recursal/Fanatic-Fandom huggingface dataset:
Povray code-tet-compound-reg.inc

```
/\* This program is free software: you can redistribute it and/or modify it
 \* under the terms of the GNU General Public License as published by the Free
 \* Software Foundation, either version 3 of the License, or (at your option)
 \* any later version.
 \*
 \* This program is distributed in the hope that it will be useful, but WITHOUT
 \* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
 \* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
 \* more details.
 \*
 \* You should have received a copy of the GNU General Public License along with
 \* this program. If not, see <http://www.gnu.org/licenses/>
 \*/

/\* Tetrahedron compound regiment \*/

#include "ike-sym.inc"

// Vertices
#declare NumVertices = NumDoeVertices;
#declare Vertices = array[NumVertices];
#local Phi3 = Phi \* Phi \* Phi;
#local Scale = 1 / sqrt(Phi3 \* sqrt(5));
#for (V, 0, NumVertices - 1, 1)
 #declare Vertices[V] = DoeVertices \* Scale;
#end

#declare Circumradius = Phi \* sqrt(3) \* Scale;

// Edges
#declare HalfNumEdges = 5 \* NumDoeFullFaces;
#declare NumEdges = 2 \* HalfNumEdges;
#declare Edges = array[NumEdges][2];

#for (F, 0, NumDoeFullFaces - 1, 1)
 #for (V, 0, 4, 1)
 #local To = 5 \* F + V;
 #declare Edges[To][0] = DoeFullFaces[F][V];
 #declare Edges[To + HalfNumEdges][0] = DoeFullFaces[F][V];
 #local Edge = mod(DoeFaceEdges[V][mod(V + 1, 5)], NumDoeEdges);
 #local AdjFace = IkeEdges[Edge][0] + IkeEdges[Edge][1] - F;
 #for (VV, 0, 4, 1)
 #if (mod(DoeFaceEdges[AdjFace][VV], NumDoeEdges) = Edge)
 #declare Edges[To][1] = DoeFullFaces[AdjFace][mod(VV + 3, 5)];
 #end

```
Nonagram

Name
Nonagram
Other name
Enneagram
Schläli symbol

{

9
2

}

{\displaystyle \left\{\frac{9}{2}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and

# ξ
tan
⁡

(

π
9

)

{\displaystyle \xi = \tan\left(\frac{\pi}{9}\right)}

.

Inradius

1
−

ξ
2

2
ξ

l

{\displaystyle \frac{1 - \xi^2}{2 \xi} l}

Circumradius

cosec
⁡

(

2
π

9

)

l

{\displaystyle \operatorname{cosec}\left(\frac{2\pi}{9}\right) l}

Circumference*full*

18
l

{\displaystyle 18l}

Circumference*shell*

9

(

1

ξ
2

)

l

{\displaystyle 9\left(1 + \xi^2\right) l}

Area*full*

9

(

1
−

ξ
2

)

2
ξ

l
2

{\displaystyle \frac{9 \left(1 - \xi^2\right)}{2 \xi} l^2}

Area*flat*

9

(

1
−

ξ
4

)

4
ξ

l
2

{\displaystyle \frac{9\left(1 - \xi^4\right)}{4 \xi} l^2}

Area*odd*

9

(

1
−

ξ
2

)

2

l
2

{\displaystyle \frac{9 \left(1 - \xi^2\right)}{2} l^2}

Internal angle

5
π

9

rad

{\displaystyle \frac{5\pi}{9} \operatorname{rad}}

 = 100°

<= Nonagon | 0D – 2D polytopes | Great nonagram =>
Great quasitruncated icosidodecahedron

Name
Great quasitruncated icosidodecahedron
Other names

Number
61
Polyhedron number
U68
Coxeter number
C87
Wenninger number
W108
Bower’s acronym
gaquatid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
xx\*’x
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
0

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
120
Number of edges
180
Number of faces
30 × {4}
20 × {6}
## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great disdyakis triacontahedron
Image

Composite image

<= Quasitruncated dodecadodecahedron | Omnitruncated polyhedra | cuboctatruncated cuboctahedron =>
Great inverted snub icosidodecahedron

Name
Great inverted snub icosidodecahedron
Other names

Number
70
Polyhedron number
U69
Coxeter number
C73
Wenninger number
W113
Bower’s acronym
gisid
Schläfli symbol

Wythoff symbol

Polyhedron symbol

Conway notation

Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
0

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
150
Number of faces
(20 + 60) × {3}
12 ×

{

5
2

}

{\displaystyle \left\{\frac{5}{2}\right\}}

## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great inverted pentagonal hexecontahedron
Image

Composite image

<= Inverted snub dodecadodecahedron | Snub polyhedra | Snub icosidodecadodecahedron =>
Hexagonal antiprism

## Basic details

Name
Hexagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
hap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Pentagrammic antiprism | Prisms and Antiprisms | Heptagonal antiprism =>
P-gon

Name
p-gon
Schläli symbol

{
p
# }

{

p
n

p
d

}

{\displaystyle \{p\} = \left\{\frac{p\_n}{p\_d}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius, ri

cot
⁡

(

π
p

)

l

{\displaystyle \cot\left(\frac{\pi}{p}\right) l}

Circumradius, rc

cosec
⁡

(

π
p

)

l

{\displaystyle \operatorname{cosec}\left(\frac{\pi}{p}\right) l}

Circumference*full*

2

p
n

l

{\displaystyle 2 p\_n l}

Circumference*shell*

2

p
n

(

1
−
tan
⁡

(

(

p
d

−
1
)
π

p
n

)

cot
⁡

(

π
p

)

