In 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... |
|2 3 5 | 3.3.3.3.5 |
I | U29 | K34 | 60 | 150 | 92 | (20 + 60){3} + 12{5} |
Kepler–Poinsot polyhedra (Regular star polyhedraStar polyhedronIn geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.There are two general kinds of star polyhedron:*Polyhedra which self-intersect in a repetitive way....
) W20, W21, W22 and W41
Index | Name | Picture | Dual name | Dual picture | Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... |
Vertex figure Vertex configuration In geometry, a vertex configuration is a short-hand notation for representing the vertex figure of a polyhedron or tiling as the sequence of faces around a vertex. For uniform polyhedra there is only one vertex type and therefore the vertex configuration fully defines the polyhedron... and Schläfli symbol |
Symmetry group | U# | K# | V | E | F | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
20 | Small stellated dodecahedron | Great dodecahedron | 5|25/2 | {5/2,5} |
Ih | U34 | K39 | 12 | 30 | 12 | 12{5/2} | ||
21 | Great dodecahedron | Small stellated dodecahedron | 5/2|2 5 | {5,5/2} |
Ih | U35 | K40 | 12 | 30 | 12 | 12{5} | ||
22 | Great stellated dodecahedron | Great icosahedron | 3|25/2 | {5/2,3} |
Ih | U52 | K57 | 20 | 30 | 12 | 12{5/2} | ||
41 | Great icosahedron (16th stellation of icosahedron) |
Great stellated dodecahedron | 5/2|2 3 | {3,5/2} |
Ih | U53 | K58 | 12 | 30 | 20 | 20{3} |
Stellations of octahedron
Index | Name | Symmetry group | Picture | Facets |
---|---|---|---|---|
2 | 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.... (regular) |
Oh | ||
19 | Stellated octahedron (Compound of two tetrahedra) |
Oh |
Stellations of dodecahedron
Index | Name | Symmetry group | Picture | Facets |
---|---|---|---|---|
5 | Dodecahedron (regular) | Ih | ||
20 | Small stellated dodecahedron (regular) (First stellation of dodecahedron) |
Ih | ||
21 | Great dodecahedron (regular) (Second stellation of dodecahedron) |
Ih | ||
22 | Great stellated dodecahedron (regular) (Third stellation of dodecahedron) |
Ih |
Stellations of icosahedron
Index | Name | Symmetry group | Picture | Facets |
---|---|---|---|---|
4 | Icosahedron Icosahedron In 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.... (regular) |
Ih | ||
26 | First stellation of icosahedron Small triambic icosahedron Small triambic icosahedron In 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.... (Triakis icosahedron) |
Ih | ||
23 | Compound of five octahedra Compound of five octahedra This 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.... (First compound stellation of icosahedron) |
Ih | ||
24 | Compound of five tetrahedra Compound of five tetrahedra This 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... (Second compound stellation of icosahedron) |
I | ||
25 | Compound of ten tetrahedra Compound of ten tetrahedra This 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... (Third compound stellation of icosahedron) |
Ih | ||
27 | Second stellation of icosahedron | Ih | ||
28 | Third stellation of icosahedron | Ih | ||
29 | Fourth stellation of icosahedron | Ih | ||
30 | Fifth stellation of icosahedron | Ih | ||
31 | Sixth stellation of icosahedron | Ih | ||
32 | Seventh stellation of icosahedron | Ih | ||
33 | Eighth stellation of icosahedron | Ih | ||
34 | Ninth stellation of icosahedron Great triambic icosahedron Great triambic icosahedron In 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... |
Ih | ||
35 | Tenth stellation of icosahedron | I | ||
36 | Eleventh stellation of icosahedron | I | ||
37 | Twelfth stellation of icosahedron | Ih | ||
