recursal/Fanatic-Fandom huggingface dataset:
Icositetrachoron regiment

## Icoic symmetry

*4. Icositetrachoron (ico)*

60. Icositetrahemicositetrachoron (**ihi**)

## Tessic symmetry

61. Octahemihexadecachoron (**oh**)

62. Small hexadecahemihexadecachoron (**shahoh**)

63. Hexadecahemihexadecachoron (**huhoh**)

64. Hexadecoctahemihexadecachoron (**hohoh**)

65. Octahexadecahemihexadecachoron (**ohuhoh**)

66. Great hexadecahemihexadecachoron (**ghahoh**)

## Demitessic symmetry

67. Rectified tesseractihemioctachoron (**ratho**)

68. Octadishemioctachoron (**odho**)

69. Hexadecoctadishemioctachoron (**hodho**)

70. Hexadecoctahemihexadecoctintercepted hemioctachoron (**hoh honho**)

71. Disoctadishemioctintercepted hemioctachoron (**dod honho**)

72. Disoctahemihexadecintercepted hemioctachoron (**doh honho**)

<= Triangular rectates | Polytopes Wiki | Pentagonal rectates =>
Hexacosichoron

## Basic details

Name
Hexacosichoron
Other names
600-cell
Number
6
Category
1
Bower’s acronym
ex
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

<= Hecatonicosachoron | Regular polychora | Faceted hexacosachoron =>
Povray code-tet-sym.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/>
 \*/

/\* Details for tetrahedral symmetry \*/

#declare Root2 = 0.707106781186548;

#declare NumTetVertices = 4;
#declare TetVertices = array[NumTetVertices] {
 < Root2, Root2, Root2>,
 <-Root2, -Root2, Root2>,
 <-Root2, Root2, -Root2>,
 < Root2, -Root2, -Root2>
};
#declare TetCircumradius = sqrt(6) / 2;

#declare NumTetEdges = 6;
#declare TetEdges = array[NumTetEdges][2] {
 {0, 1}, {0, 2}, {0, 3},
 {1, 2}, {1, 3}, {2, 3}
};

#declare NumTetFaces = 4;
#declare TetFaces = array[NumTetFaces][3] {
 {0, 1, 2},
 {0, 1, 3},
 {0, 2, 3},
 {1, 2, 3}
};

