Prismatic uniform polychoron
Encyclopedia
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 prism
s and antiprism
uniform polyhedra, but add a third category called duoprism
s, constructed as a product of two regular polygons.
The prismatic uniform polychora consist of two infinite families:
. The cells of such a polychoron are two identical uniform polyhedra lying in parallel hyperplane
s (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra
(of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).
There are 18 convex polyhedral prisms created from 5 Platonic solid
s and 13 Archimedean solid
s as well as for the infinite families of three-dimensional prism
s and antiprism
s. The symmetry number of a polyhedral prism is twice that of the base polyhedron.
The second is the infinite family of uniform duoprisms
, products of two regular polygon
s.
Their Coxeter-Dynkin diagram
is of the form
This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.
The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:
There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprism
s.
Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:
The infinite sets of uniform antiprismatic prism
s are constructed from two parallel uniform antiprism
s: (p≥3) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.
A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, a prismatic uniform polytope is a 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....
with a nonconnected Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
symmetry group. These figures are analogous to the set of prism
Prism
-Science and mathematics:* Prism , a transparent object which refracts light** Dispersive prism, the most familiar type of optical prism* Prism , a kind of polyhedron* Prism , a type of sedimentary deposit-Books, comics and magazines:...
s and 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...
uniform polyhedra, but add a third category called duoprism
Duoprism
In 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...
s, constructed as a product of two regular polygons.
The prismatic uniform polychora consist of two infinite families:
- Polyhedral prisms: products of a line segment and a uniform polyhedron. This family is infinite because it includes prisms built on 3-dimensional prisms and 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...
s. - 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...
s: product of two regular polygons.
Convex polyhedral prisms
The most obvious family of prismatic polychora is the polyhedral prisms, i.e. products of a polyhedron with a line segmentLine segment
In 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...
. The cells of such a polychoron are two identical uniform polyhedra lying in parallel hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
(of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract).
There are 18 convex polyhedral prisms created from 5 Platonic solid
Platonic solid
In 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...
s and 13 Archimedean solid
Archimedean 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...
s as well as for the infinite families of three-dimensional prism
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...
s and 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...
s. The symmetry number of a polyhedral prism is twice that of the base polyhedron.
Tetrahedral prisms: A3 × A1
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... and Schläfli symbols |
Cells by type | Element counts | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| Cells | Faces | Edges | Vertices | |||||||
| 48 | Tetrahedral prism (tepe) |  |
t0{3,3}×{} |
2  3.3.3 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... |
4  3.4.4 Triangular prism In 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.... |
6 | 8 {3} 6 {4} |
16 | 8 | |
| 49 | Truncated tetrahedral prism Truncated tetrahedral prism In 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... (tuttip) |
 |
t0,1{3,3}×{} |
2  3.6.6 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 :... |
4 3.4.4 Triangular prism In 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.... |
4  4.4.6 Hexagonal prism In 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... |
10 | 8 {3} 18 {4} 8 {6} |
48 | 24 |
| [51] | Rectified tetrahedral prism (Same as octahedral prism Octahedral prism In geometry, a octahedral prism is a convex uniform polychoron . This polychoron has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.- Related polytopes :... ) (ope) |
 |
t1{3,3}×{} |
2  3.3.3.3 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.... |
4 3.4.4 Triangular prism In 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.... |
6 | 16 {3} 12 {4} |
30 | 12 | |
| [50] | Cantellated tetrahedral prism (Same as cuboctahedral prism Cuboctahedral prism In 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.... ) (cope) |
 |
t0,2{3,3}×{} |
2  3.4.3.4 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,... |
8 3.4.4 Triangular prism In 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.... |
6  4.4.4 Cube In 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... |
16 | 16 {3} 36 {4} |
60 | 24 |
| [54] | Cantitruncated tetrahedral prism (Same as truncated octahedral prism Truncated octahedral prism In 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... ) (tope) |
 |
t0,1,2{3,3}×{} |
2  4.6.6 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.... |
8 3.4.4 Hexagonal prism In 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... |
6 4.4.4 Cube In 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... |
16 | 48 {4} 16 {6} |
96 | 48 |