38 | Thirteenth stellation of icosahedron | I | ||
39 | Fourteenth stellation of icosahedron | I | ||
40 | Fifteenth stellation of icosahedron | I | ||
41 | Great icosahedron (regular) (Sixteenth stellation of icosahedron) |
Ih | ||
42 | Final stellation of the icosahedron Final stellation of the icosahedron In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram.... |
Ih |
Stellations of cuboctahedron
Index | Name | Symmetry group | Picture | Facets (octahedral planes) | Facets (cube planes) |
---|---|---|---|---|---|
11 | Cuboctahedron Cuboctahedron In 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,... (regular) |
Oh | |||
43 | Compound of cube and octahedron Compound of cube and octahedron 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 octahedron and a cube... (First stellation of cuboctahedron) |
Oh | |||
44 | Second stellation of cuboctahedron | Oh | |||
45 | Third stellation of cuboctahedron | Oh | |||
46 | Fourth stellation of cuboctahedron | Oh |
Stellations of icosidodecahedron
Index | Name | Symmetry group | Picture | Facets (icosahedral planes) | Facets (dodecahedral planes) |
---|---|---|---|---|---|
12 | Icosidodecahedron Icosidodecahedron In 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... (regular) |
Ih | |||
47 | (First stellation of icosidodecahedron) Compound of dodecahedron and icosahedron Compound of dodecahedron and icosahedron In 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... |
Ih | |||
48 | Second stellation of icosidodecahedron | Ih | |||
49 | Third stellation of icosidodecahedron | Ih | |||
50 | Fourth stellation of icosidodecahedron (Compound of small stellated dodecahedron and triakis icosahedron) |
Ih | |||
51 | Fifth stellation of icosidodecahedron (Compound of small stellated dodecahedron and five octahedra) |
Ih | |||
52 | Sixth stellation of icosidodecahedron | Ih | |||
53 | Seventh stellation of icosidodecahedron | Ih | |||
54 | Eighth stellation of icosidodecahedron (Compound of five tetrahedra and great dodecahedron) |
I | |||
55 | Ninth stellation of icosidodecahedron | Ih | |||
56 | Tenth stellation of icosidodecahedron | Ih | |||
57 | Eleventh stellation of icosidodecahedron | Ih | |||
58 | Twelfth stellation of icosidodecahedron | Ih | |||
59 | Thirteenth stellation of icosidodecahedron | Ih | |||
60 | Fourteenth stellation of icosidodecahedron | Ih | |||
61 | Compound of great stellated dodecahedron and great icosahedron | Ih | |||
62 | Fifteenth stellation of icosidodecahedron | Ih | |||
63 | Sixteenth stellation of icosidodecahedron | Ih | |||
64 | Seventeenth stellation of icosidodecahedron | Ih | |||
65 | Eighteenth stellation of icosidodecahedron | Ih | |||
66 | Nineteenth stellation of icosidodecahedron | Ih |
Uniform nonconvex solids W67 to W119
Index | Name | Picture | Dual name | Dual picture | Wythoff symbol Wythoff symbol In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles.... |
Vertex figure Vertex figure In 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:... |
Symmetry group | U# | K# | V | E | F | Faces by type |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
67 | Tetrahemihexahedron Tetrahemihexahedron In 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 .... |
Tetrahemihexacron Tetrahemihexacron In 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... |
3/23|2 | 4.3/2.4.3 |
Td | U04 | K09 | 6 | 12 | 7 | 4{3}+3{4} | ||
68 | Octahemioctahedron Octahemioctahedron In 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 :... |
Octahemioctacron Octahemioctacron In geometry, the octahemioctacron is the dual of the octahemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the hexahemioctacron.... |