#declare TetFaceEdges = array[NumTetFaces][4];
#for (F, 0, NumTetFaces - 1, 1)
 #local NextV = 0;
 #for (E, 0, NumTetEdges - 1, 1)
 #if (TetEdges[E][0] = F | TetEdges[E][1] = F)
 #declare TetFaceEdges[F][NextV] = E;
 #declare NextV = NextV + 1;
 #end
 #end
#end

```
Great dirhombicosidodecahedron

Name
Great dirhombicosidodecahedron
Other names

Number
75
Polyhedron number
U75
Coxeter number
C92
Wenninger number
W119
Bower’s acronym
gidrid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

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}
24 ×

{

5
2

}

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

## Dual

Name
Great dirhombicosidodecacron
Image

Composite image

<= Small inverted retrosnub icosicosidodecahedron | Snub polyhedra | Triangular prism =>
Quasitruncated dodecadodecahedron

Name
Quasitruncated dodecadodecahedron
Other names

Number
60
Polyhedron number
U59
Coxeter number
C75
Wenninger number
W98
Bower’s acronym
quitdid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
x^x\*’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
120
Number of edges
180
Number of faces
30 × {4}
12 × {10}
12 ×

{

10
3

}

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

## Schwarz triangle

22
5/25/2
55

## Dual

Name
Medial disdyakis triacontahedron
Image

Composite image

<= Quasitruncated cuboctahedron | Omnitruncated polyhedra | Great quasitruncated icosidodecahedron =>
Prisms and Antiprisms

There are an infinite number of prisms and antiprisms, one prism and one antiprism for each regular polygon. On top of that, if

3
2

≤
p
≤
2

{\displaystyle \frac{3}{2} \le p \le 2}

, where p is the number of sides divided by the density of the polygon, there is a retroprism.

## Prisms

P1. Triangular prism (**trip**)

*2. Cube (cube)*

P2. Pentagonal prism (**pip**)

P3. Pentagrammic prism (**stip**)

P5. Hexagonal prism (**hip**)

P6. Heptagonal prism (**hep**)

P7. Heptagrammic prism (**ship**)

P8. Great heptagrammic prism (**giship**)

P9. Octagonal prism (**op**)

P9. Octagrammic prism (**stop**)

P10. Enneagonal prism (**ep**)

P11. Enneagrammic prism (**step**)

P12. Great enneagrammic prism (**gistep**)

P13. Decagonal prism (**dip**)

P14. Decagrammic prism (**stiddip**)

…

P(p) p-gonal prism

…

## Antiprisms

*1. Octahedron (oct)*

AP1. Square antiprism (**squap**)

AP2. Pentagonal antiprism (**pap**)

AP3. Pentagrammic antiprism (**stap**)

AP4. Hexagonal antiprism (**hap**)

AP5. Heptagonal antiprism (**heap**)

AP6. Heptagrammic antiprism (**sithap**)

AP7. Great heptagrammic antiprism (**gisthap**)

AP8. Octagonal antiprism (**oap**)

AP9. Octagrammic antiprism (**stoap**)

AP10. Enneagonal antiprism (**eap**)

AP11. Enneagrammic antiprism (**steap**)

AP12. Great enneagrammic antiprism (**gisteap**)

AP13. Decagonal antiprism (**dap**)

AP14. Decagrammic antiprism (**stiddap**)

…

AP(p). p-gonal antiprism

…

## Retroprisms

RP1. Pentagrammic retroprism (**sterp**)

RP2. Great heptagrammic retroprism (**gisthirp**)

RP3. Octagrammic retroprism (**storp**)

RP4. Great enneagrammic retroprism (**gisterp**)

…

RP(p). p-gonal retroprism

…

<= Snub polyhedra | Polytopes Wiki | Polyhedron degenerates =>
Heptagrammic prism

## Basic details

Name
Heptagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
ship
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

<= Heptagonal prism | Prisms and Antiprisms | Great heptagrammic prism =>
Small ditrigonary icosidodecahedron

Name
Small ditrigonary icosidodecahedron
Other names

Number
33
Polyhedron number
U30
Coxeter number
C39
Wenninger number
W70
Bower’s acronym
sidtid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(ox\*o)
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
20
Number of edges
6
Number of faces
12 × {5}
12 ×

{

5
2

}

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

## Schwarz triangle

5/25/2
33
33

5/25/2
5/25/2
5/25/2

## Dual

Name
Small triambic icosahedron
Image

Composite image

<= Great icosihemidodecahedron | Quasiregular polyhedra | Ditrigonary dodecadodecahedron =>
Rhombicosahedron

## Basic details

Name
Rhombicosahedron
Other names

Number
44
Polyhedron number
U56
Coxeter number
C72
Wenninger number
W96
Bower’s acronym
ri
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
6
### 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}
30 × {4}

## Dual

Name
Rhombicosacron
Image

Composite image

<= Icosidodecadodecahedron | Trapezivert polyhedra | Great cubicuboctahedron =>
Miscellaneous polychora 1

## Antiprisms

963. Grand antiprism (**gap**)

964. Pentagrammattic double antiprismoid (**padiap**)

965. Great duoantiprism (**gudap**)

966. Small ditrigonary icosidodecahedron (sidtid) antiprism (**sidtidap**)

967. Ditrigonary dodecadodecahedron (ditdid) antiprism (**ditdidap**)

968. Great ditrigonary icosidodecahedron (gidtid) antiprism (**gidtidap**)

## Snubs

969. Snub disicositetrachoron (**sadi**)

970. Retrosnub disicositetrachoron (**rasdi**)

971. Retroantiprismatosnub disicositetrachoron (**rappisdi**)

972. Small retrohemiantiprismatosnub prismatodisicositetrachoron (**sirhapsippady**)

973. Great retrohemiantiprismatosnub prismatodisicositetrachoron (**girhapsippady**)

974. Retroantiprismatosnub dishecatonicosachoron (**rapsady**)

## Swirlprisms

975. Small swirlprism (**sisp**)

976. Great swirlprism (**gisp**)

977. Swirlprismatic hecatonicosahexacosachoron (**sphix**)

978. Swirlprismatic dicositetracontatetracosioctacontachoron (**spiddit**)

979. Small swirlprismatic tricosihexacontatricosihexacontachoron (**sispatit**)

980. Great swirlprismatic tricosihexacontatricosihexacontachoron (**gispatit**)

981. Swirlprismatic tetracosioctacontadicositetracontachoron (**spittid**)

982. Swirlprismatic hexacosahecatonicosachoron (**spaxhi**)

## Inverted quasiprismatodishexadecachoron type

983. Inverted quasiprismatodishexadecachoron (**iquipadah**)

984. Great quasiprismatodishexadecachoron (**gaquipadah**)

## Octagonal spinoduoprism type

*see Miscellaneous polychora B.*

<= Polychoral prisms | Polytopes Wiki | Prismated dishecatonicosachoron regiment =>
Pentachoron

## Basic details

Name
Pentachoron
Other names
4D simplex
5-cell
Number
1
Category
1
Bower’s acronym
pen
Sequence number
α4
Conway notation

Polychoron notation

Coxeter diagram

Dynkin symbol

Vertex figure

Flags
Convex
### 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
Enneagonal prism

## Basic details

Name
Enneagonal prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
ep
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 prism | Prisms and Antiprisms | Enneagrammic prism =>
Small cubicuboctahedron

Name
Small cubicuboctahedron
Other names

Number
37
Polyhedron number
U13
Coxeter number
C38
Wenninger number
W69
Bower’s acronym
socco
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
x’o’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
24
Number of edges
48
Number of faces
8 × {3}
6 × {4}
6 × {8}
## Schwarz triangle

3/23/2
44
44

## Dual

Name
Small hexacronic icositetrahedron
Image

Composite image

<= Small rhombicuboctahedron | Trapezivert polyhedra | Small rhombihexahedron =>
Povcode:Tet