| [59] | Snub tetrahedral prism (Same as icosahedral prism Icosahedral prism In geometry, an icosahedral prism is a convex uniform polychoron . This polychoron has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles... ) (ipe) |
 |
s{3,3}×{} |
2  3.3.3.3.3 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.... |
20 3.4.4 Triangular prism In 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.... |
22 | 40 {3} 30 {4} |
72 | 24 |
Octahedral prisms: BC3 × A1
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... and Schläfli symbols |
Cells by type | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Cells | Faces | Edges | Vertices | ||||||||
| [10] | Cubic prism (Same as tesseract Tesseract In 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... ) (Same as 4-4 duoprism) (tes) |
 |
t0{4,3}×{} |
2 4.4.4 Cube In 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... |
6 4.4.4 Cube In 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... |
8 | 24 {4} | 32 | 16 | ||
| 50 | Cuboctahedral prism Cuboctahedral prism In 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.... (Same as cantellated tetrahedral prism) (cope) |
|
t1{4,3}×{} |
2 3.4.3.4 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,... |
8 3.4.4 Triangular prism In 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.... |
6 4.4.4 Cube In 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... |
16 | 16 {3} 36 {4} |
60 | 24 | |
| 51 | Octahedral prism Octahedral prism In geometry, a octahedral prism is a convex uniform polychoron . This polychoron has 10 polyhedral cells: 2 octahedra connected by 8 triangular prisms.- Related polytopes :... (Same as rectified tetrahedral prism) (Same as triangular antiprismatic prism) (ope) |
|
t2{4,3}×{} |
2 3.3.3.3 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.... |
8 3.4.4 Triangular prism In 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.... |
10 | 16 {3} 12 {4} |
30 | 12 | ||
| 52 | Rhombicuboctahedral prism Rhombicuboctahedral prism In 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... (sircope) |
 |
t0,2{4,3}×{} |
2  3.4.4.4 Rhombicuboctahedron In 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... |
8 3.4.4 Triangular prism In 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.... |
18 4.4.4 Cube In 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... |
28 | 16 {3} 84 {4} |
120 | 96 | |
| 53 | Truncated cubic prism Truncated cubic prism In 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... (ticcup) |
 |
t0,1{4,3}×{} |
2  3.8.8 Truncated cube In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces , 36 edges, and 24 vertices.... |
8 3.4.4 Triangular prism In 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.... |
6  4.4.8 Octagonal prism In 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 :... |
16 | 16 {3} 36 {4} 12 {8} |
96 | 48 | |
| 54 | Truncated octahedral prism Truncated octahedral prism In 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... (Same as cantitruncated tetrahedral prism) (tope) |
|
t1,2{4,3}×{} |
2 4.6.6 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.... |
6 4.4.4 Cube In 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... |
8 4.4.6 Hexagonal prism In 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... |
16 | 48 {4} 16 {6} |
96 | 48 | |
| 55 | Truncated cuboctahedral prism Truncated cuboctahedral prism In 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... (gircope) |
 |
t0,1,2{4,3}×{} |
2  4.6.8 Truncated cuboctahedron In 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... |
12 4.4.4 Cube In 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... |
8 4.4.6 Hexagonal prism In 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... |
6 4.4.8 Octagonal prism In 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 :... |
28 | 96 {4} 16 {6} 12 {8} |
192 | 96 |
| 56 | Snub cubic prism Snub cubic prism In 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... (sniccup) |
 |
s{4,3}×{} |
2  3.3.3.3.4 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... |
32 3.4.4 Triangular prism In 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.... |
6 4.4.4 Cube In 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... |
40 | 64 {3} 72 {4} |
144 | 48 |
Icosahedral prisms: H3 × A1
| # | Johnson Name (Bowers style acronym) | Picture | Coxeter-Dynkin Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... and Schläfli symbols |
Cells by type | Element counts | ||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Cells | Faces | Edges | Vertices | ||||||||
| 57 | Dodecahedral prism (dope) |  |
t0{5,3}×{} |
2  5.5.5 |
12  4.4.5 Pentagonal prism In 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 :... |
14 | 30 {4} 24 {5} |
80 | 40 | ||
| 58 | Icosidodecahedral prism Icosidodecahedral prism In 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.... (iddip) |
 |
t1{5,3}×{} |
2  3.5.3.5 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... |
20 3.4.4 Triangular prism In 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.... |
12 4.4.5 Pentagonal prism In 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 :... |
34 | 40 {3} 60 {4} 24 {5} |
150 | 60 | |
| 59 | Icosahedral prism Icosahedral prism In geometry, an icosahedral prism is a convex uniform polychoron . This polychoron has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles... (same as snub tetrahedral prism) (ipe) |
|
t2{5,3}×{} |
2 3.3.3.3.3 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.... |
20 3.4.4 Triangular prism In 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.... |
22 | 40 {3} 30 {4} |
72 | 24 | ||
| 60 | Truncated dodecahedral prism Truncated dodecahedral prism In 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... (tiddip) |