3/23|3 | 6.3/2.6.3 |
Oh | U03 | K08 | 12 | 24 | 12 | 8{3}+4{6} | ||
69 | Small cubicuboctahedron Small cubicuboctahedron In 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 hexacronic icositetrahedron Small hexacronic icositetrahedron In geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron.... |
3/24|4 | 8.3/2.8.4 |
Oh | U13 | K18 | 24 | 48 | 20 | 8{3}+6{4}+6{8} | ||
70 | Small ditrigonal icosidodecahedron Small ditrigonal icosidodecahedron In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron... |
Small triambic icosahedron Small triambic icosahedron In 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.... |
3|5/23 | (5/2.3)3 |
Ih | U30 | K35 | 20 | 60 | 32 | 20{3}+12{5/2} | ||
71 | Small icosicosidodecahedron Small icosicosidodecahedron In 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 icosacronic hexecontahedron Small icosacronic hexecontahedron In geometry, the small icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small icosicosidodecahedron. A Hexacontahedron has 60 faces.... |
5/23|3 | 6.5/2.6.3 |
Ih | U31 | K36 | 60 | 120 | 52 | 20{3}+12{5/2}+20{6} | ||
72 | Small dodecicosidodecahedron Small dodecicosidodecahedron In geometry, the small dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U33. Its vertex figure is a crossed quadrilateral.-Related polyhedra:... |
Small dodecacronic hexecontahedron Small dodecacronic hexecontahedron In geometry, the small dodecacronic hexecontahedron is the dual of the small dodecicosidodecahedron.-External links:... |
3/25|5 | 10.3/2.10.5 |
Ih | U33 | K38 | 60 | 120 | 44 | 20{3}+12{5}+12{10} | ||
73 | Dodecadodecahedron | Medial rhombic triacontahedron Medial rhombic triacontahedron In 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.... |
2|5/25 | (5/2.5)2 |
Ih | U36 | K41 | 30 | 60 | 24 | 12{5}+12{5/2} | ||
74 | Small rhombidodecahedron Small rhombidodecahedron In geometry, the small rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U39. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
Small rhombidodecacron Small rhombidodecacron In 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 :*... |
25/25| | 10.4.10/9.4/3 |
Ih | U39 | K44 | 60 | 120 | 42 | 30{4}+12{10} | ||
75 | Truncated great dodecahedron | Small stellapentakis dodecahedron Small stellapentakis dodecahedron In 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 :*... |
25/2|5 | 10.10.5/2 |
Ih | U37 | K42 | 60 | 90 | 24 | 12{5/2}+12{10} | ||
76 | Rhombidodecadodecahedron | Medial deltoidal hexecontahedron Medial deltoidal hexecontahedron In 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 :*... |
5/25|2 | 4.5/2.4.5 |
Ih | U38 | K43 | 60 | 120 | 54 | 30{4}+12{5}+12{5/2} | ||
77 | Great cubicuboctahedron Great cubicuboctahedron In 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 hexacronic icositetrahedron Great hexacronic icositetrahedron In geometry, the great hexacronic icositetrahedron is the dual of the great cubicuboctahedron.... |
3 4|4/3 | 8/3.3.8/3.4 |
Oh | U14 | K19 | 24 | 48 | 20 | 8{3}+6{4}+6{8/3} | ||
78 | Cubohemioctahedron Cubohemioctahedron In 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... |
Hexahemioctacron Hexahemioctacron In geometry, the hexahemioctacron is the dual of the cubohemioctahedron, and is one of nine dual hemipolyhedra. It appears visually indistinct from the octahemioctacron.... |
4/34|3 | 6.4/3.6.4 |
Oh | U15 | K20 | 12 | 24 | 10 | 6{4}+4{6} | ||