```
/* Tetrahedron */
#version 3.7;
#include "functions.inc"
#include "golds.inc"
#include "metals.inc"
#include "glass.inc"
#include "textures.inc"
#include "colors.inc"

#include "..\\tet-sym.inc"
#declare NumVertices = NumTetVertices;
#declare Vertices = TetVertices;
#declare Circumradius = TetCircumradius;
#declare NumEdges = NumTetEdges;
#declare Edges = TetEdges;
#declare NumFaces = NumTetFaces;
#declare Faces = TetFaces;

#include "..\\stats.inc"
#include "..\\scene.inc"
```
Icosidodecatruncated icosidodecahedron

## Basic details

Name
Icosidodecatruncated icosidodecahedron
Other names

Number
63
Polyhedron number
U45
Coxeter number
C57
Wenninger number
W84
Bower’s acronym
idtid
Schläfli symbol

Wythoff symbol

Conway notation

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

Dynkin symbol

Vertex figure

Kernel

Flags

Density

Genus
9
### Army

General
Icosidodecatruncated icosidodecahedron
Colonel
Icosidodecatruncated 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
120
Number of edges
180
Number of faces
20 × {6}
12 × {10}
12 ×

{

10
3

}

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

## Schwarz triangle

5/35/3
33
55

## Dual

Name
Tridyakis icosahedron
Image

Composite image

<= Cuboctatruncated cuboctahedron | Omnitruncated polyhedra | Snub cube =>
Blend polychora

## Normal cases

1462. Small blendic disprismatodishecatonicosachoron (**sabbadipady**)

1463. Great blendic disprismatodishecatonicosachoron (**gabbadipady**)

1464. Small blendic prismatodishecatonicosachoron (**sabippady**)

1465. Great blendic prismatodishecatonicosachoron (**gabippady**)

1466. Small blendic trishecatonicosachoron (**sabbathi**)

1467. Great blendic trishecatonicosachoron (**gabbathi**)

1468. Small spinoblendic prismatodishecatonicosachoron (**sanbippady**)

1469. Great spinoblendic prismatodishecatonicosachoron (**ganbippady**)

## Quasitruncated dodecadodecahedron cases

1470. Small spinoblendic trishecatonicosachoron (**sanbathi**)

1471. Great spinoblendic trishecatonicosachoron (**ganbathi**)

1472. Blendic tetrakishecatonicosachoron (**bathi**)

1473. Blendic dishecatonicosachoron (**badohi**)

<= Great tritrigonary hexacosatrishecatonicosachoron regiment | Polytopes Wiki | Disnubs =>
Great inverted retrosnub icosidodecahedron

Name
Great inverted retrosnub icosidodecahedron
Other names

Number
73
Polyhedron number
U74
Coxeter number
C90
Wenninger number
W117
Bower’s acronym
girsid
Schläfli symbol

Wythoff symbol

Conway notation
s\*’s,s
Polyhedron symbol

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\}}

## Dual

Name
Great pentagrammic hexecontahedron
Image

Composite image

<= Great snub dodecicosidodecahedron | Snub polyhedra | Small inverted retrosnub icosicosidodecahedron =>
Compounds

There are two types of compounds here:

* a number of congruent polyhedra having the vertices of another polyhedron, and
* a polyhedron and its dual.

## Polyhedra with duals

* a cube and an octahedron
* a dodecahedron and an icosahedron

## Congruent polyhedra

* Stella octangula (Two tetrahedra in a cube)
* Five cubes in a dodecahedron File:5cube-thumb.gif
* Five tetrahedra in a dodecahedron File:5tet-thumb.gif
* Ten tetrahedra in a dodecahedron File:10tet-thumb.gif

<= Degenerates | Polytopes Wiki | Polyhedron stellations =>
Great ditrigonary dodecicosidodecahedron

Name
Great ditrigonary dodecicosidodecahedron
Other names

Number
48
Polyhedron number
U42
Coxeter number
C54
Wenninger number
W81
Bower’s acronym
gidditdid
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
