Uniform antiprismatic prism

From Wikipedia, the free encyclopedia
  (Redirected from Octagonal antiprismatic prism)
Jump to: navigation, search
Set of uniform antiprismatic prisms
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,p}×{}
Coxeter diagram CDel node.pngCDel 2x.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
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 Uniform antiprismatic prism verf.png
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.

Example 15-gonal antiprismatic prism
15-gonal antiprismatic prism.png
Schlegel diagram
15-gonal antiprismatic prism verf.png
Net

Convex uniform antiprismatic prisms[edit]

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
CDel node.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 10.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 12.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 14.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 7.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 16.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 2x.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node h.pngCDel p.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Image Digonal antiprismatic prism.png Triangular antiprismatic prism.png Square antiprismatic prism.png Pentagonal antiprismatic prism.png Hexagonal antiprismatic prism.png Heptagonal antiprismatic prism.png Octagonal antiprismatic prism.png 15-gonal antiprismatic prism.png
Vertex
figure
Tetrahedral prism verf.png Tetratetrahedral prism verf.png Square antiprismatic prism verf2.png Pentagonal antiprismatic prism verf.png Hexagonal antiprismatic prism verf.png Heptagonal antiprismatic prism verf.png Octagonal antiprismatic prism verf.png Uniform antiprismatic prism verf.png
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 Tetrahedron prism net.png Octahedron prism net.png 4-antiprismatic prism net.png 5-antiprismatic prism net.png 6-antiprismatic prism net.png 7-antiprismatic prism net.png 8-antiprismatic prism net.png 15-gonal antiprismatic prism verf.png

Star antiprismatic prisms[edit]

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
CDel node h.pngCDel 5.pngCDel rat.pngCDel d2.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
2 pentagrammic antiprisms
2 pentagrammic prisms
10 triangular prisms
Pentagrammic antiprismatic prism.png Pentagrammic antiprismatic prism net.png
Pentagrammic crossed antiprismatic prism
5/3 antiduoprism
CDel node h.pngCDel 5.pngCDel rat.pngCDel d3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 10.pngCDel rat.pngCDel d3.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
2 pentagrammic crossed antiprisms
2 pentagrammic prisms
10 triangular prisms
Crossed pentagrammic antiprismatic prism.png Crossed pentagrammic antiprismatic prism net.png
...

Square antiprismatic prism[edit]

Square antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,4}x{}
Coxeter-Dynkin CDel node h.pngCDel 4.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.4)Square antiprism.png
8 (3.4.4)Triangular prism.png
2 4.4.4 Tetragonal prism.png
Faces 16 {3}, 20 {4}
Edges 40
Vertices 16
Vertex figure Square antiprismatic prism verf2.png
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.

Square antiprismatic prism
Square antiprismatic prism.png
Schlegel diagram
Square antiprismatic prism net.png
Net

Pentagonal antiprismatic prism[edit]

Pentagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,5}x{}
Coxeter-Dynkin CDel node h.pngCDel 5.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 10.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.5) Pentagonal antiprism.png
10 (3.4.4) Triangular prism.png
2 (4.4.5) Pentagonal prism.png
Faces 20 {3}, 20 {4}, 4 {5}
Edges 50
Vertices 20
Vertex figure Pentagonal antiprismatic prism verf.png
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.

Pentagonal antiprismatic prism
Pentagonal antiprismatic prism.png
Schlegel diagram
Pentagonal antiprismatic prism net.png
Net

Hexagonal antiprismatic prism[edit]

Hexagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,6}x{}
Coxeter-Dynkin CDel node h.pngCDel 6.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 12.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.6) Hexagonal antiprism.png
12 (3.4.4) Triangular prism.png
2 (4.4.6) Hexagonal prism.png
Faces 24 {3}, 24 {4}, 4 {6}
Edges 60
Vertices 24
Vertex figure Hexagonal antiprismatic prism verf.png
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.

Hexagonal antiprismatic prism
Hexagonal antiprismatic prism.png
Schlegel diagram
Hexagonal antiprismatic prism net.png
Net

Heptagonal antiprismatic prism[edit]

Heptagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,7}×{}
Coxeter-Dynkin CDel node h.pngCDel 7.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 14.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.7) Antiprism 7.png
14 (3.4.4) Triangular prism.png
2 (4.4.7) Prism 7.png
Faces 28 {3}, 28 {4}, 4 {7}
Edges 70
Vertices 28
Vertex figure Heptagonal antiprismatic prism verf.png
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.

Heptagonal antiprismatic prism
Heptagonal antiprismatic prism.png
Schlegel diagram
Heptagonal antiprismatic prism net.png
Net

Octagonal antiprismatic prism[edit]

Octagonal antiprismatic prism
Type Prismatic uniform 4-polytope
Schläfli symbol s{2,8}×{}
Coxeter-Dynkin CDel node h.pngCDel 8.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
CDel node.pngCDel 16.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2.pngCDel node 1.png
Cells 2 (3.3.3.8) Octagonal antiprism.png
16 (3.4.4) Triangular prism.png
2 (4.4.8) Octagonal prism.png
Faces 32 {3}, 32 {4}, 4 {8}
Edges 80
Vertices 32
Vertex figure Octagonal antiprismatic prism verf.png
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.

Octagonal antiprismatic prism
Octagonal antiprismatic prism.png
Schlegel diagram
Octagonal antiprismatic prism net.png
Net

See also[edit]

References[edit]

External links[edit]