79 | Cubitruncated cuboctahedron Cubitruncated cuboctahedron In geometry, the cubitruncated cuboctahedron is a nonconvex uniform polyhedron, indexed as U16.- Convex hull :Its convex hull is a nonuniform truncated cuboctahedron.- Cartesian coordinates :... (Cuboctatruncated cuboctahedron) |
Tetradyakis hexahedron Tetradyakis hexahedron In 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:*... |
4/33 4| | 8/3.6.8 |
Oh | U16 | K21 | 48 | 72 | 20 | 8{6}+6{8}+6{8/3} | ||
80 | Ditrigonal dodecadodecahedron Ditrigonal dodecadodecahedron In geometry, the Ditrigonal dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U41.- Related polyhedra :Its convex hull is a regular dodecahedron... |
Medial triambic icosahedron | 3|5/35 | (5/3.5)3 |
Ih | U41 | K46 | 20 | 60 | 24 | 12{5}+12{5/2 | ||
81 | Great ditrigonal dodecicosidodecahedron Great ditrigonal dodecicosidodecahedron In 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 dodecacronic hexecontahedron Great ditrigonal dodecacronic hexecontahedron In geometry, the great ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great ditrigonal dodecicosidodecahedron.... |
3 5|5/3 | 10/3.3.10/3.5 |
Ih | U42 | K47 | 60 | 120 | 44 | 20{3}+12{5}+12{10/3} | ||
82 | Small ditrigonal dodecicosidodecahedron Small ditrigonal dodecicosidodecahedron In 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 dodecacronic hexecontahedron Small ditrigonal dodecacronic hexecontahedron In geometry, the small ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform small ditrigonal dodecicosidodecahedron.... |
5/33|5 | 10.5/3.10.3 |
Ih | U43 | K48 | 60 | 120 | 44 | 20{3}+12{5/2}+12{10} | ||
83 | Icosidodecadodecahedron Icosidodecadodecahedron In 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... |
Medial icosacronic hexecontahedron Medial icosacronic hexecontahedron In geometry, the medial icosacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform icosidodecadodecahedron.... |
5/35|3 | 6.5/3.6.5 |
Ih | U44 | K49 | 60 | 120 | 44 | 12{5}+12{5/2}+20{6} | ||
84 | Icositruncated dodecadodecahedron Icositruncated dodecadodecahedron In geometry, the icositruncated dodecadodecahedron is a nonconvex uniform polyhedron, indexed as U45.- Convex hull :Its convex hull is a nonuniform great rhombicosidodecahedron.- Cartesian coordinates :... (Icosidodecatruncated icosidodecahedron) |
Tridyakis icosahedron Tridyakis icosahedron In 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... |
5/33 5| | 10/3.6.10 |
Ih | U45 | K50 | 120 | 180 | 44 | 20{6}+12{10}+12{10/3} | ||
85 | Nonconvex great rhombicuboctahedron (Quasirhombicuboctahedron) |
Great deltoidal icositetrahedron Great deltoidal icositetrahedron In geometry, the great deltoidal icositetrahedron is the dual of the uniform great rhombicuboctahedron.-External links:... |
3/24|2 | 4.3/2.4.4 |
Oh | U17 | K22 | 24 | 48 | 26 | 8{3}+(6+12){4} | ||
86 | Small rhombihexahedron Small rhombihexahedron In 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 rhombihexacron Small rhombihexacron In geometry, the small rhombihexacron is the dual of the small rhombihexahedron. Its faces are antiparallelograms formed by pairs of coplanar triangles.... |
3/22 4| | 4.8.4/3.8 |
Oh | U18 | K23 | 24 | 48 | 18 | 12{4}+6{8} | ||
87 | Great ditrigonal icosidodecahedron | Great triambic icosahedron Great triambic icosahedron In 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... |
3/2|3 5 | (5.3.5.3.5.3)/2 |
Ih | U47 | K52 | 20 | 60 | 32 | 20{3}+12{5} | ||
88 | Great icosicosidodecahedron Great icosicosidodecahedron In 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 icosacronic hexecontahedron Great icosacronic hexecontahedron In geometry, the great icosacronic hexecontahedron is the dual of the great icosicosidodecahedron.- External links :... |