3

}

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

## Schwarz triangle

5/35/3
33
55

## Dual

Name
Great ditrigonal dodecacronic hexecontahedron
Image

Composite image

<= Great rhombihexahedron | Trapezivert polyhedra | Great icosicosidodecahedron =>
Hendecagon

Name
Hendecagon
Schläli symbol
{11}
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

ξ

−
1

l

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

Circumradius

cosec
⁡

(

π
11

)

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

Circumference

22
l

{\displaystyle 22 l}

Area

11

ξ

−
1

l

2

{\displaystyle 11\xi ^{-1}l^{2}}

Internal angle

9
π

11

rad

{\displaystyle {\frac {9\pi }{11}}\operatorname {rad} }

 = 147 3/11°

<= Decagram | 0D – 2D polytopes | Hendecagram =>
Great dodecahedron

## Basic details

Name
Great dodecahedron
Other names

Conjugate
Small stellated dodecahedron
Number
6
Mæder number
U35
Coxeter number
C44
Wenninger number
W21 (page 39)
Bower’s acronym
gad (*GAD*)
Schläfli symbol

{

5
,

5
2

}

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

Wythoff symbol
5/2 | 2 5
Conway notation
gT
Polyhedron symbol
o\*o^x
Coxeter diagram

Dynkin symbol
x5/2o5o
Vertex description

5

5
2

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

Vertex figure

Kernel
Dodecahedron
Flags
Regular
Nonconvex
Colonel
Orientable
Achiral
Density
3
Genus
4
### Army

Regiments in army
Icosahedron (general)
Small stellated dodecahedron
Army vertex figure
Polyhedra in regiment
Icosahedron (colonel)
**Great dodecahedron**
Regiment vertex figure

## Metrics

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

Circumradius, *rc*

ϕ

5

l

{\displaystyle \sqrt{\phi \sqrt{5}} l}

Midradius, *rm*

ϕ
l

{\displaystyle \phi l}

Inradii, *ri*

Dihedral angles

Vertex volume

Coordinates
Even permutations of

(

±
ϕ
,

±
1
,

0

)

{\displaystyle \left(\pm\phi,\ \pm1,\ 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
12 × {5}
Number of pieces
## Schwarz triangle

22
5/25/2
55

## Dual

Name
Small stellated dodecahedron
Image

<= Icosahedron | Regular polyhedra | Small stellated dodecahedron =>
Enneagonal antiprism

## Basic details

Name
Enneagonal antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
eap
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 antiprism | Prisms and Antiprisms | Enneagrammic antiprism =>
Decagon

Name
Decagon
Other name
Truncated pentagon
Schläli symbols
{10}
t{5}
Coxeter diagrams

## Measurements

In the following,

l

{\displaystyle l}

 is the semiedge length, and the golden section

# ϕ

1
s
q
r
t

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
l

{\displaystyle 2\phi l}

Circumference

20
l

{\displaystyle 20l}

Area

2

ϕ

3
2

×

5

5
4

l

2

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

Internal angle

4
π

5

rad

{\displaystyle {\frac {4\pi }{5}}\operatorname {rad} }

 = 144°

<= Great nonagram | 0D – 2D polytopes | Decagram =>
Quasitruncated great stellated dodecahedron

Name
Quasitruncated great stellated dodecahedron
Other names
Great stellated truncated dodecahedron
Conjugate
Truncated dodecahedron
Number
19
Mæder number
U66
Coxeter number
C83
Wenninger number
W104 (page 161)
Bower’s acronym
quit gissid (*quit GIS sid*)
Schläfli symbol

Wythoff symbol
2 3 | 5/3
Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Vertex description
10/32.3
Vertex figure

Kernel

Flags
Archimedean
Nonconvex
General
Colonel
Orientable
Achiral
Density
13
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
90
Number of faces
20 × {3}