)

l

{\displaystyle 2 p\_n \left(1 - \tan\left(\frac{(p\_d-1) \pi}{p\_n}\right) \cot\left(\frac{\pi}{p}\right)\right) l}

Area*full*

p
n

cot
⁡

(

π
p

)

l
2

{\displaystyle p\_n \cot\left(\frac{\pi}{p}\right) l^2}

Area*flat*

p
n

(

cot
⁡

(

π
p

)

−
tan
⁡

(

(

p
d

−
1
)
π

p
n

)

(

cot
⁡

(

π
p

)

)

2

)

l
2

{\displaystyle p\_n \left(\cot\left(\frac{\pi}{p}\right) - \tan\left(\frac{(p\_d-1) \pi}{p\_n}\right) \left(\cot\left(\frac{\pi}{p}\right)\right)^2\right) l^2}

Area*odd*

p
n

(

tan
⁡

(

π
p

)

2

∑

# i
1

p
d

−
1

(
−
1

)

p
d

−
i

tan
⁡

(

i
π

p
n

)

)

(

cot
⁡

(

π
p

)

)

2

l
2

{\displaystyle p\_n \left(\tan\left(\frac{\pi}{p}\right) + 2\sum\_{i=1}^{p\_d - 1}(-1)^{p\_d - i} \tan\left(\frac{i \pi}{p\_n}\right)\right)\left(\cot\left(\frac{\pi}{p}\right)\right)^2 l^2}

Internal angle

(
p
−
2
)
π

p

rad

{\displaystyle \frac{(p - 2)\pi}{p} \operatorname{rad}}

 = 180 (p-2)/p°

<= Dodecagram | 0D – 2D polytopes | Triangular tesselation =>
Great heptagrammic antiprism

## Basic details

Name
Great heptagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
gisthap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Heptagrammic antiprism | Prisms and Antiprisms | Octagonal antiprism =>
Inverted snub dodecadodecahedron

Name
Inverted snub dodecadodecahedron
Other names

Number
69
Polyhedron number
U60
Coxeter number
C76
Wenninger number
W114
Bower’s acronym
isdid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
s\*’s^s
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
4

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
150
Number of faces
60 × {3}
12 × {5}
12 ×

{

5
2

}

{\displaystyle \left\{\frac{5}{2}\right\}}

## Schwarz triangle

22
5/25/2
55

## Dual

Name
Medial inverted pentagonal hexecontahedron
Image

Composite image

<= Small snub icosicosidodecahedron | Snub polyhedra | Great inverted snub icosidodecahedron =>
Small dodecahemidodecahedron

## Basic details

Name
Small dodecahemidodecahedron
Other names

Number
25
Polyhedron number
U51
Coxeter number
C65
Wenninger number
W91
Bower’s acronym
sidhid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(x^o^’x^)/2
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
7
### Army

General
Icosidodecahedron
Colonel
Icosidodecahedron
Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
30
Number of edges
60
Number of faces
12 × {5}
6 × {10}
## Schwarz triangle

5/45/4
55
55

## Dual

Name
Small dodecahemidodecacron
Image

Composite image

<= Icosidodecahedron | Quasiregular polyhedra | Small icosihemidodecahedron =>
Hendecagram

Name
Hendecagram
Schläli symbol

{

11
2

}

{\displaystyle \left\{\frac{11}{2}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and

# ξ
t
a
n

(

π
11

)

{\displaystyle \xi = tan\left(\frac{\pi}{11}\right)}

.

Inradius

1
−

ξ
2

2
ξ

l

{\displaystyle \frac{1 - \xi^2}{2 \xi} l}

Circumradius

cosec
⁡

(

2
π

11

)

{\displaystyle \operatorname{cosec}\left(\frac{2 \pi}{11}\right)}

Circumference*full*

22
l

{\displaystyle 22 l}

Circumference*shell*

11

(

1

ξ
2

)

l

{\displaystyle 11 \left(1 + \xi^2\right) l}

Area*full*

11

(

1
−

ξ
2

)

2
ξ

l
2

{\displaystyle \frac{11 \left(1 - \xi^2\right)}{2 \xi} l^2}

Area*flat*

11

(

1
−

ξ
4

)

4
ξ

l
2

{\displaystyle \frac{11 \left(1 - \xi^4\right)}{4 \xi} l^2}

Area*odd*

11
ξ

(

1
−

ξ
2

)

2

l
2

{\displaystyle \frac{11 \xi \left(1 - \xi^2\right)}{2} l^2}

Internal angle

7
π

11

rad

{\displaystyle \frac{7\pi}{11} \operatorname{rad}}

 = 114 6/11°

<= Hendecagon | 0D – 2D polytopes | Great hendecagram =>
Icositetrachoron

## Basic details

Name
Icositetrachoron
Other names
24-cell
Number
4
Category
1
Bower’s acronym
ico
Conway notation

Polychoron notation

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

### Army

General

Colonel

Sergeant

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius,

r

0

{\displaystyle r\_{0}}

Mid-edge radius,

r
1

{\displaystyle r\_1 }

Face-centre radius,

r
2

{\displaystyle r\_2 }

Inradius,

r
3

{\displaystyle r\_3}

Dihedral angle

Vertex volume

Coordinates

Surface volume

Hypervolume

## Components

Number of vertices

Number of edges

Number of faces

Number of cells

## Cross sections

<= Hexadecachoron | Regular polychora | Hecatonicosachoron =>
Pentagon

Name
Pentagon
Schläli symbol
{5}
Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length and the golden section,

# ϕ

1

5

2

{\displaystyle \phi=\frac{1+\sqrt{5}}{2}}

.