3/25|3 | 6.3/2.6.5 |
Ih | U48 | K53 | 60 | 120 | 52 | 20{3}+12{5}+20{6} | ||
89 | Small icosihemidodecahedron Small icosihemidodecahedron In 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 icosihemidodecacron Small icosihemidodecacron In 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.... |
3/23|5 | 10.3/2.10.3 |
Ih | U49 | K54 | 30 | 60 | 26 | 20{3}+6{10} | ||
90 | Small dodecicosahedron Small dodecicosahedron In 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 dodecicosacron Small dodecicosacron In geometry, the small dodecicosacron is the dual of the small dodecicosahedron . It has 60 intersecting bow-tie-shaped faces.-External links:*... |
3/23 5| | 10.6.10/9.6/5 |
Ih | U50 | K55 | 60 | 120 | 32 | 20{6}+12{10} | ||
91 | Small dodecahemidodecahedron Small dodecahemidodecahedron In 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 dodecahemidodecacron Small dodecahemidodecacron In 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.... |
5/45|5 | 10.5/4.10.5 |
Ih | U51 | K56 | 30 | 60 | 18 | 12{5}+6{10} | ||
92 | Stellated truncated hexahedron (Quasitruncated hexahedron) |
Great triakis octahedron Great triakis octahedron In geometry, the great triakis octahedron is the dual of the stellated truncated hexahedron . It has 24 intersecting isosceles triangle faces.... |
2 3|4/3 | 8/3.8/3.3 |
Oh | U19 | K24 | 24 | 36 | 14 | 8{3}+6{8/3} | ||
93 | Great truncated cuboctahedron (Quasitruncated cuboctahedron) |
Great disdyakis dodecahedron Great disdyakis dodecahedron In geometry, the great disdyakis dodecahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great truncated cuboctahedron.... |
4/32 3| | 8/3.4.6 |
Oh | U20 | K25 | 48 | 72 | 26 | 12{4}+8{6}+6{8/3} | ||
94 | Great icosidodecahedron | Great rhombic triacontahedron Great rhombic triacontahedron In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron... |
2|5/23 | (5/2.3)2 |
Ih | U54 | K59 | 30 | 60 | 32 | 20{3}+12{5/2} | ||
95 | Truncated great icosahedron | Great stellapentakis dodecahedron Great stellapentakis dodecahedron In 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 :*... |
25/2|3 | 6.6.5/2 |
Ih | U55 | K60 | 60 | 90 | 32 | 12{5/2}+20{6} | ||
96 | Rhombicosahedron Rhombicosahedron In geometry, the rhombicosahedron is a nonconvex uniform polyhedron, indexed as U56. Its vertex figure is an antiparallelogram.- Related polyhedra :... |
Rhombicosacron Rhombicosacron In 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 :*... |
25/23| | 6.4.6/5.4/3 |
Ih | U56 | K61 | 60 | 120 | 50 | 30{4}+20{6} | ||
97 | Small stellated truncated dodecahedron (Quasitruncated small stellated dodecahedron) |
Great pentakis dodecahedron Great pentakis dodecahedron In 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.... |
2 5|5/3 | 10/3.10/3.5 |
Ih | U58 | K63 | 60 | 90 | 24 | 12{5}+12{10/3} | ||
98 | Truncated dodecadodecahedron (Quasitruncated dodecahedron) |
Medial disdyakis triacontahedron Medial disdyakis triacontahedron In geometry, the medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron.... |
5/32 5| | 10/3.4.10 |
Ih | U59 | K64 | 120 | 180 | 54 | 30{4}+12{10}+12{10/3} | ||
99 | Great dodecicosidodecahedron Great dodecicosidodecahedron In 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 dodecacronic hexecontahedron Great dodecacronic hexecontahedron In geometry, the great dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great dodecicosidodecahedron.... |
5/23|5/3 | 10/3.5/2.10/3.3 |
Ih | U61 | K66 | 60 | 120 | 44 | 20{3}+12{5/2}+12{10/3 } | ||