12 ×

{

10
3

}

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

Number of pieces

## Schwarz triangle

22
5/25/2
33

## Dual

Name
Great triakis icosahedron
Image

Composite image

<= Quasitruncated small stellated dodecahedron | Truncated polyhedra | Tetrahemihexahedron =>
Code

Various code has been used to generate these images. They can be found from the links below:

## Povray includes

### General includes

* dump.inc
* macros.inc
* poly-texture.inc
* scene.inc
* stats.inc

### Code for different families

#### Prisms & antiprisms

* ap.inc Used for the antiprisms
* prism.inc

#### Regular symmetries

* ike-sym.inc
* oct-sym.inc
* tet-sym.inc

#### Regiments

* co-reg.inc
* did-reg.inc
* gaddid-reg.inc
* gidditdid-reg.inc
* gid-reg.inc
* gocco-reg.inc
* id-reg.inc
* raded-reg.inc
* sidtid-reg.inc
* siid-reg.inc
* sirco-reg.inc
* srid-reg.inc
* tet-compound-reg.inc

#### Johnson solid

* j-aug.inc
* j-doe.inc
* j-ike.inc
* j-prism.inc
* j-srid.inc
* j-tic.inc
* j-tid.inc
* j-tut.inc

## Povray sources

### Prisms

* trip.pov
* pip.pov
* trip.pov
* hip.pov
* hep.pov
* ship.pov
* giship.pov
* op.pov
* stop.pov
* ep.pov
* step.pov
* gistep.pov
* dip.pov
* stiddip.pov

### Antiprisms

* squap.pov
* pap.pov
* hap.pov
* heap.pov
* oap.pov
* eap.pov
* dap.pov
* stap.pov
* sithap.pov
* steap.pov
* stiddap.pov
* sterp.pov
* gisthap.pov
* stoap.pov
* gisthirp.pov
* gisteap.pov
* storp.pov
* gisterp.pov

### Regulars

* tet.pov
* cube.pov
* oct.pov
* doe.pov
* ike.pov
* gad.pov
* sissid.pov
* gike.pov
* gissid.pov

### Truncates

* tut.pov
* tic.pov
* toe.pov
* tid.pov
* ti.pov
* tigid.pov
* tiggy.pov
* quith.pov
* quit-sissid.pov
* quit-gissid.pov

### Quasiregulars

* thah.pov
* co.pov
* oho.pov
* cho.pov
* id.pov
* sidhid.pov
* seihid.pov
* did.pov
* sidhei.pov
* gidhei.pov
* gid.pov
* gidhid.pov
* geihid.pov
* sidtid.pov
* ditdid.pov
* gidtid.pov

## Bash shell scripts

render.sh Used to generate all the images

## C code

<= Goursat tetrahedra | Polytopes Wiki | =>
Icosahedron

Name
Icosahedron
Other names
Biaugmented pentagonal antiprism
Snub tetrahedron
Snub tetratetrahedron
Conjugate
Great icosahedron
Number
5
Mæder number
U22
Coxeter number
C25
Wenninger number
W4 (page 17)
Bower’s acronym
ike (*IKE*)
Schläfli symbol
{3, 5}
s{3, 4}
sr{3, 3}
s

{

3

3

}

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

Wythoff symbol
5 | 2 3
Conway notation
I
sT
Polyhedron symbol
xo^o
Coxeter diagram

Dynkin symbol
x3o5o
Vertex description
35
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, *rc*

ϕ

5

l

{\displaystyle \sqrt{\phi \sqrt{5}} l}

Midradius, *rm*

ϕ
l

{\displaystyle \phi l}

Inradii, *ri*

ϕ

2

3

3

l

{\displaystyle {\frac {\phi ^{2}{\sqrt {3}}}{3}}l}

Dihedral angles

Vertex volume

Coordinates
Even permutations of

(

±
ϕ
,

±
1
,

0

)

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

where

# ϕ

1

5

2

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

Surface area

20

3

l
2

{\displaystyle 20\sqrt{3}l^2}

Volume

20

ϕ

2

l

3

3

{\displaystyle {\frac {20\phi ^{2}l^{3}}{3}}}

## Components

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

22
33
55

33
33
22

## Dual

Name
Dodecahedron
Image

Composite image

## Stellations

Facial planes
1
Number of stellations
7 (Primary)
8 (Mainline)
11 (Monoacral) (1 chiral)
18 (Fully supported) (2 chiral)
59 (Miller’s Rules) (27 chiral)
Click here for the complete list.

## References

1. ↑ Coxeter, H.S.M., et al. *The Fifty-Nine Icosahedra* 3rd ed., Tarquin Publications, 1999