Inradius

ϕ

3
2

×

5

−

1
4

l

{\displaystyle \phi^\frac{3}{2} \times 5^{-\frac{1}{4}} l}

Circumradius

2

ϕ

1
2

×

5

−

1
4

{\displaystyle 2 \phi^\frac{1}{2} \times 5^{-\frac{1}{4}}}

Circumference

10
l

{\displaystyle 10l}

Area

ϕ

3
2

×

5

3
4

{\displaystyle \phi^\frac{3}{2} \times 5^\frac{3}{4}}

Internal angle

3
π

5

rad

{\displaystyle \frac{3\pi}{5} \operatorname{rad}}

 = 1080°

<= Square | 0D – 2D polytopes | Pentagram =>
Great enneagrammic antiprism

## Basic details

Name
Great enneagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
gisteap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Enneagrammic antiprism | Prisms and Antiprisms | Decagonal antiprism =>
Dodecagram

Name
Dodecagram
Other name
Quasitruncated hexagon
Schläli symbols

{

12
5

}

{\displaystyle \left\{\frac{12}{5}\right\}}

t

{

6
5

}

{\displaystyle \left\{\frac{6}{5}\right\}}

Coxeter diagrams

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

(

2
−

3

)

l

{\displaystyle \left(2 - \sqrt{3}\right) l}

Circumradius

2

2
−

3

l

{\displaystyle 2 \sqrt{2 - \sqrt{3}} l}

Circumference*full*

24
l

{\displaystyle 24 l}

Circumference*shell*

48

(

2
−

3

)

l

{\displaystyle 48\left(2 - \sqrt{3}\right) l}

Area*full*

12

(

2
−

3

)

l
2

{\displaystyle 12\left(2 - \sqrt{3}\right) l^2}

Area*flat*

24

(

7
−
4

3

)

l
2

{\displaystyle 24\left(7-4\sqrt{3}\right) l^2}

Area*odd*

4

(

24
−
11

3

)

(

7
−
4

3

)

l
2

{\displaystyle 4 \left(24 - 11 \sqrt{3}\right)\left(7 - 4 \sqrt{3}\right) l^2}

Internal angle

π
6

rad

{\displaystyle \frac{\pi}{6} \operatorname{rad}}

 = 30°

<= Dodecagon | 0D – 2D polytopes | p-gon =>
Quasirhombicosidodecahedron

Name
Quasirhombicosidodecahedron
Other names

Number
55
Polyhedron number
U67
Coxeter number
C84
Wenninger number
W105
Bower’s acronym
qrid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
xo\*’x
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
0

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
120
Number of faces
20 × {3}
30 × {4}
12 × {5}
## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great deltoidal hexecontahedron
Image

Composite image

<= Great dodecicosidodecahedron | Trapezivert polyhedra | Great rhombidodecahedron =>
Great cubicuboctahedron

## Basic details

Name
Great cubicubboctahedron
Other names

Number
45
Polyhedron number
U14
Coxeter number
C50
Wenninger number
W77
Bower’s acronym
gocco
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(o’x’x)
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
3
### Army

General
Truncated cube
Colonel
Great cubicuboctahedron
Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
24
Number of edges
48
Number of faces
8 × {3}
6 × {4}
6 ×

{

8
3

}

{\displaystyle \left\{\frac{8}{3}\right\}}

## Schwarz triangle

3/23/2
44
44

## Dual

Name
Great hexacronic icositetrahedron
Image

Composite image

<= Rhombicosahedron | Trapezivert polyhedra | Quasirhombicuboctahedron =>
Definitions

Army
A collection of polytopes sharing (possibly after scaling and rotating) the same vertices. An army is lead by a *general*.
Base
The polytope, where the components are multiplied by the density.
Cell
A 3-dimensional unit in a polytope of four or more dimensions.
Colonel
The leader of a *regiment*. Its conjugate is a *lieutenant*. If a polytope is both a colonel and a lieutenant, it is a lieutenant colonel.
Components
The vertices, edges, etc. of the polytope.
Dual
The dual of a polytope is a polytope that is inverted in a sphere, causing the vertices to interchange with the *facets*, etc.
Facet
In an n-dimentional polytope, a facet is a (n-1)-unit. For example, a facet of a polyhedron is a polygon, and a facet of a polychoron is a polyhedron.
Flat
Similar to *Base*, but the components are counted singly. Unless there are holes in the polytope, the *flat* volumne will equal the *shell* volume, although the surface areas may differ.
General
The leader of an *army*.
Honeycomb
A uniform packing of n-dimensional polytopes in an n+1 dimensional space. A 2-dimensional honeycomb is called a *tesselation*.
Kernel
The portion of a polytope that is within all *facets*.
Lieutenant
The conjugate of a *colonel*.
Lieutenant colonel
A *colonel* that is self-conjugate. *See lieutenant*.
Major
The conjugate of a *general*. If a general is self-conjugating, it is a *major general*.
Major general
A self-conjugating *general*. *See major*.
Odd
The polytope, when rendered using an odd winding rule.
Regiment
A collection of polytopes sharing (possibly after scaling and rotating) the same vertices and edges. A regiment is lead by a *colonel*.
Shell
The exposed parts of a polytope. The densities are not considered.
Verf
See *Vertex figure*.
Vertex figure
A diagram formed by selecting a vertex and taking the midpoints of the edges connected to it. The edges of the vertex figure represent the faces of the polytope, and are marked with the type of face.
Vertex volume
Consider an n-sphere, of radius *l*, around a vertex. The vertex volume is the volume that is within both the n-sphere, and the polytope.
Octagrammic antiprism

## Basic details

Name
Octagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stoap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Octagonal antiprism | Prisms and Antiprisms | Enneagonal antiprism =>
Great heptagram

Name
Great heptagram
Schläli symbol

{

7
3

}

{\displaystyle \left\{\frac{7}{3}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and

# ξ
tan
⁡

(

π
7

)

{\displaystyle \xi = \tan\left(\frac{\pi}{7}\right)}

.