100 | Small dodecahemicosahedron Small dodecahemicosahedron In 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 dodecahemicosacron Small dodecahemicosacron In 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.... |
5/35/2|3 | 6.5/3.6.5/2 |
Ih | U62 | K67 | 30 | 60 | 22 | 12{5/2}+10{6} | ||
101 | Great dodecicosahedron Great dodecicosahedron In 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 dodecicosacron Great dodecicosacron In geometry, the great dodecicosacron is the dual of the great dodecicosahedron . It has 60 intersecting bow-tie-shaped faces.- External links :*... |
5/35/23| | 6.10/3.6/5.10/7 |
Ih | U63 | K68 | 60 | 120 | 32 | 20{6}+12{10/3} | ||
102 | Great dodecahemicosahedron Great dodecahemicosahedron In 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 dodecahemicosacron Great dodecahemicosacron In 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.... |
5/45|3 | 6.5/4.6.5 |
Ih | U65 | K70 | 30 | 60 | 22 | 12{5}+10{6} | ||
103 | Great rhombihexahedron Great rhombihexahedron In 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 rhombihexacron Great rhombihexacron In 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.... |
4/33/22| | 4.8/3.4/3.8/5 |
Oh | U21 | K26 | 24 | 48 | 18 | 12{4}+6{8/3} | ||
104 | Great stellated truncated dodecahedron (Quasitruncated great stellated dodecahedron) |
Great triakis icosahedron Great triakis icosahedron In geometry, the great triakis icosahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great stellated truncated dodecahedron.... |
2 3|5/3 | 10/3.10/3.3 |
Ih | U66 | K71 | 60 | 90 | 32 | 20{3}+12{10/3} | ||
105 | Nonconvex great rhombicosidodecahedron (Quasirhombicosidodecahedron) |
Great deltoidal hexecontahedron Great deltoidal hexecontahedron In 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.... |
5/33|2 | 4.5/3.4.3 |
Ih | U67 | K72 | 60 | 120 | 62 | 20{3}+30{4}+12{5/2} | ||
106 | Great icosihemidodecahedron Great icosihemidodecahedron In 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 icosihemidodecacron Great icosihemidodecacron In 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.... |
3 3|5/3 | 10/3.3/2.10/3.3 |
Ih | U71 | K76 | 30 | 60 | 26 | 20{3}+6{10/3} | ||
107 | Great dodecahemidodecahedron Great dodecahemidodecahedron In geometry, the great dodecahemidodecahedron is a nonconvex uniform polyhedron, indexed as U70. Its vertex figure is a crossed quadrilateral.... |
Great dodecahemidodecacron Great dodecahemidodecacron In geometry, the great dodecahemidodecacron is the dual of the great dodecahemidodecahedron, and is one of nine dual hemipolyhedra. It is appears indistinct from the great icosihemidodecacron.... |
5/35/2|5/3 | 10/3.5/3.10/3.5/2 |
Ih | U70 | K75 | 30 | 60 | 18 | 12{5/2}+6{10/3} | ||
108 | Great truncated icosidodecahedron (Great quasitruncated icosidodecahedron) |
Great disdyakis triacontahedron Great disdyakis triacontahedron In geometry, the great disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the great truncated icosidodecahedron.... |
5/32 3| | 10/3.4.6 |
Ih | U68 | K73 | 120 | 180 | 62 | 30{4}+20{6}+12{10/3} | ||
109 | Great rhombidodecahedron Great rhombidodecahedron In geometry, the great rhombidodecahedron is a nonconvex uniform polyhedron, indexed as U73. Its vertex figure is a crossed quadrilateral.- Related polyhedra :... |
Great rhombidodecacron Great rhombidodecacron In geometry, the great rhombidodecacron is a nonconvex isohedral polyhedron. It is the dual of the great rhombidodecahedron. Its faces are antiparallelograms.... |
3/25/32| | 4.10/3.4/3.10/7 |
Ih | U73 | K78 | 60 | 120 | 42 | 30{4}+12{10/3} | ||
110 | Small snub icosicosidodecahedron Small snub icosicosidodecahedron In 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 hexagonal hexecontahedron Small hexagonal hexecontahedron In 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.... |
|5/23 3 | 3.3.3.3.3.5/2 |
Ih | U32 | K37 | 60 | 180 | 112 | (40+60){3}+12{5/2} | ||
111 | Snub dodecadodecahedron | Medial pentagonal hexecontahedron Medial pentagonal hexecontahedron In 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 :*... |
|25/25 | 3.3.5/2.3.5 |
I | U40 | K45 | 60 | 150 | 84 | 60{3}+12{5}+12{5/2} | ||
112 | Snub icosidodecadodecahedron Snub icosidodecadodecahedron In 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... |
Medial hexagonal hexecontahedron Medial hexagonal hexecontahedron In geometry, the small ditrigonal dodecacronic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform snub icosidodecadodecahedron.... |
|5/33 5 | 3.3.3.3.5.5/3 |
I | U46 | K51 | 60 | 180 | 104 | (20+6){3}+12{5}+12{5/2} | ||
113 | Great inverted snub icosidodecahedron | Great inverted pentagonal hexecontahedron Great inverted pentagonal hexecontahedron In 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.... |
|5/32 3 | 3.3.3.3.5/3 |
I | U69 | K74 | 60 | 150 | 92 | (20+60){3}+12{5/2} | ||
114 | Inverted snub dodecadodecahedron | Medial inverted pentagonal hexecontahedron Medial inverted pentagonal hexecontahedron In geometry, the medial inverted pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron.... |
|5/32 5 | 3.5/3.3.3.5 |
I | U60 | K65 | 60 | 150 | 84 | 60{3}+12{5}+12{5/2} | ||
115 | Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron In 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 hexagonal hexecontahedron Great hexagonal hexecontahedron In geometry, the great hexagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform great snub dodecicosidodecahedron.... |
|5/35/23 | 3.5/3.3.5/2.3.3 |
I | U64 | K69 | 60 | 180 | 104 | (20+60){3}+(12+12){5/2} | ||
116 | Great snub icosidodecahedron | Great pentagonal hexecontahedron Great pentagonal hexecontahedron In 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 :*... |
|25/25/2 | 3.3.3.3.5/2 |
I | U57 | K62 | 60 | 150 | 92 | (20+60){3}+12{5/2} | ||
117 | Great retrosnub icosidodecahedron | Great pentagrammic hexecontahedron Great pentagrammic hexecontahedron In geometry, the great pentagrammic hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron.... |
|3/25/32 | (3.3.3.3.5/3)/2 |
I | U74 | K79 | 60 | 150 | 92 | (20+60){3}+12{5/2} | ||
118 | Small retrosnub icosicosidodecahedron Small retrosnub icosicosidodecahedron In 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 hexagrammic hexecontahedron Small hexagrammic hexecontahedron In 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.... |
|3/23/25/2 | (3.3.3.3.3.5/2)/2 |
Ih | U72 | K77 | 180 | 60 | 112 | (40+60){3}+12{5/2} | ||
119 | Great dirhombicosidodecahedron Great dirhombicosidodecahedron In 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 dirhombicosidodecacron Great dirhombicosidodecacron In 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... |
|3/25/335/2 | (4.5/3.4.3.4.5/2.4.3/2)/2 |
Ih | U75 | K80 | 60 | 240 | 124 | 40{3}+60{4}+24{5/2} |
External links
- Magnus J. Wenninger
- Software used to generate images in this article:
- Stella: Polyhedron Navigator Stella (software)Stella (software)Stella, a computer program available in three versions was created by Robert Webb of Australia...
- Can create and print nets for all of Wenninger's polyhedron models. - Vladimir Bulatov's Polyhedra Stellations Applet
- Stella: Polyhedron Navigator Stella (software)
- M. Wenninger, Polyhedron Models, Errata: known errors in the various editions.