# Uniform antiprismatic prism

(Redirected from Octagonal antiprismatic prism)
 Set of uniform antiprismatic prisms Type Prismatic uniform 4-polytope Schläfli symbol s{2,p}×{} Coxeter diagram Cells 2 p-gonal antiprisms, 2 p-gonal prisms and 2p triangular prisms Faces 4p {3}, 4p {4} and 4 {p} Edges 10p Vertices 4p Vertex figure Trapezoidal pyramid Symmetry group [2p,2+,2], order 8p [(p,2)+,2], order 4p Properties convex if the base is convex

In 4-dimensional geometry, a uniform antiprismatic prism or antiduoprism is a uniform 4-polytope with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. The symmetry of a p-gonal antiprismatic prism is [2p,2+,2], order 8p.

A p-gonal antiprismatic prism or p-gonal antiduoprism has 4p equilateral triangle, 4p square and 4 regular p-gon faces. It has 10p edges, and 4p vertices.

 Schlegel diagram Net

## Convex uniform antiprismatic prisms

There is an infinite series of convex uniform antiprismatic prisms, starting with the digonal antiprismatic prism is a tetrahedral prism, with two of the tetrahedral cells degenerated into squares. The triangular antiprismatic prism is the first nondegenerate form, which is also an octahedral prism. The remainder are unique uniform 4-polytopes.

Convex p-gonal antiprismatic prisms
Name s{2,2}×{} s{2,3}×{} s{2,4}×{} s{2,5}×{} s{2,6}×{} s{2,7}×{} s{2,8}×{} s{2,p}×{}
Coxeter
diagram

Image
Vertex
figure
Cells 2 s{2,2}
(2) {2}×{}={4}
4 {3}×{}
2 s{2,3}
2 {3}×{}
6 {3}×{}
2 s{2,4}
2 {4}×{}
8 {3}×{}
2 s{2,5}
2 {5}×{}
10 {3}×{}
2 s{2,6}
2 {6}×{}
12 {3}×{}
2 s{2,7}
2 {7}×{}
14 {3}×{}
2 s{2,8}
2 {8}×{}
16 {3}×{}
2 s{2,p}
2 {p}×{}
2p {3}×{}
Net

## Star antiprismatic prisms

There are also star forms following the set of star antiprisms, starting with the pentagram {5/2}:

Name Coxeter
diagram
Cells Image Net
Pentagrammic antiprismatic prism
5/2 antiduoprism

2 pentagrammic antiprisms
2 pentagrammic prisms
10 triangular prisms
Pentagrammic crossed antiprismatic prism
5/3 antiduoprism

2 pentagrammic crossed antiprisms
2 pentagrammic prisms
10 triangular prisms
...

## Square antiprismatic prism

Square antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,4}x{}
Coxeter-Dynkin
Cells 2 (3.3.3.4)
8 (3.4.4)
2 4.4.4
Faces 16 {3}, 20 {4}
Edges 40
Vertices 16
Vertex figure
Trapezoidal pyramid
Symmetry group [(4,2)+,2], order 16
[8,2+,2], order 32
Properties convex

A square antiprismatic prism or square antiduoprism is a convex uniform 4-polytope It is formed as two parallel square antiprisms connected by cubes and triangular prisms. The symmetry of a square antiprismatic prism is [8,2+,2], order 32. It has 16 triangle, 16 square and 4 square faces. It has 40 edges, and 16 vertices.

 Schlegel diagram Net

## Pentagonal antiprismatic prism

Pentagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,5}x{}
Coxeter-Dynkin
Cells 2 (3.3.3.5)
10 (3.4.4)
2 (4.4.5)
Faces 20 {3}, 20 {4}, 4 {5}
Edges 50
Vertices 20
Vertex figure
Trapezoidal pyramid
Symmetry group [(5,2)+,2], order 20
[10,2+,2], order 40
Properties convex

A pentagonal antiprismatic prism or pentagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel pentagonal antiprisms connected by cubes and triangular prisms. The symmetry of a pentagonal antiprismatic prism is [10,2+,2], order 40. It has 20 triangle, 20 square and 4 pentagonal faces. It has 50 edges, and 20 vertices.

 Schlegel diagram Net

## Hexagonal antiprismatic prism

Hexagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,6}x{}
Coxeter-Dynkin
Cells 2 (3.3.3.6)
12 (3.4.4)
2 (4.4.6)
Faces 24 {3}, 24 {4}, 4 {6}
Edges 60
Vertices 24
Vertex figure
Trapezoidal pyramid
Symmetry group [(2,6)+,2], order 24
[12,2+,2], order 48
Properties convex

A hexagonal antiprismatic prism or hexagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel hexagonal antiprisms connected by cubes and triangular prisms. The symmetry of a hexagonal antiprismatic prism is [12,2+,2], order 48. It has 24 triangle, 24 square and 4 hexagon faces. It has 60 edges, and 24 vertices.

 Schlegel diagram Net

## Heptagonal antiprismatic prism

Heptagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,7}×{}
Coxeter-Dynkin
Cells 2 (3.3.3.7)
14 (3.4.4)
2 (4.4.7)
Faces 28 {3}, 28 {4}, 4 {7}
Edges 70
Vertices 28
Vertex figure
Trapezoidal pyramid
Symmetry group [(7,2)+,2], order 28
[14,2+,2], order 56
Properties convex

A heptagonal antiprismatic prism or heptagonal antiduoprism is a convex uniform 4-polytope. It is formed as two parallel heptagonal antiprisms connected by cubes and triangular prisms. The symmetry of a heptagonal antiprismatic prism is [14,2+,2], order 56. It has 28 triangle, 28 square and 4 heptagonal faces. It has 70 edges, and 28 vertices.

 Schlegel diagram Net

## Octagonal antiprismatic prism

Octagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,8}×{}
Coxeter-Dynkin
Cells 2 (3.3.3.8)
16 (3.4.4)
2 (4.4.8)
Faces 32 {3}, 32 {4}, 4 {8}
Edges 80
Vertices 32
Vertex figure
Trapezoidal pyramid
Symmetry group [(8,2)+,2], order 32
[16,2+,2], order 64
Properties convex

A octagonal antiprismatic prism or octagonal antiduoprism is a convex uniform 4-polytope (four-dimensional polytope). It is formed as two parallel octagonal antiprisms connected by cubes and triangular prisms. The symmetry of an octagonal antiprismatic prism is [16,2+,2], order 64. It has 32 triangle, 32 square and 4 octagonal faces. It has 80 edges, and 32 vertices.

 Schlegel diagram Net