Inradius

1
−
3

ξ
2

3
ξ
−

ξ
3

l

{\displaystyle \frac{1 - 3 \xi^2}{3 \xi - \xi ^ 3} l}

Circumradius

cosec
⁡

(

3
π

7

)

l

{\displaystyle \operatorname{cosec}\left(\frac{3 \pi}{7}\right) l}

Circumference*full*

14
l

{\displaystyle 14 l}

Circumference*shell*

14

(

1

ξ
2

)

2

(

1
−

ξ
2

)

(

3
−

ξ
2

)

l

{\displaystyle \frac{14 \left(1 + \xi^2\right)^2}{\left(1 - \xi^2\right)\left(3 - \xi^2\right)} l}

Area*full*

7

(

1
−
3

ξ
2

)

ξ

(

3
−

ξ
2

)

l
2

{\displaystyle \frac{7 \left(1 - 3 \xi^2\right)}{\xi \left(3 - \xi^2\right)} l^2}

Area*flat*

7

(

1
−
3

ξ
2

)

(

1

ξ
2

)

2

(

1
−

ξ
2

)

(

3
−

ξ
2

)

2

ξ

l
2

{\displaystyle \frac{7 \left(1 - 3 \xi^2\right)\left(1 + \xi^2\right)^2}{\left(1 - \xi^2\right)\left(3 - \xi^2\right)^2 \xi} l^2}

Area*odd*

7

(

1
7

ξ
4

)

(

1
−
3

ξ
2

)

2

(

1
−

ξ
2

)

(

3
−

ξ
2

)

2

ξ

l
2

{\displaystyle \frac{7\left(1 + 7 \xi^4\right)\left(1 - 3\xi^2\right)^2}{\left(1 - \xi^2\right)\left(3 - \xi^2\right)^2 \xi} l^2}

Internal angle

π
7

rad

{\displaystyle \frac{\pi}{7} \operatorname{rad}}

 = 25 5</sup/7°

<= Heptagram | 0D – 2D polytopes | Octagon =>
Pentagonal antiprism

## Basic details

Name
Pentagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Schwarz triangle

22
22
55

## Dual

Name

Image

Composite image

<= Square antiprism | Prisms and Antiprisms | Pentagrammic antiprism =>
Grand hendecagram

Name
Grand hendecagram
Schläli symbol

{

11
4

}

{\displaystyle \left\{\frac{11}{4}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and

# ξ
tan
⁡

(

π
11

)

{\displaystyle \xi = \tan\left(\frac{\pi}{11}\right)}

.

Inradius

(

1
−
6

ξ
2

ξ
4

)

4
ξ

(

1
−

ξ
2

)

l

{\displaystyle \frac{\left(1 - 6 \xi^2 + \xi^4\right)}{4 \xi\left(1 - \xi^2\right)} l}

Circumradius

cosec
⁡

(

4
π

11

)

l

{\displaystyle \operatorname{cosec}\left(\frac{4\pi}{11}\right) l}

Circumference*full*

22
l

{\displaystyle 22 l}

Circumference*shell*

11

(

1

ξ
2

)

3

2

(

1
−
3

ξ
2

)

(

1
−

ξ
2

)

l

{\displaystyle \frac{11\left(1 + \xi^2\right)^3}{2 \left(1 - 3 \xi^2\right)\left(1 - \xi^2\right)} l}

Area*full*

11

(

1
−
6

ξ
2

ξ
4

)

4
ξ

(

1
−

ξ
2

)

l
2

{\displaystyle \frac{11 \left(1 - 6 \xi^2 + \xi^4\right)}{4 \xi\left(1 - \xi^2\right)} l^2}

Area*flat*

11

(

1

ξ
2

)

3

(

1
−
6

ξ
2

ξ
4

)

16
ξ

(

1
−
3

ξ
2

)

(

1
−

ξ
2

)

2

l
2

{\displaystyle \frac{11\left(1 + \xi^2\right)^3 \left(1 - 6 \xi^2 + \xi^4\right)}{16 \xi \left(1 - 3 \xi^2\right)\left(1 - \xi^2\right)^2} l^2}

Area*odd*

11

(

tan
⁡

(

4
π

11

)

−
2
tan
⁡

(

3
π

11

)

2
tan
⁡

(

2
π

11

)

−
2
ξ

)

l
2

{\displaystyle 11\left(\tan\left(\frac{4 \pi}{11}\right) -2\tan\left(\frac{3\pi}{11}\right) + 2\tan\left(\frac{2\pi}{11}\right) - 2\xi\right) l^2}

Internal angle

3
π

11

rad

{\displaystyle \frac{3\pi}{11} \operatorname{rad}}

 = 49 1/11°

<= Great hendecagram | 0D – 2D polytopes | Great grand hendecagram =>
Great enneagrammic retroprism

## Basic details

Name
Great enneagrammic retroprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
gisterp
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Octagrammic retroprism | Prisms and Antiprisms | p-gonal retroprism =>
Podiumverts

1. ↑ 1.0 1.1 1.2 Rit, firt and hinnit where first encountered as part of the Triangular rectates
2. ↑ 2.0 2.1 2.2 2.3 Sto and gotto appear twice as they can each be orientated in two different ways

## Rectified tesseract regiment

*42. Rectified tesseract (rit)*

*43. Facetorectified tesseract (firt)*

*44. Hexadexintercepted tesseract (hinnit)*

482. Small triakisoctachoron (**sto**)

483. Great triakisoctachoron (**gotto**)

## Great tesseractitesseractihexadecachoron regiment

484. Great tesseractitesseractihexadecachoron (**gittith**)

485. Quasidisprismatotesseractihexadecachoron (**quidpith**)

486. Prismatocubintercepted tesseract (**picnut**)

487. Great tesseractifaceted cubitesseractihexadecachoron (**gittifcoth**)

488. Great hexadecafaceted prismatotesseractioctachoron (**gahfipto**)

489. Great spinoprismatotesseractihexadecachoron (**gnappoth**)

490. Great spinoprismatotesseractioctachoron (**gnipto**)

## Great ditrigonary hecatonicosahexacosahecatonicosachoron regiment

491. Great ditrigonary hecatonicosahexacosahecatonicosachoron (**gidthixhi**)

492. Great dishexacosadishecatonicosachoron (**gadixady**)

493. Hexacosahecatonicosintercepted hexacosachoron (**xhinhi**)

494. Great hexacosafaceted hexacosadishecatonicosachoron (**gix fixady**)

495. Great hecatonicosafaceted hexacosadishecatonicosachoron (**gohi fixady**)

496. Great spinotriakishecatonicosachoron (**gnathi**)

497. Great spinohecatonicosadishexacosachoron (**ganhiddix**)

## Great hexacosahecatonicosahecatonicosachoron regiment

498. Great hexacosahecatonicosahecatonicosachoron (**gixhihy**)

499. Quasidisprismatohexacosahecatonicosachoron (**quidpixhi**)

500. Prismatoprismatohecatonicosachoron (**dippany**)

501. Great hecatonicosafaceted prismatohecatonicosahexacosachoron (**ghif phix**)

502. Great hexacosafaceted prismatodishecatonicosachoron (**gix fipady**)

503. Great spinoprismatohexacosahecatonicosachoron (**ganpixhi**)

504. Great spinoprismatodishecatoncosachoron (**gnappady**)

## Small hecatonicosahecatonicosahexacosachoron regiment

505. Small hecatonicosahecatonicosahexacosachoron (**shihix**)

506. Small hexacosahecatonicosadishecatonicosachoron (**sixhidy**)

507. Hecatonicosahecatonicosintercepted hecatonicosachoron (**hihiny**)

508. Small hecatonicosafaceted hecatonicosahexacosahecatonicosachoron (**shi fohixhi**)

509. Great hexacosafaceted triakishecatonicosachoron (**saxifthi**)

510. Small spinohexacosadishecatonicosachoron (**snaxady**)

511. Small spinohecatonicosadishecatonicosachoron (**sun hidy**)

<= Antipodiumverts | Polytopes Wiki | Squapverts =>
Regular polychora

There are 16 regular polychora, six being convex (the 4D equivalent of the Platonic solids) and the other ten being stellar (the 4D equivalent of the Kepler-Poinsot polyhedrra)

## Convex polychora

1. Pentachoron (**pen**)

2. Tesseract (**tes**)

3. Hexadecachoron (**hex**)

4. Icositetrachoron (**ico**)

5. Hecatonicosachoron (**hi**)

6. Hexacosichoron (**ex**)

## Stellar polychora

7. Faceted hexacosachoron (**fix**)

8. Great hecatonicosachoron (**gohi**)

9. Grand hecatonicosachoron (**gahi**)

10. Small stellated hecatonicosachoron (**sishi**)

11. Great grand hecatonicosachoron (**gaghi**)

12. Great stellated hecatonicosachoron (**gishi**)

13. Grand stellated hecatonicosachoron (**gashi**)

14. Great faceted hexacosachoron (**gofix**)

15. Grand hexacosachoron (**gax**)

16. Great grand stellated hecatonicosachoron (**gogishi**)

## Miscellaneous

The following polychoron, although not regular, has been placed in this category, for want of a better place to put it.

17. Tesseractihemioctachoron (**tho**)

<= | Polytopes Wiki | Truncated polychora =>
Great dodecahemicosahedron

Name
Great dodecahemicosahedron
Other names

Number
29
Polyhedron number
U65
Coxeter number
C81
Wenninger number
W102
Bower’s acronym
gidhei
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(x^o^’x)/2
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
5

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
30
Number of edges
60
Number of faces
12 × {5}
10 × {6}
## Schwarz triangle

5/25/2
33
33

3/23/2
55
55

## Dual

Name
Great dodecahemicosacron
Image

Composite image

<= Small dodecahemicosahedron | Quasiregular polyhedra | Great icosidodecahedron =>
Small ditrigonary dodecicosidodecahedron

Name
Small ditrigonary dodecicosidodecahedron
Other names

Number
52
Polyhedron number
U43
Coxeter number
C55
Wenninger number
W82
Bower’s acronym
sidditdid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(o\*’x^x)
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
9

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
120
Number of faces
20 × {3}
12 × {10}
12 ×

{

5
2

}

{\displaystyle \left\{\frac{5}{2}\right\}}

## Schwarz triangle

5/35/3
33
55

## Dual

Name
Small ditrigonal dodecacronic hexecontahedron
Image

Composite image

<= Small icosicosidodecahedron | Trapezivert polyhedra | Small dodecicosahedron =>
Octagrammic prism

## Basic details

Name
Octagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stop
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Schwarz triangle

22
22
44

## Dual

Name

Image

Composite image

<= Octagonal prism | Prisms and Antiprisms | Enneagonal prism =>
Great heptagrammic prism

## Basic details

Name
Great heptagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
giship
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Heptagrammic prism | Prisms and Antiprisms | Octagonal prism =>
Great rhombicuboctahedron

Name
Great rhombicuboctahedron
Other names

Number
57
Polyhedron number
U57
Coxeter number
C23
Wenninger number
W15
Bower’s acronym
girco
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
xx’x
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags
Semiregular
Archimedean
Convex
Zonohedron
Density

Genus
0

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
48
Number of edges
72
Number of faces
12 × {4}
8 × {6}
6 × {8}
## Schwarz triangle

22
33
44

## Dual

Name
Disdyakis dodecahedron
Image

Composite image

<= Great rhombidodecahedron | Omnitruncated polyhedra | Great rhombicosidodecahedron =>
Quasitruncated cuboctahedron

Name
Quasitruncated cuboctahedron
Other names

Number
59
Polyhedron number
U20
Coxeter number
C67
Wenninger number
W93
Bower’s acronym
quitco
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
xx’x
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
0

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
48
Number of edges
72
Number of faces
12 × {4}
8 × {6}
6 ×

{

8
3

}

{\displaystyle \left\{\frac{8}{3}\right\}}

## Schwarz triangle

22
33
44

## Dual

Name
Great disdyakis dodecahedron
Image

Composite image

<= Great rhombicosidodecahedron | Omnitruncated polyhedra | Quasitruncated dodecadodecahedron =>
Omnitruncated polyhedra

57. Great rhombicuboctahedron (**girco**)

58. Great rhombicosidodecahedron (**grid**)

59. Quasitruncated cuboctahedron (**quitco**)

60. Quasitruncated dodecadodecahedron (**quitdid**)

61. Great quasitruncated icosidodecahedron (**gaquatid**)

62. Cuboctatruncated cuboctahedron (**cotco**)

63. Icosidodecatruncated icosidodecahedron (**idtid**)

<= Trapezovert polyhedra | Polytopes Wiki | Snub polyhedra =>
Great dodecicosahedron

Name
Great dodecicosahedron
Other names

Number
50
Polyhedron number
U63
Coxeter number
C79
Wenninger number
W101
Bower’s acronym
giddy
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
15

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
120
Number of faces
20 × {6}
12 ×

{

10
3

}

{\displaystyle \left\{\frac{10}{3}\right\}}

## Dual

Name
Great dodecicosacron
Image

Composite image

<= Great icosicosidodecahedron | Trapezivert polyhedra | Small icosicosidodecahedron =>
Octagon

Name
Octagon
Other name
Truncated square
Schläli symbols
{8}
t{4}
Coxeter diagrams

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

(

1

2

)

l

{\displaystyle \left(1 + \sqrt{2}\right) l}

Circumradius

2

(

2

2

)

l

{\displaystyle \sqrt{2\left(2+\sqrt{2}\right)} l}

Circumference

16
l

{\displaystyle 16 l}

Area

8

(

1

2

)

l
2

{\displaystyle 8 \left(1 + \sqrt{2}\right) l^2}

Internal angle

3
π

4

rad

{\displaystyle \frac{3\pi}{4} \operatorname{rad}}

 = 135°

<= Great heptagram | 0D – 2D polytopes | Octagram =>
Octagram

Name
Octagram
Other name
Quasitruncated square
Schläli symbols

{

8
3

}

{\displaystyle \left\{\frac{8}{3}\right\}}

t

{

4
3

}

{\displaystyle \left\{\frac{4}{3}\right\}}

Coxeter diagrams

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

(

2

−
1

)

l

{\displaystyle \left(\sqrt{2} - 1\right) l}

Circumradius

2

(

2
−

2

)

l

{\displaystyle \sqrt{2\left(2-\sqrt{2}\right)} l}

Circumference*full*

16
l

{\displaystyle 16 l}

Circumference*shell*

16

(

2
−

2

)

l

{\displaystyle 16 \left(2 - \sqrt{2}\right) l}

Area*full*

8

(

2

−
1

)

l
2

{\displaystyle 8 \left(\sqrt{2} - 1\right) l^2}

Area*flat*

8

(

3

2

−
4

)

l
2

{\displaystyle 8 \left(3\sqrt 2 - 4\right) l^2}

Area*odd*

24

(

5

2

−
7

)

l
2

{\displaystyle 24 \left(5 \sqrt 2 - 7 \right) l^2}

Internal angle

π
4

rad

{\displaystyle \frac{\pi}{4} \operatorname{rad}}

 = 45°

<= Octagon | 0D – 2D polytopes | Nonagon =>
Great icosahedron

Name
Great icosahedron
Other names

Conjugate
Icosahedron
Number
8
Mæder number
U53
Coxeter number
C69
Wenninger number
W41 (page 63)
Bower’s acronym
gike (*GIKE*)
Schläfli symbol

{

3
,

5
2

}

{\displaystyle \left\{3,\ \frac{5}{2}\right\}}

Wythoff symbol
5/2 | 2 3
Conway notation

Polyhedron symbol
xo\*o
Coxeter diagram

Dynkin symbol
x3o5/2o
Vertex description

3

5
2

{\displaystyle 3^\frac{5}{2}}

Vertex figure

Kernel
Icosahedron
Flags
Regular
Nonconvex
Deltahedron
Orientable
Achiral
Density
7
Genus
0

Polyhedra in regimet
Great dodecahedron (colonel)
**Great icosahedron**
Regiment vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates
Even permutations of

(

±
1
,

±

ϕ

−
1

,

0

)

{\displaystyle \left(\pm1,\ \pm\phi^{-1},\ 0\right)}

where

# ϕ

1

5

2

{\displaystyle \phi=\frac{1+\sqrt{5}}{2}}

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
12 × 5/2
Number of edges
30
Number of faces
20 × {3}
## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great stellated dodecahedron
Image

<= Small stellated dodecahedron | Regular polyhedra | Great stellated dodecahedron =>
Octagonal antiprism

Name
Octagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
oap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Great heptagrammic antiprism | Prisms and Antiprisms | Octagrammic antiprism =>
Disnubs

## Small disnubs

1474. Small disnub dishexacosachoron (**sadsadox**)

1475. Small inverted disnub dishexacosachoron (**sidsadox**)

1476. Small disnub hexacosahexacosachoron (**sidsoxix**)

1477. Small disnub hexacosadishexacosachoron (**sidsaxadox**)

1478. Small disnub dishexacosahexacosachoron (**sidsadixox** )

1479. Small snubdisnub dishexacosachoron (**sisdasadex**)

1480. Small disnub trishexacosachoron (**sadistex**)

1481. Small disnub hexacosahexacosahexacosachoron (**sidsaxixox**)

1482. Small disnubsnub dishexacosachoron (**sidsosadex**)

## Great disnubs

1483. Great disnub dishexacosachoron (**gadsadox**)

1484. Great inverted disnub dishexacosachoron (**gidsadox**)

1485. Great disnub hexacosahexacosachoron (**gidsoxix**)

1486. Great disnub hexacosadishexacosachoron (**gidsaxadox**)

1487. Great disnub dishexacosahexacosachoron (**gidsadixox**)

1488. Great snubdisnub dishexacosachoron (**gisdasadex**)

1489. Great disnub trishexacosachoron (**gadistex**)

1490. Great disnub hexacosahexacosahexacosachoron (**gidsaxixox**)

1491. Great disnubsnub dishexacosachoron (**gidsosadex**)

<= Blends | Polytopes Wiki | Inverted dicuboctisnub doubles =>
Omnitruncated polychora

## Convex omnitruncates

330. Great prismatodecachoron (**gippid**)

331. Great disprismatotesseractihexadecachoron (**gidpith**)

332. Great prismatotetracontoctachoron (**gippic**)

333. Great disprismatohexacosahecatonicosachoron (**gidpixhi**)

## Great quasiprismates

334. Great quasidisprismatotesseractihexadecachoron (**gaquidpoth**)

335. Great quasiprismatotetracontoctachoron (**gaquapac**)

336. Grand quasidisprismatodishecatonicosachoron (**gaquidipdy**)

337. Great quasidisprismatodishecatonicosachoron (**goquidipdy**)

338. Great quasidisprismatohecatonicosahecatonicosachoron (**gaquidphihi**)

339. Great quasidisprismatohexacosahecatonicosachoron (**gaquidapixhi**)

## Cotcoics

340. Tesseractihexadecatruncated prismatotesseractihexadecachoron (**thatpath**)

341. Tesseractihexadecaquasitruncated prismatotesseractihexadecachoron (**thaquitpath**)

342. Icositetraquasitruncated prismatotetracontoctachoron (**iquatpic**)

343. Hecatonicosahexacosatruncated prismatohecatonicosahexacosachoron (**hixtaphix**)

344. Dishecatonicosatruncated prismatodishecatonicosachoron (**dohitipady**)

345. Hecatonicosahexacosaquasitruncated prismatohecatonicosahexacosachoron (**hixquitphix**)

346. Small dishecatonicosaquasitruncated prismatodishecatonicosachoron (**sidhiquit paddy**)

347. Great dishecatonicosaquasitruncated prismatodishecatonicosachoron (**gidhiquit paddy**)

## Others

348. Distetracontoctachoron (**doc**)

349. Tetracontoctarhombated tetracontoctachoron (**croc**)

350. Tetracontoctaquasirhombated tetracontoctachoron (**coqroc**)

351. Hexacosatrishecatonicosachoron (**xithi**)

<= Grombates | Polytopes Wiki | Prismatorhombates =>
Nonagon

Name
Nonagon
Other name
Enneagon
Schläli symbol
{9}
Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and

# ξ
tan
⁡

(

π
9

)

{\displaystyle \xi = \tan\left(\frac{\pi}{9}\right)}

Inradius

ξ

−
1

l

{\displaystyle \xi^{-1} l}

Circumradius

cosec
⁡

(

π
9

)

l

{\displaystyle \operatorname{cosec}\left(\frac{\pi}{9}\right) l}

Circumference

18
l

{\displaystyle 18l}

Area

9

ξ

−
1

l
2

{\displaystyle 9 \xi^{-1} l^2}

Internal angle

7
π

9

rad

{\displaystyle \frac{7\pi}{9} \operatorname{rad}}

 = 140°

<= Octagram | 0D – 2D polytopes | Nonagram =>
Cube

## Basic details

Name
Cube
Other names
Hexahedron
Square prism
Measure polyhedron
Conjugate
**Cube**
Number
2
Mæder number
U06
Coxeter number
C18
Wenninger number
W3 (page 16)
Bower’s acronym
cube (*KYOOB*)
Schläfli symbols
{4, 3}
t{2, 4}
{4} × {}
tr{2, 2}
{} × {} × {}
Wythoff symbol
3 | 2 4
Sequece symbol

γ

3

{\displaystyle \gamma \_{3}}

Conway notation
C
Polyhedron symbol
oo’x
Coxeter diagram

Dynkin symbol
x4o3o
Vertex description
43
Vertex figure

Flags
Regular
Convex
Parallelohedron
Zonohedron
General
Colonel
Orientable
Achiral
Density
1
Genus
0
### Army

Regiments in army
**Cube** (Major general)
Stella octangula
Polyhedra in regiment
**Cube** (Lieutenant colonel)
Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius,

r
c

{\displaystyle r\_c}

3

l

{\displaystyle \sqrt{3} l}

Midradius,

r

m

{\displaystyle r\_{m}}

2

l

{\displaystyle \sqrt{2} l}

Inradius,

r
i

{\displaystyle r\_i}

l

{\displaystyle l}

Dihedral angles
90° =

π
2

{\displaystyle \frac{\pi}{2}}

Vertex volume

π
2

{\displaystyle \frac{\pi}{2}}

Coordinates

(

±
1
,

±
1
,

±
1

)

{\displaystyle \left(\pm 1,\ \pm 1,\ \pm 1\right)}

Surface area

24

l
2

{\displaystyle 24l^2}

Volume

8

l
3

{\displaystyle 8l^3}

## Components

Number of vertices
8 × {3}
Number of edges
12
Number of faces
6 × {4}
Number of pieces
1
## Schwarz triangle

22
33
44

22
22
44

## Dual

Name
Octahedron
Image

Composite image

## Stellations

The cube does not have any stellations.

<= Tetrahedron | Regular polyhedra | Octahedron =>
Great snub dodecicosidodecahedron

Name
Great snub dodecicosidodecahedron
Other names

Number
72
Polyhedron number
U64
Coxeter number
C80
Wenninger number
W115
Bower’s acronym
gisdid
Schläfli symbol

Wythoff symbol

conway notation

Polyhedron symbol
(x\*’s\*s)
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
9

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
60
Number of edges
180
Number of faces
(20 + 60) × {3}
24 ×

{

5
2

}

{\displaystyle \left\{\frac{5}{2}\right\}}

## Schwarz triangle

5/35/3
5/25/2
33

## Dual

Name
Great hexagonal hexecontahedron
Image

Composite image

<= Snub icosidodecadodecahedron | Snub polyhedra | Great inverted retrosnub icosidodecahedron =>
Decagram

Name
Decagram
Other names
Quasitruncated pentagram
Quasitruncated pentangle
Schläli symbols

{

10
3

}

{\displaystyle \left\{\frac{10}{3}\right\}}

t

{

5
3

}

{\displaystyle \left\{\frac{5}{3}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

ϕ

−

3
2

×

5

1
4

l

{\displaystyle \phi^{-\frac{3}{2}} \times 5^\frac{1}{4} l}

Circumradius

2

ϕ

−
1

l

{\displaystyle 2 \phi^{-1} l}

Circumference*full*

20
l

{\displaystyle 20l}

Circumference*shell*

40

ϕ

−
3

l

{\displaystyle 40 \phi^{-3} l}

Area*full*

2

ϕ

−

3
2

×

5

5
4

l
2

{\displaystyle 2 \phi^{-\frac{3}{2}} \times 5^\frac{5}{4} l^2}

Area*flat*

4

ϕ

−

9
2

×

5

5
4

l
2

{\displaystyle 4 \phi^{-\frac{9}{2}} \times 5^\frac{5}{4} l^2}

Area*odd*

(

155
−
60

5

)

ϕ

×

5

3
4

l
2

{\displaystyle \left(155 - 60 \sqrt{5}\right) \sqrt{\phi}\times 5^\frac{3}{4} l^2}

Internal angle

2
π

5

rad

{\displaystyle \frac{2\pi}{5} \operatorname{rad}}

 = 72°

<= Decagon | 0D – 2D polytopes | Hendecagon =>
Octagrammic retroprism

Name
Octagrammic retroprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
storp
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Great heptagrammic retroprism | Prisms and Antiprisms | Great enneagrammic retroprism =>
Hexagon

Name
Hexagon
Other name
Truncated triangle
Schläli symbols
{6}
t{3}
Coxeter diagrams

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

3

l

{\displaystyle \sqrt{3} l}

Circumradius

2
l

{\displaystyle 2 l}

Circumference

12
l

{\displaystyle 12 l}

Area

6

3

l
2

{\displaystyle 6 \sqrt{3} l^2}

Internal angle

2
π

3

rad

{\displaystyle \frac{2\pi}{3} \operatorname{rad}}

 = 120°

<= Pentagram | 0D – 2D polytopes | Heptagon =>
Cuboctatruncated cuboctahedron

Name
Cuboctitruncated cuboctahedron
Other names

Number
62
Polyhedron number
U16
Coxeter number
C52
Wenninger number
W79
Bower’s acronym
cotco
Schläfli symbol

Wythoff symbol

conway notation

Polyhedron symbol
(x’x’x)
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
3

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices
48
Number of edges
72
Number of faces
8 × {6}
6 × {8}
6 ×

{

8
3

}

{\displaystyle \left\{\frac{8}{3}\right\}}

## Schwarz triangle

3/23/2
44
44

## Dual

Name
Tetradyakis hexahedron
Image

Composite image

<= Great quasitruncated icosidodecahedron | Omnitruncated polyhedra | Icosidodecatruncated icosidodecahedron =>
Heptagonal antiprism

## Basic details

Name
Hexagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
heap
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

Genus

### Army

General

Colonel

Army vertex figure

## Metrics

In the following, *l* is half the edge length.

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

{\displaystyle }

## Components

Number of vertices

Number of edges

Number of faces

## Dual

Name

Image

Composite image

<= Hexagonal antiprism | Prisms and Antiprisms | Heptagrammic antiprism =>