 |
t0,1{5,3}×{} |
2  3.10.10 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 :... |
20 3.4.4 Triangular prism In 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.... |
12  4.4.5 Decagonal prism In 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.... |
34 | 40 {3} 90 {4} 24 {10} |
240 | 120 | |
| 61 | Rhombicosidodecahedral prism Rhombicosidodecahedral prism In 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... (sriddip) |
 |
t0,2{5,3}×{} |
2  3.4.5.4 Rhombicosidodecahedron In 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.... |
20 3.4.4 Triangular prism In 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.... |
30 4.4.4 Cube In 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... |
12 4.4.5 Pentagonal prism In 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 :... |
64 | 40 {3} 180 {4} 24 {5} |
300 | 120 |
| 62 | Truncated icosahedral prism Truncated icosahedral prism In 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... (tipe) |
 |
t1,2{5,3}×{} |
2  5.6.6 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.... |
12 4.4.5 Pentagonal prism In 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 :... |
20 4.4.6 Hexagonal prism In 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... |
34 | 90 {4} 24 {5} 40 {6} |
240 | 120 | |
| 63 | Truncated icosidodecahedral prism Truncated icosidodecahedral prism In 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... (griddip) |
 |
t0,1,2{5,3}×{} |
2  4.6.4.10 Truncated icosidodecahedron In 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.... |
30 4.4.4 Cube In 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... |
20 4.4.6 Hexagonal prism In 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... |
12 4.4.10 Decagonal prism In 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.... |
64 | 240 {4} 40 {6} 24 {5} |
480 | 240 |
| 64 | Snub dodecahedral prism Snub dodecahedral prism In 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... (sniddip) |
 |
s{5,3}×{} |
2  3.3.3.3.5 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.... |
80 3.4.4 Triangular prism In 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.... |
12 4.4.5 Pentagonal prism In 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 :... |
94 | 240 {4} 40 {6} 24 {10} |
360 | 120 |
Duoprisms: [p] × [q]
Duoprism
In 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...
, products of two regular polygon
Regular polygon
A regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
s.
Their Coxeter-Dynkin diagram
Coxeter-Dynkin diagram
In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
is of the form
This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.
The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:
- Cells: p q-gonal prisms, q p-gonal prisms
- Faces: pq squares, p q-gons, q p-gons
- Edges: 2pq
- Vertices: pq
There is no uniform analogue in four dimensions to the infinite family of three-dimensional 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...
s.
Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms:
- 3-3 duoprism - - 6 triangular prisms
- 3-4 duoprism - - 3 cubes, 4 triangular prisms
- 4-4 duoprism - - 8 cubes (same as tesseract)
- 3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms
- 4-5 duoprism - - 4 pentagonal prisms, 5 cubes
- 5-5 duoprism - - 10 pentagonal prisms
- 3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms
- 4-6 duoprism - - 4 hexagonal prisms, 6 cubes
- 5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms
- 6-6 duoprism - - 12 hexagonal prisms
- ...
Polygonal prismatic prisms: [p] × [ ] × [ ]
The infinte set of uniform prismatic prisms overlaps with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism)- Triangular prismatic prism - - 3 cubes and 4 triangular prisms - (same as 3-4 duoprism)
- Square prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract)
- Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
- Hexagonal prismatic prism - - 6 cubes and 4 hexagonal prisms - (same as 4-6 duoprism)
- Heptagonal prismatic prism - - 7 cubes and 4 heptagonal prisms - (same as 4-7 duoprism)
- Octagonal prismatic prism - - 8 cubes and 4 octagonal prisms - (same as 4-8 duoprism)
- ...
The infinite sets of 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....
s are constructed from two parallel uniform 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...
s: (p≥3) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms.
- Triangular antiprismatic prism - - 2 octahedras connected by 8 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....
s (same as the octahedral 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 :...
- - 2 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...
s connected by 2 cubes and 8 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....
s
- Pentagonal antiprismatic prism - - 2 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...
s connected by 2 pentagonal prisms and 10 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....
s
- Hexagonal antiprismatic prism - - 2 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...
s connected by 2 hexagonal prisms and 12 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....
s - Heptagonal antiprismatic prism - - 2 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...
s connected by 2 heptagonal prisms and 14 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....
s - Octagonal antiprismatic prism - - 2 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...
s connected by 2 octagonal prisms and 16 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....
s - ...
A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices.