recursal/Fanatic-Fandom huggingface dataset:
Povray code-poly-textures.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/>
 \*/

/\* Polyhedra textures \*/
#ifndef (POLY\_TEXTURES\_INC)
 #declare POLY\_TEXTURES\_INC = 1;

 #include "golds.inc"
 #include "metals.inc"
 #include "glass.inc"
 #include "textures.inc"
 #include "colors.inc"

 #declare BaseTexture = 0;
 #declare DualTexture = 1;
 #declare RhombTexture = 4;
 #declare MedialTexture = 5;
 #declare SnubTexture = 2;
 #declare SnubTexture2 = 3;
 #declare NullTexture = 10;

 #ifndef (FaceTextures)
 #declare FaceTextures = array[11] {
 texture { // 0
 pigment {
 color rgbt <1.00, 0.00, 0.00, 0.50>
 }
 },
 texture { // 1
 pigment {
 color rgbt <0.00, 1.00, 0.00, 0.50>
 }
 },
 texture { // 2
 pigment {
 color rgbt <0.00, 0.50, 0.50, 0.50>
 }
 },
 texture { // 3
 pigment {
 colour rgbt <0.00, 0.00, 1.00, 0.50>
 }
 },
 texture { // 4
 pigment {
 colour rgbt <0.50, 0.50, 0.00, 0.50>
 }
 },
 texture { // 5
 pigment {
 colour rgbt <0.50, 0.00, 0.50, 0.50>
 }
 },
 texture { // 6
 pigment {
 colour rgbt <0.30, 0.50, 0.70, 0.50>
 }
 },
 texture { // 7
 pigment {
 colour rgbt <0.50, 0.70, 0.30, 0.50>
 }
 },
 texture { // 8
 pigment {
 colour rgbt <0.70, 0.30, 0.50, 0.50>
 }
 },
 texture { // 9
 pigment {
 colour rgbt <0.30, 0.70, 0.50, 0.50>
 }
 },
 texture { // 10
 pigment {
 colour rgbt <1,1,1,1>
 }
 }
 }
 #end
#end // ifndef POLY\_TEXTURES\_INC

```
Cylinder in the sink

**c**ylinder in the sink

c**y**linder in the sink

cy**l**inder in the sink

cyl**i**nder in the sink

cyli**n**der in the sink

cylin**d**er in the sink

cylind**e**r in the sink

cylinde**r** in the sink

cylinder **i**n the sink

cylinder i**n** the sink

cylinder in **t**he sink

cylinder in t**h**e sink

cylinder in th**e** sink

cylinder in the **s**ink

cylinder in the s**i**nk

cylinder in the si**n**k

cylinder in the sin**k**
Dodecagon

Name
Dodecagon
Other name
Truncated hexagon
Schläli symbols
{12}
t{6}
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 \left(2 + \sqrt{3}\right) l}

Circumference

24
l

{\displaystyle 24 l}

Area

12

(

2

3

)

l
2

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

Internal angle

5
π

6

rad

{\displaystyle \frac{5\pi}{6} \operatorname{rad}}

 = 150°

<= Great grand hendecagram | 0D – 2D polytopes | Dodecagram =>
Great rhombidodecahedron

Name
Great rhombidodecahedron
Other names

Number
56
Polyhedron number
U73
Coxeter number
C89
Wenninger number
W109
Bower’s acronym
gird
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Vertex figure

Flags

Density

Genus
10

## 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
30 × {4}
12 ×

{

10
3

}

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

## Dual

Name
Great rhombidodecacron
Image

Composite image

<= Quasirhombicosidodecahedron | Trapezivert polyhedra | Great rhombicuboctahedron =>
Quasitruncated hexahedron

Name
Quasitruncated hexahedron
Other names
Stellated truncated hexahedron
Stellated truncated cube
Conjugate
Truncated cube
Number
17
Mæder number
U19
coxeter number
C66
Wenninger number
W92 (page 144)
Bower’s acronym
quith (*QUITH*)
Schläfli symbol

Wythoff symbols
2 3 | 4/3
2 3/2 | 4/3
Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Vertex description
8/32.3
Vertex figure

Kernel

Flags
Archimedean
Nonconvex
General
Colonel
Orientable
Achiral
Density
7
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
24
Number of edges
36
Number of faces
8 × {3}
6 ×

{

8
3

}

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

Number of pieces

## Schwarz triangle

22
33
44

## Dual

Name
Great triakis octahedron
Image

Composite image

<= Truncated great icosahedron | Truncated polyhedra | Quasitruncated small stellated dodecahedron =>
Cuboctahedron

## Basic details

Name
Cuboctahedron
Other names

Conjugate
**Cuboctahedron**
Number
21
Mæder number
U07
Coxeter number
C19
Wenninger number
W11 (page 25)
Bower’s acronym
co (*CO*)
Schläfli symbol

{

4

3

}

{\displaystyle \left\{\begin{array}{c}4\\3\end{array}\right\} }

Wythoff symbol
2 | 3 4
Conway notation

Polyhedron symbol
ox’o
Coxeter diagram

Dynkin symbol

Vertex description
(4.3)2
Vertex figure

Flags
Semiregular
Archimedean
Convex
General
Colonel
Orientable
Achiral
Density
1
Genus
0
### Army

Regiments in army
[[Cuboctahedron (general)
Polyhedra in regiment
Octahemioctahedron
Cubohemioctahedron
Regiment 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
12
Number of edges
24
Number of faces
8 × {3}
6 × {4}
Number of pieces
1
## Schwarz triangle

22
33
44

33
33
22

## Dual

Name
Rhombic dodecahedron
Image

Composite image

<= Tetrahemihexahedron | Quasiregular polyhedra | Octahemioctahedron =>
Quasitruncated small stellated dodecahedron

Name
Quasitruncated small stellated dodecahedron
Other names
Small stellated truncated dodecahedron
Conjugate : Truncated great dodecahedron
Number
18
M&aeelig;der number
U58
Coxeter number
C74
Wenninger number
W97 (page 151)
Bower’s acronym
quit sissid (*quit SIS sid*)
Schläfli symbol

Wythoff symbol
2 5 | 5/3
2 5/4 | 5/3
Conway notation

Polyhedron symbol
o^x\*’x
Coxeter diagram

Dynkin symbol

Vertex description
10/32.5
Vertex figure

Kernel

Flags
Archimedean
Nonconvex
General
Colonel
Orientable
Achiral
Density
9
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
90
Number of faces
12 × {5}
12 ×

{

10
3

}

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

Number of pieces

## Schwarz triangle

22
5/25/2
55

## Dual

Name
Great pentakis dodecahedron
Image

Composite image

<= Quasitruncated hexahedron | Truncated polyhedra | Quasitruncated great stellated dodecahedron =>
Decagrammic antiprism

## Basic details

Name
Decagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stiddap
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

<= Decagonal antiprism | Prisms and Antiprisms | p-gonal antiprism =>
Pentagrammic antiprism

## Basic details

Name
Pentagrammic 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

## Dual

Name

Image

Composite image

<= Pentagonal antiprism | Prisms and Antiprisms | Hexagonal antiprism =>
Great nonagram

Name
Great nonagram
Other name
Great enneagram
Schläli symbol

{

9
4

}

{\displaystyle \left\{\frac{9}{4}\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
−
6

ξ
2

ξ
4

4
ξ

(

1
−

ξ
2

)

l

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

Circumradius

cosec
⁡

(

4
π

9

)

l

{\displaystyle \operatorname{cosec}\left(\frac{4\pi}{9}\right) l}

Circumference*full*

18
l

{\displaystyle 18l}

Circumference*shell*

18

(

1
−

3

cot
⁡

(

4
π

9

)

)

l

{\displaystyle 18\left(1 - \sqrt{3} \cot\left(\frac{4 \pi}{9}\right)\right) l}

Area*full*

9

(

1
−
6

ξ
2

ξ
4

)

4
ξ

(

1
−

ξ
2

)

l
2

{\displaystyle \frac{9 \left(1 - 6 \xi^2 + \xi^4\right)}{4 \xi \left(1 - \xi^2\right)} l^2}

Area*flat*

9

(

1
−

3

cot
⁡

(

4
π

9

)

)

2
tan
⁡

(

4
π

9

)

l
2

{\displaystyle \frac{9\left(1 - \sqrt{3} \cot\left(\frac{4 \pi}{9}\right)\right)}{2 \tan\left(\frac{4 \pi}{9}\right)} l^2}

Area*odd*

9

(

tan
⁡

(

4
π

9

)

−
2

3

2
tan
⁡

(

2
π

9

)

−
2
ξ

)

(

cot
⁡

(

4
π

9

)

)

2

l
2

{\displaystyle 9 \left(\tan\left(\frac{4\pi}{9}\right)-2 \sqrt{3} + 2 \tan\left(\frac{2 \pi}{9}\right) - 2\xi\right)\left(\cot\left(\frac{4\pi}{9}\right)\right)^2 l^2}

Internal angle

π
9

rad

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

 = 20°

<= Nonagram | 0D – 2D polytopes | Decagon =>
Bitruncated polychora

## Convex birtuncates

298. Decachoron (**deca**)

299. Tesseractihexadecachoron (**tah**)

300. Tetracontoctachoron (**cont**)

301. Hexacosahecatonicosachoron (**xhi**)

## Stellar birtuncates

302. Small hecatonicosahecatonicosachoron (**shihi**)

303. Dishecatonicosachoron (**dahi**)

304. Great hexacosahecatonicosachoron (**gixhi**)

## Biquasitruncates

305. Great tetracontoctachoron (**gic**)

306. Great hecatonicosahecatonicosachoron (**ghihi**)

<= Sphenoverts | Polytopes Wiki | Grombates =>
Pentagram

Name
Pentagram
Other name
Pentangle
Schläli symbol

{

5
2

}

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

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

l

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

Circumference*full*

10
l

{\displaystyle 10l}

Circumference*flat*

20

ϕ

−
2

l

{\displaystyle 20 \phi^{-2} l}

Area*full*

5

3
4

×

ϕ

−

3
2

l
2

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

Area*flat*

2

ϕ

−

7
2

×

5

3
4

l
2

{\displaystyle 2 \phi^{-\frac{7}{2}} \times 5^\frac{3}{4} l^2}

Area*odd*

ϕ

−

9
2

×

5

5
4

l
2

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

Internal angle

π
5

rad

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

 = 36 °

<= Pentagon | 0D – 2D polytopes | Hexagon =>
Great grand hendecagram

Name
Great grand hendecagram
Schläli symbol

{

11
5

}

{\displaystyle \left\{\frac{11}{5}\right\}}

Coxeter diagram

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length.

Inradius

cot
⁡

(

5
π

11

)

l

{\displaystyle \cot\left(\frac{5\pi}{11}\right) l}

Circumradius

cosec
⁡

(

5
π

11

)

l

{\displaystyle \operatorname{cosec}\left(\frac{5\pi}{11}\right) l}

Circumference*full*

22
l

{\displaystyle 22 l}

Circumference*shell*

11

(

1
−
tan
⁡

(

4
π

11

)

cot
⁡

(

5
π

11

)

)

l

{\displaystyle 11\left(1 - \tan\left(\frac{4 \pi}{11}\right) \cot\left(\frac{5\pi}{11}\right)\right) l}

Area*full*

11
cot
⁡

(

5
π

11

)

l
2

{\displaystyle 11 \cot\left(\frac{5 \pi}{11}\right) l^2}

Area*flat*

11

(

1
−
tan
⁡

(

4
π

11

)

cot
⁡

(

5
π

11

)

)

cot
⁡

(

5
π

11

)

l
2

{\displaystyle 11\left(1 - \tan\left(\frac{4 \pi}{11}\right) \cot\left(\frac{5 \pi}{11}\right)\right) \cot\left(\frac{5 \pi}{11}\right) l^2}

Area*odd*

11

(

tan
⁡

(

5
π

11

)

−
2
tan
⁡

(

4
π

11

)

2
tan
⁡

(

3
π

11

)

−
2
tan
⁡

(

2
π

11

)

2
tan
⁡

(

π
11

)

)

(

cot
⁡

(

5
π

11

)

)

2

l
2

{\displaystyle 11\left(\tan\left(\frac{5 \pi}{11}\right) - 2 \tan\left(\frac{4 \pi}{11}\right) +2 \tan\left(\frac{3 \pi}{11}\right) -2 \tan\left(\frac{2 \pi}{11}\right) +2\tan\left(\frac{\pi}{11}\right)\right) \left(\cot\left(\frac{5 \pi}{11}\right)\right)^2 l^2}

Internal angle

π
11

rad

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

 = 16 4/11°

<= Grand hendecagram | 0D – 2D polytopes | Dodecagon =>
Polytopes Wiki

**Welcome to the Polytopes Wiki!**
This wiki contains details of the different polytopes

## 0-dimensions to 2-dimensions

The low-dimensional polytopes (points to polygons) are the simplest. As such, they have been collated under one sub-page.

## Polyhedra

### Uniform Polyhedra

1. Regulars (1-9)
2. Truncates (10-19)
3. Quasiregulars (20-35)
4. Trapeziverts (36-56)
5. Omnitruncates (57-63)
6. Snubs (64-75)
7. Prisms and Antiprisms

* Degenerates
* Compounds

### Non-uniform Finite Polyhedra

* Johnson solids
* Archimedean duals
* Stellations
* Facets

## Polychora

### Uniform Polychora

1. Regulars (1 - 17)
2. Truncates (18 - 38)
3. Triangular rectates (39-59)
4. Icositetrachoron regiment (60 - 72)
5. Pentagonal rectates (73 - 132)
6. Sphenoverts (133 - 297)
7. Bitruncates (298 - 306)
8. Grombates (307 - 329)
9. Omnitruncates (330 - 351)
10. Prismatorhombates (352 - 441)
11. Antipodiumverts (442 - 481)
12. Podiumverts (482 - 511)
13. Squapverts (512 - 551)
14. Skewverts (552 - 611)
15. Antifrustary distetracontoctachoron regiment (612 - 664)
16. Antifrustary hexacosatrishecatonicosachoron regiment (665 - 763)
17. Small stellated hecatonicosachoron regiment (764 - 777)
18. Ditetrahedrals (778 - 888)
19. Prisms (889 - 962)
20. Miscellaneous polychora 1 (963 - 984)
21. Prismated dishecatonicosachoron regiment (985 - 1065)
22. Great dipentary trishecatonicosachoron regiment (1066 - 1146)
23. Rectified small ditrigonary hexacosahecatonicosachoron regiment (1147 - 1303)
24. Small tritrigonary prismatohecatonicosadishexacosachoron regiment (1304 - 1382)
25. Great tritrigonary hexacosatrishecatonicosachoron regiment (1383 - 1461)
26. Blends (1462 - 1473)
27. Disnubs (Sidtaps and gidtaps) (1474 - 1491)
28. Inverted dicuboctisnub doubles (idcossids) (1492 - 1668)
29. Diretrocuboctisnubprismatodiprismatodoubles (dircospids) (1669 - 1845)
30. Miscellaneous polychora 2 (1846-1855)
31. Snub disoctachora (1856-2188)

#### Infinite series

As well as making a prism from the 3D prisms and antiprisms, there are two other infinite categories of uniform polychora.

A. Duoprisms

B. Antiduoprisms

### CRF polychora

The CRF (Convex Regular-Faced) polychora are the 4D analgue of the 3D Johnson polyhedra.

## Annexes

Other pages available are:

* Definitions
* List of uniform polyhedra by army
* Schwarz triangles
* List of Goursat tetrahedra
* Code used to generate the images, etc.

## Other polytopes websites

* www.polytope.net: This is a comprehensive web site, giving lots of useful information
* bendwavy.org
* eusebeia
* verse-and-dimensions.fandom.com: wiki site
* hi.gher.space: wiki site
* polytope.miraheze.org: wiki site, “The Polytope Wiki is a wiki dedicated to the classification, description, and discovery of polytopes.”
* wikipedia
* polytope discord
Octahedron

## Basic details

Name
Octahedron
Other names
Cross polyhedron
Tetratetrahedron
Rectified tetrahedron
Conjugate
**Octahedron**
Number
3
Mæder number
U05
Coxeter number
C17
Wenninger number
W2 (page 15)
Bower’s acronym
oct (*OCT*)
Schläfli symbol
{3, 4}

{

3

3

}

{\displaystyle \left\{\begin{array}{c}3\\3\end{array}\right\} }

Wythoff symbol
4 | 2 3
Sequece symbol

β
3

{\displaystyle \beta\_3}

Conway notations
O
aT
Polyhedron symbol
xo’o
Coxeter diagram

Dynkin symbol
x3o4o
Vertex description
34
Vertex figure

Flags
Regular
Convex
Deltahedron
General
Colonel
Orientable
Achiral
Density
1
Genus
0

## Metrics

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

Circumradius,

r
c

{\displaystyle r\_c}

2

l

{\displaystyle \sqrt{2} l}

Midradius,

r

m

{\displaystyle r\_{m}}

l

{\displaystyle l}

Inradius,

r
i

{\displaystyle r\_i}

6

l

3

{\displaystyle \frac{\sqrt{6} l}{3}}

Dihedral angles

arccos
⁡

(

−

1
3

)

{\displaystyle \arccos\left(-\frac{1}{3}\right)}

 ≈ 109.47° ≈ 0.608 π radians
Vertex volume

4
arccos
⁡

(

−

1
3

)

−
2
π

{\displaystyle 4 \arccos\left(-\frac{1}{3}\right) - 2 \pi}

 ≈ 1.359
Coordinates
Permutations of

(

±
2
,

0
,

0

)

{\displaystyle \left(\pm2,\ 0,\ 0\right)}

Surface area

8

3

l
2

{\displaystyle 8\sqrt{3}l^2}

Volume

8

6

l
3

9

{\displaystyle \frac{8 \sqrt{6} l^3}{9}}

## Components

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

33
33
22

22
22
33

## Dual

Name
Cube
Image

Composite image

## Stellations

Facial planes
1
Number of stellations
1 (+ kernel)

<= Cube | Regular polyhedra | Dodecahedron =>
Small dodecicosidodecahedron

Name
Small dodecicosidodecahedron
Other names

Number
40
Polyhedron number
U33
Coxeter number
C42
Wenninger number
W72
Bower’s acronym
saddid
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 × {5}
12 × {10}
## Schwarz triangle

3/23/2
55
55

## Dual

Name
Small dodecacronic hexecontahedron
Image

Composite image

<= Small rhombicosidodecahedron | Trapezivert polyhedra | Small rhombidodecahedron =>
Miscellaneous polychora 2

*For the antiprisms, snubs, swirlprisms and inverted quasiprismatodishexadecachoron type polychora (polychora 963–984), see Miscellaneous polychora 1.*

## Octagonal spinoduoprismchoron type

1846. Octagonal spinoduoprismchoron (**ondip**)

1847. Small ditetragonal spinoduoprismchoron (**sidtindip**)

1848. Great octagonal spinoduoprismchoron (**gondip**)

1849. Great ditetragonal spinoduoprismchoron (**gidtindip**)

## Swirlprisms 2

1850. Small dicositetraconta-tricosihexacontaswirlprismatic hecatonicosa-tricosihexacontachoron (**sidditsphit**)

1851. Great dicositetraconta-tricosihexacontaswirlprismatic hecatonicosa-tricosihexacontachoron (**gidditsphit**)

## Ixitics

1852. Small tristetracosioctacontachoron (**setut**)

1853. Great tristetracosioctacontachoron (**getut**)

## Antiprisms 2

1854. Pentacubic extended duofacetoantiprism (**pecuexidfap**)

1855. Pentacubic extended duoantiprism (**pecuexdap**)

<= Diretrocuboctisnubprismatodiprismato doubles | Polytopes Wiki | Snub disoctachora =>
Great icosidodecahedron

Name
Great icosidodecahedron
Other names

Number
30
Polyhedron number
U54
Coxeter number
C70
Wenninger number
W94
Bower’s acronym
gid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
ox\*o
Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

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
30
Number of edges
60
Number of faces
20 × {3}
12 ×

{

5
2

}

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

## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great rhombic triacontahedron
Image

Composite image

<= Great dodecahemicosahedron | Quasiregular polyhedra | Great dodecahemidodecahedron =>
Faceted hexacosachoron

## Basic details

Name
Faceted hexacosichoron
Other names

Number
7
Category
1
Bower’s acronym
fix
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

<= Hexacosichoron | Regular polychora | Great hecatonicosachoron =>
Grombates

## Convex great rhombates

307. Great rhombated pentachoron (**grip**)

308. Great rhombated tesseract (**grit**)

309. Great rhombated icositetrachoron (**grico**)

310. Great rhombated hecatonicosachoron (**grahi**)

311. Great rhombated hexacosachoron (**grix**)

## Stellar great rhombates

312. Great rhombated faceted hexacosachoron (**girfix**)

313. Great rhombated grand hecatonicosachoron (**graghi**)

314. Great rhombated grand hexacosachoron (**graggix**)

## Great quasirhombates

315. Great quasirhombated tesseract (**gaqrit**)

316. Great quasirhombated icositetrachoron (**gaqri**)

317. Great quasirhombated great grand hecatonicosachoron (**gaqrigaghi**)

318. Great quasirhombated great faceted hexacosachoron (**gaqrigafix**)

319. Great quasirhombated small stellated hecatonicosachoron (**gaqrisashi**)

320. Great quasirhombated great grand stellated hecatonicosachoron (**gaqrigagishi**)

321. Great quasirhombated great stellated hecatonicosachoron (**gaqrigashi**)

## Cotcoics

322. Tesseractihexadecatruncated tesseractihexadecachoron (**thatoth**)

323. Tesseractihexadecaquasitruncated tesseractihexadecachoron (**thaquitoth**)

324. Icositetraquasitruncated tetracontoctachoron (**iquatoc**)

325. Hecatonicosahexacosatruncated hecatonicosahexacosachoron (**hixthix**)

326. Dishecatonicosatruncated dishecatonicosachoron (**dahitady**)

327. Hecatonicosahecatonicosaquasitruncated dishecatonicosachoron (**hihiquatady**)

328. Dishecatonicosaquasitruncated dishecatonicosachoron (**dahiquatady**)

329. Hecatonicosahexacosaquasitruncated hecatonicosahexacosachoron (**hixquathix**)

<= Bitruncated polychora | Polytopes Wiki | Omnitruncated polychora =>
P-gonal antiprism

## Basic details

Name
p-gonal antiprism
Other names

Number
AP(p)
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

p
d

{\displaystyle p\_d}

 where

# p

p
n

p
d

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

Genus
0
### 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

<= Decagrammic antiprism | Prisms and Antiprisms | Pentagrammic retroprism =>
Regular polyhedra

A polyhedron is regular if it is both vertex regular and face regular.  There are five convex regular polyhedra and 4 regular star polyhedra.

## Convex polyhedra

1. Tetrahedron (**tet**)

2. Cube (**cube**)

3. Octahedron (**oct**)

4. Dodecahedron (**doe**)

5. Icosahedron (**ike**)

## Star polyhedra

6. Great dodecahedron (**gad**)

7. Small stellated dodecahedron (**sissid**)

8. Great icosahedron (**gike**)

9. Great stellated dodecahedron (**gissid**)

<= 0D – 2D polytopes | Polytopes Wiki | Truncated polyhedra =>
Icosidodecahedron

Name
Icosidodecahedron
Other names

Conjugate
Great icosidodecahedron
Number
24
Mæder number
U24
Coxeter number
C28
Wenninger number
W12 (page 26)
Bower’s acronym
id (*ID*)
Schläfli symbol

{

5

3

}

{\displaystyle \left\{\begin{array}{c}5\\3\end{array}\right\} }

Wythoff symbol
2 | 3 5
Conway notation

Polyhedron symbol
ox^o
Coxeter diagram

Dynkin symbol

Vertex description
(5.3)2
Vertex figure

Flags
Semiregular
Archimedean
Convex
General
Colonel
Orientable
Achiral
Density
1
Genus
0

## Metrics

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

Circumradius, *rc*

Midradius, *rm*

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates

Surface area

Volume

## Components

Number of vertices
30
Number of edges
60
Number of faces
20 × {3}
12 × {5}
Number of pieces : 1
## Schwarz triangle

22
33
55

## Dual

Name
Rhombic triacontahedron
Image

Composite image

<= Cubohemioctahedron | Quasiregular polyhedra | Small dodecahemidodecahedron =>
Octagonal prism

## Basic details

Name
Octagonal prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
oc
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

<= Great heptagrammic prism | Prisms and Antiprisms | Octagrammic prism =>
Quasirhombicuboctahedron

## Basic details

Name
Quasirhombicuboctahedron
Other names

Number
46
Polyhedron number
U17
Coxeter number
C59
Wenninger number
W85
Bower’s acronym
querco
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
xo’x
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
0
### 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}
18 × {4}
## Schwarz triangle

22
33
44

## Dual

Name
Great deltoidal icositetrahedron
Image

Composite image

<= Great cubicuboctahedron | Trapezivert polyhedra | Great rhombihexahedron =>
Quasiregular polyhedra

The Quasiregulars are characterised by the vertex figure being rectangular. Each regiment contains three polyhedra—the colonel, having the complete rectangle (or in the case of the *sidtid regiment*, a equiangular hexagon, with sides alternating between two lengths), with the other two polyhedra having one pair of parallel sides, along with the two diagonals.

## Octahedron regiment

*4. Octahedron (oct)*

20. Tetrahemihexahedron (**thah**)

## Cuboctahedron regiment

21. Cuboctahedron (**co**)

22. Octahemioctahedron (**oho**)

23. Cubohemioctahedron (**cho**)

## Icosidodecahedron regiment

24. Icosidodecahedron (**id**)

25. Small dodecahemidodecahedron (**sidhid**)

26. Small icosihemidodecahedron (**seihid**)

## Dodecadodecahedron regiment

27. Dodecadodecahedron (**did**)

28. Small dodecahemicosahedron (**sidhei**)

29. Great dodecahemicosahedron (**gidhei**)

## Great icosidodecahedron regiment

30. Great icosidodecahedron (**gid**)

31. Great dodecahemidodecahedron (**gidhid**)

32. Great icosihemidodecahedron (**geihid**)

## Small ditrigonary icosidodecahedron regiment

33. Small ditrigonary icosidodecahedron (**sidtid**)

34. Ditrigonary dodecadodecahedron (**ditdid**)

35. Great ditrigonary icosidodecahedron (**gidtid**)

## Footnotes

1. ↑ The tetrahemihexahedron would be formed by replacing alternate triangles of an octahedron with central squares in two ways

<= Truncated polyhedra | Polytopes Wiki | Trapezovert polyhedra =>
Decagrammic prism

## Basic details

Name
Decagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stiddip
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

<= Decagonal prism | Prisms and Antiprisms | p-gonal prism =>
Small dodecahemicosahedron

## Basic details

Name
Small dodecahemicosahedron
Other names

Number
28
Polyhedron number
U62
coxeter number
C78
Wenninger number
W100
Bower’s acronym
sidhei
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(x\*o\*’x)/2
Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
5
### Army

General
Icosidodecahedron
Colonel
Dodecadodecahedron
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}
10 × {6}
## Schwarz triangle

5/35/3
5/25/2
33

## Dual

Name
Small dodecahemicosacron
Image

Composite image

<= Dodecadodecahedron | Quasiregular polyhedra | Great dodecahemicosahedron =>
Heptagrammic antiprism

Name
Heptagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
sithap
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

<= Heptagonal antiprism | Prisms and Antiprisms | Great heptagrammic antiprism =>
Polyhedra by army

## Tetrahedral symmetry

*Both of the tetrahedral polyhedra are achiral.*

* Tetrahedron
* Truncated tetrahedron

## Octahedral symmetry

### Achiral

* Cube
* Cuboctahedron army and regiment
	+ Cubihemioctahedron
	+ Cuboctahedron
	+ Octahemioctahedron
* Cuboctitruncated cuboctahedron
* Great rhombicuboctahedron
* Octahedron army and regiment
	+ Octahedron
	+ Tetrahemihexahedron
* Quasitruncated cuboctahedron
* Quasitruncated hexahedron
* Small rhombicuboctahedron army and regiment
	+ Small cubicuboctahedron
	+ Small rhombicuboctahedron
	+ Small rhombihexahedron
* Truncated cube army
	+ Great cubicuboctahedron regiment
		- Great cubicuboctahedron
		- Great rhombihexahedron
		- Quasirhombicuboctahedron
	+ Truncated cube
* Truncated octahedron

## Icosahedral symmetry

### Achiral

* Dodecahedron army
	+ Dodecahedron regiment
		- Dodecahedron
		- Great dodecahedron
	+ Great stellated dodecahedron
	+ Small ditrigonary icosidodecahedron regiment
		- Ditrigonary dodecadodecahedron
		- Great ditrigonary icosidodecahedron
		- Small ditrigonary icosidodecahedron
* Great dirhombicosidodecahedron
* Great dodecicosidodecahedron army and regiment
	+ Great dodecicosidodecahedron
	+ Great rhombidodecahedron
	+ Quasirhombicosidodecahedron
* Great quasitruncated icosidodecahedron
* Great rhombicosidodecahedron
* Icosahedron army
	+ Icosahedron
	+ Small stellated dodecahedron regiment
		- Great icosahedron
		- Small stellated dodecahedron
* Icosidodecahedron army
	+ Dodecadodecahedron regiment
		- Dodecadodecahedron
		- Great dodecahemicosahedron
		- Small dodecahemicosahedron
	+ Great icosidodecahedron regiment
		- Great dodecahemidodecahedron
		- Great icosahemidodecahedron
		- Great icosidodecahedron
	+ Icosidodecahedron regiment
		- Icosidodecahedron
		- Small dodecahemidodecahedron
		- Small icosahemidodecahedron
* Icosidodecatruncated icosidodecahedron
* Quasitruncated dodecadodecahedron
* Quasitruncated great stellated dodecahedron
* Quasitruncated small stellated dodecahedron
* Rhombidodecadodecahedron army and regiment
	+ Icosidodecadodecahedron
	+ Rhombicosahedron
	+ Rhombidodecadodecahedron
* Small icosicosidodecahedron army and regiment
	+ Small ditrigonary dodecicosidodecahedron
	+ Small dodecicosahedron
	+ Small icosicosidodecahedron
* Small inverted retrosnub icosicosidodecahedron
* Small rhombicosidodecahedron army and regiment
	+ Small dodecicosidodecahedron
	+ Small rhombicosidodecahedron
	+ Small rhombidodecahedron
* Small snub icosicosidodecahedron
* Truncated dodecahedron army
	+ Great ditrigonary dodecicosidodecahedron regiment
		- Great ditrigonary dodecicosidodecahedron
		- Great dodecicosahedron
		- Great icosicosidodecahedron
	+ Truncated dodecahedron
* Truncated great dodecahedron
* Truncated great icosahedron
* Truncated icosahedron

### Chiral

* Great inverted retrosnub icosidodecahedron
* Great inverted snub icosidodecahedron
* Great snub dodecicosidodecahedron
* Great snub icosidodecahedron
* Inverted snub dodecadodecahedron
* Snub dodecadodecahedron
* Snub dodecahedron
* Snub icosidodecadodecahedron

<= Definitions | Polytopes Wiki | Schwarz triangles =>
Advanced Screaming Byte

AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA"BRO STOP SCREAMING-"AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAaaaaaaaaaaaaaaaaaaaaaaaaa

Creator: Mrmissingno10
Antipodiumverts

1. ↑ 1.0 1.1 1.2 1.3 Piphid and nipdip are represented twice in this regiment, as they can ecah be rotated into the corresponding polychoron

## Small prismatodecachoron regiment

442. Small prismatodecachoron (**spid**)

443. Prismatopentahemidecachoron (**piphid**)

444. Decahemidecachoron (**duhd**)

445. Prismatohemidecachoron (**phud**)

446. Spinoprismatodispentachoron (**nipdip**)

## Small disprismatotesseractihexadecachoron regiment

447. Small disprismatotesseractihexadecachoron (**sidpith**)

448. Small tesseractifaceted cubitesseractihexadecachoron (**stefacoth**)

449. Small hexadecafaceted prismatotesseractioctachoron (**shafipto**)

450. Small tesseractitesseractihexadecachoron (**steth**)

451. Disprismatotesseract (**dippit**)

452. Small spinoprismatotesseractihexadecachoron (**snappoth**)

453. Small spinoprismatotesseractioctachoron (**snipto**)

## Small disprismatohexacosahecatonicosachoron regiment]]

454. Small disprismatohexacosahecatonicosachoron (**sidpixhi**)

455. Small hecatonicosafaceted prismatohecatonicosahexacosachoron (**shif phix**)

456. Small hexacosafaceted prismatodishecatonicosachoron (**six fipady**)

457. Small hexacosahecatonicosahecatonicosachoron (**sixhihy**)

458. Prismatoprismatohecatonicosachoron (**paphi**)

459. Small spinoprismatohexacosahecatonicosachoron (**snapaxhi**)

460. Small spinoprismatodishecatonicosachoron (**snappady**)

## Small disprismatodishecatonicosachoron regiment

461. Small disprismatodishecatonicosachoron (**siddapady**)

462. Hecatonicosafaceted prismatodishecatonicosachoron (**hi fipady**)

463. Medial hecatonicosafaceted prismatodishecatonicosachoron (**mohi fipady**)

464. Triakishecatonicosachoron (**tahi**)

465. Disprismatohecatonicosachoron (**daphi**)

466. Small spinoprismatohecatonicosahecatonicosachoron (**snaphihu**)

467. Great spinoprismatohecatonicosahecatonicosachoron (**gnaphihy**)

## Small dishexacosadishecatonicosachoron regiment

468. Small dishexacosadishecatonicosachoron (**sadixady**)

469. Small hexacosafaceted hexacosadishecatonicosachoron (**six fixady**)

470. Small hecatonicosafaceted hexacosadishecatonicosachoron (**shi fixady**)

471. Small ditrigonary hecatonicoshexacosahecatonicosachoron (**sidthixhi**)

472. Hecatonicosahecatonicosahexacosachoron (**hihix**)

473. Small spinotriakishecatonicosachoron (**snathi**)

474. Small spinohecatonicosadishexacosachoron (**sanhiddix**)

## Great hexacosahecatonicosadishecatonicosachoron regiment

475. Great hexacosahecatonicosadishecatonicosachoron (**gixhady**)

476. Great hecatonicosafaceted hecatonicosahexacosahecatonicosachoron (**gohi fohixhi**)

477. Great hexacosafaceted triakishecatonicosachoron (**gaxifthi**)

478. Great hecatonicosahecatonicosahexacoschoron (**gohihix**)

479. Hecatonicosadishecatonicosachoron (**hidy**)

480. Great spinohexacosadishecatonicosachoron (**ganxady**)

481. Great spinohecatonicosadishecatonicosachoron (**gun hidy**)

<= Prismatorhombates | Polytopes Wiki | Podiumverts =>
Dodecahedron

## Basic details

Name
Dodecahedron
Conjugate
Great stellated dodecahedron
Number
4
Mæder number
U23
Coxeter number
C26
Wenninger number
W5 (page 19)
Bower’s acronym
doe
Schläfli symbol
{5, 3}
Wythoff symbol
3 | 2 5
Conway notation
D
Polyhedron symbol
oo^x
Coxeter diagram

Dynkin symbol
x5o3o
Vertex description
53
Vertex figure

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

Regiments in army
**Dodecahedron** (general)
Great stellated dodecahedron
Army vertex figure

Polyhedra in regiment
**Dodecahedron** (colonel)
Regiment vertex figure

## Metrics

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

Circumradius, *rc*

ϕ

3

{\displaystyle \phi \sqrt{3}}

Midradius, *rm*

ϕ
2

{\displaystyle \phi^2}

Inradii, *ri*

ϕ
5

5

{\displaystyle \sqrt{\frac{\phi^5}{\sqrt{5}}}}

where

# ϕ

1

5

2

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

Dihedral angles

Vertex volume

Coordinates
even permutations of

(
0
,

±
1
,

±

ϕ
2

)

{\displaystyle (0,\ \pm1,\ \pm\phi^2)}

 and

(
±
ϕ
,

±
ϕ
,

±
ϕ
)

{\displaystyle (\pm\phi,\ \pm\phi,\ \pm\phi)}

Surface area

12

ϕ

3
2

×

5

3
4

{\displaystyle 12 \phi^\frac{3}{2} \times 5^\frac{3}{4}}

Volume

4

ϕ
4

5

{\displaystyle 4 \phi^4 \sqrt{5}}

## Components

Number of vertices
20 × {3}
Number of edges
30
Number of faces
12 × {5}
Number of pieces
1
## Schwarz triangle

22
33
55

## Dual

Name
Icosahedron
Image

Composite image

## Stellations

Facial planes
1
Number of stellations
3 (+ kernel)

<= Octahedron | Regular polyhedra | Icosahedron =>
Enneagrammic prism

## Basic details

Name
Enneagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
step
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

<= Enneagonal prism | Prisms and Antiprisms | Great enneagrammic prism =>
Rhombidodecadodecahedron

Name
Rhombidodecdodecahedron
Other names

Number
42
Polyhedron number
U38
Coxeter number
C48
Wenninger number
W76
Bower’s acronym
raded
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
x\*o^x
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
120
Number of faces
30 × {4}
12 × {5}
12 ×

{

5
2

}

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

## Schwarz triangle

22
5/25/2
55

## Dual

Name
Medial deltoidal hexecontahedron
Image

Composite image

<= Small rhombidodecahedron | Trapezivert polyhedra | Icosidodecadodecahedron =>
Octahemioctahedron

## Basic details

Name
Octahemioctahedron
Other names

Conjugate
**Octahemioctahedron**
Number
22
Mæder number
U03
Coxeter number
C37
Wenninger number
W68 (page 103)
Bower’s acronym
oho (*O ho*)
Schläfli symbol

Wythoff symbol
3/2 3 | 3
Conway notation

Polyhedron symbol
(xo,x)
Coxeter diagram

Dynkin symbol

Vertex description
3.6.3/2.6
Vertex figure

Flags
Nonconvex
Nonorientable
Achiral
Density
2 (4 × ½)
Genus
1
### Army

Regiments in army
Cuboctahedron (general)
Polyhedra in regiment
Cuboctahedron (colonel)
**Octahemioctahedron**
Cubohemioctahedron
Regiment 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
12
Number of edges
24
Number of faces
8 × {3}
4 × {6}
Number of pieces
8 (of 1 type)
## Schwarz triangle

3/23/2
33
33

## Dual

Name
Octahemioctacron
Image

Composite image

<= Cuboctahedron | Quasiregular polyhedra | Cubohemioctahedron =>
Heptagonal prism

## Basic details

Name
Heptagonal prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
help
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 prism | Prisms and Antiprisms | Heptagrammic prism =>
Pentagonal prism

## Basic details

Name

Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym

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

<= Triangular prism | Prisms and Antiprisms | Pentagrammic prism =>
Decagonal antiprism

Name
Decagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
dap
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 enneagrammic antiprism | Prisms and Antiprisms | Decagrammic antiprism =>
Pentagrammic retroprism

## Basic details

Name
Pentagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
sterp
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

<= p-gonal antiprism | Prisms and Antiprisms | Great heptagrammic retroprism =>
Icosidodecadodecahedron

## Basic details

Name
Icosidodecadodecahedron
Other names

Number
43
Polyhedron number
U44
Coxeter number
C56
Wenninger number
W83
Bower’s acronym
ided
Schläfli symbol

Wythoff symbol

Conway notation

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

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
9
### Army

General
Rhombidodecadodecahedron
Colonel
Rhombidodecadodecahedron
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
60
Number of edges
120
Number of faces
20 × {6}
12 × {5}
12 ×

{

5
2

}

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

## Schwarz triangle

5/35/3
33
55

## Dual

Name
Medial icosacronic hexecontahedron
Image

Composite image

<= Rhombidodecadodecahedron | Trapezivert polyhedra | Rhombicosahedron =>
Decagonal prism

## Basic details

Name
Decagonal prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
dip
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

<= Great enneagrammic prism | Prisms and Antiprisms | decagrammic prism =>