<= Dodecahedron | Regular polyhedra | Great dodecahedron =>
Heptagram

Name
Hexagram
Schläli symbol

{

7
2

}

{\displaystyle \left\{\frac{7}{2}\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
−

ξ
2

2
ξ

l

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

Circumradius

cosec
⁡

(

2
π

7

)

l

{\displaystyle \operatorname{cosec}\left(\frac{2 \pi}{7}\right) l}

Circumference*full*

14
l

{\displaystyle 14 l}

Circumference*shell*

7

(

1

ξ
2

)

l

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

Area*full*

7

(

1
−

ξ
2

)

2
ξ

l
2

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

Area*flat*

7

(

1
−

ξ
4

)

4
ξ

l
2

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

Area*odd*

7
ξ

(

1
−

ξ
2

)

2

l
2

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

Internal angle

3
π

7

rad

{\displaystyle \frac{3\pi}{7} \operatorname{rad}}

 = 77 1/7°

<= Heptagon | 0D – 2D polytopes | Great heptagram =>
Enneagrammic antiprism

Name
Enneagrammic antiprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
steap
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

<= Enneagonal antiprism | Prisms and Antiprisms | Great enneagrammic antiprism =>
Povray code-id-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/>
 \*/

/\* Icosidodecahedron regiment \*/

#include "ike-sym.inc"

// Vertices
#declare NumVertices = NumIkeEdges;
#declare Vertices = array[NumVertices];

#for (V, 0, NumVertices - 1, 1)
 #declare Vertices[V] = IkeVertices[IkeEdges[V][0]] +
 IkeVertices[IkeEdges[V][1]];
#end

#declare Circumradius = 2 \* Phi;

// Edges
#declare NumEdges = 2 \* NumIkeEdges;
#declare Edges = array[NumEdges][2];

#for (F, 0, NumIkeFaces - 1, 1)
 #for (E, 0, 2, 1)
 #declare Edges[3 \* F + E][0] = mod(IkeFaceEdges[F][E], NumVertices);
 #declare Edges[3 \* F + E][1] = mod(IkeFaceEdges[F][mod(E + 1, 3)],
 NumVertices);
 #end
#end

// Full Faces
#local NumDecSides = sqrt(2 \* NumEdges + 1) - 1;
#local NumHalfDecSides = NumDecSides / 2;
#declare PentFullFacesStart = 0;
#declare NumPentFullFaces = NumDoeFullFaces;
#declare TriFullFacesStart = PentFullFacesStart + NumPentFullFaces;
#declare NumTriFullFaces = NumIkeFaces;
#declare DecFullFacesStart = TriFullFacesStart + NumTriFullFaces;
#declare NumDecFullFaces = NumIkeEdges / NumHalfDecSides;
#declare NumFullFaces = DecFullFacesStart + NumDecFullFaces;
#declare FullFaces = array[NumFullFaces][NumDecSides];
#declare FullFaceSizes = array[NumFullFaces];

#include "macros.inc"

// Pentagons
#for (F, 0, NumPentFullFaces - 1, 1)
 #local To = PentFullFacesStart + F;
 #for (V, 0, 4, 1)
 #declare FullFaces[To][V] = mod(DoeFaceEdges[F][V], NumDoeEdges);
 #end
 #declare FullFaceSizes[To] = 5;
#end

// Triangles
#for (F, 0, NumTriFullFaces - 1, 1)
 #local To = TriFullFacesStart + F;
 #for (V, 0, 2, 1)
 #declare FullFaces[To][V] = mod(IkeFaceEdges[F][V], NumIkeEdges);
 #end
 #declare FullFaceSizes[To] = 3;
#end

// Decagons
GetMedialPolygons(DecFullFacesStart, NumDecFullFaces, NumDecSides)

// Faces
#declare NumPentagonFaces = 3 \* NumPentFullFaces;
#declare NumTriangleFaces = NumTriFullFaces;
#declare NumDecagonFaces = (NumDecSides - 2) \* NumDecFullFaces;

#declare PentagonFaces = array[NumPentagonFaces][3];
#declare TriangleFaces = array[NumTriangleFaces][3];
#declare DecagonFaces = array[NumDecagonFaces][3];

TriangulateConvex(PentagonFaces, 0, PentFullFacesStart, NumPentFullFaces, 5)
TriangulateConvex(TriangleFaces, 0, TriFullFacesStart, NumTriFullFaces, 3)
TriangulateConvex(DecagonFaces, 0, DecFullFacesStart, NumDecFullFaces,
 NumDecSides)

```
Great heptagrammic retroprism

## Basic details

Name
Great heptagrammic retroprism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
gisthirp
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 retroprism | Prisms and Antiprisms | Octagrammic retroprism =>
Truncated dodecahedron

## Basic details

Name
Truncated dodecahedron
Other names

Conjugate
Quasitruncated great stellated dodecahedron
Number
13
Mæder number
U26
Coxeter number
C29
Wenninger number
W10 (page 24)
Bower’s acronym
tid (*TID*)
Schläfli symbols
t{5, 3}
t0,1{5, 3}
Wythoff symbol
2 3 | 5
Conway notation
tD
Polyhedron symbol
ox^x
Coxeter diagram

Dynkin symbol
x5x3o
Vertex description
102.3
Vertex figure

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

The truncated dodecahedron forms its own army.

## 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
60
Number of edges
90
Number of faces
20 × {3}
12 × {10}
## Schwarz triangle

22
33
55

## Dual

Name
Triakis icosahedron
Image

Composite image

<= Truncated octahedron | Truncated polyhedra | Truncated icosahedron =>
Great icosicosidodecahedron

Name
Great icosicosidodecahedron
Other names

Number
49
Polyhedra number
U48
Coxeter number
C62
Wenninger number
W88
Bower’s acronym
giid
Schläfli symbol

Wythoff symbol

conway notation

Polyhedron symbol
(x,o^x)
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
60
Number of edges
120
Number of faces
20 × {3}
12 × {5}
20 × {6}
## Schwarz triangle

3/23/2
33
55

## Dual

Name
Great icosacronic hexecontahedron
Image

Composite image

<= Great ditrigonary dodecicosidodecahedron | Trapezivert polyhedra | Great dodecicosahedron =>
Great dodecicosidodecahedron

Name

Other names

Number
54
Polyhedron number
U61
Coxeter number
C77
Wenninger number
W99
Bower’s acronym
gaddid
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
2

}

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

12 ×

{

10
3

}

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

## Schwarz triangle

5/35/3
5/25/2
33

## Dual

Name
Great dodecacronic hexecontahedron
Image

Composite image

<= Small dodecicosahedron | Trapezivert polyhedra | Quasirhombicosidodecahedron =>
Great rhombihexahedron

Name
Great rhombihexahedron
Other names

Number
47
Polyhedron number
U21
Coxeter number
C82
Wenninger number
W103
Bower’s acronym
groh
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

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
24
Number of edges
48
Number of faces
12 × {4}
6 ×

{

8
3

}

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

## Dual

Name
Great rhombihexacron
Image

Composite image

<= Quasirhombicuboctahedron | Trapezivert polyhedra | Great ditrigonary dodecicosidodecahedron =>
Small dodecicosahedron

Name
Small dodecicosahedron
Other names

Number
53
Polyhedron number
U50
Coxeter number
C64
Wenninger number
W90
Bower’s acronym
siddy
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}

## Dual

Name
Small dodecicosacron
Image

Composite image

<= Small ditrigonary dodecicosidodecahedron | Trapezivert polyhedra | Great dodecicosidodecahedron =>
P-gonal prism

## Basic details

Name
p-gonal prism
Other names

Number
P(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 prism | Prisms and Antiprisms | Square antiprism =>
P-gonal retroprism

## Basic details

Name
p-gonal retroprism
Other names

Number
RP(p)
Polyhedron number

Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol

Coxeter diagram

Dynkin symbol

Kernel

Vertex figure

Flags

Density

p
n

−

p
d

{\displaystyle p\_n - 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

2

p
n

{\displaystyle 2 p\_n}

Number of edges

4

p
n

{\displaystyle 4 p\_n}

Number of faces
2 × {p}

2

p
n

{\displaystyle 2 p\_n}

 × {3}

## Dual

Name

Image

Composite image

<= Great enneagrammic retroprism | Prisms and Antiprisms | =>
Great snub icosidodecahedron

Name
Great snub icosidodecahedron
Other names

Number
67
Polyhedron number
U57
Coxeter number
C88
Wenninger number
W116
Bower’s acronym
gosid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
ss\*s
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 pentagonal hexecontahedron
Image

Composite image

<= Snub dodecadodecahedron | Snub polyhedra | Small snub icosicosidodecahedron =>
Ditrigonary dodecadodecahedron

Name
Ditrigonary dodecadodecahedron
Other names

Number
34
Polyhedron number
U41
Coxeter number
C53
Wenninger number
W80
Bower’s acronym
ditdid
Schläfli symbol

Wythoff symbol

Conway number

Polyhedron symbol
(o\*’x^o)
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
20
Number of edges
60
Number of faces
20 × {3}
12 × {5}
## Schwarz triangle

5/35/3
33
55

## Dual

Name
Medial triambic icosahedron
Image

Composite image

<= Small ditrigonary icosidodecahedron | Quasiregular polyhedra | Great ditrigonary icosidodecahedron =>
Hexagonal prism

## Basic details

Name
Hexagonal prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
hip
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
33

## Dual

Name

Image

Composite image

<= Pentagrammic prism | Prisms and Antiprisms | Heptagonal prism =>
Great rhombicosidodecahedron

Name
Great rhombicosidodecahedron
Other names

Number
58
Polyhedron number
U28
Coxeter number
C31
Wenninger number
W16
Bower’s acronym
grid
Schläfli symbol
r

{

5

3

}

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

Wythoff symbol

Conway notation

Polyhedron symbol
xx^x
Coxeter diagram

Dynkin symbol

Vertex figure

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
120
Number of edges
180
Number of faces
30 × {4}
20 × {6}
12& × {10}
## Schwarz triangle

22
33
55

## Dual

Name
Disdyakis triacontahedron
Image

Composite image

<= Great rhombicuboctahedron | Omnitruncated polyhedra | Quasitruncated cuboctahedron =>
Great hecatonicosachoron

## Basic details

Name
Great hecatonicosachoron
Other names

Number
8
Category
1
Bower’s acronym
gohi
Schläfli symbol

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

<= Faceted hexacosachoron | Regular polychora | Grand hecatonicosachoron =>
Great enneagrammic prism

## Basic details

Name
Great enneagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
gistep
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 prism | Prisms and Antiprisms | Decagonal prism =>
Snub icosidodecadodecahedron

Name
Snub icosidodecadodecahedron
Other names

Number
71
Polyhedron number
U46
Coxeter number
C58
Wenninger number
W112
Bower’s acronym
sided
Schläfli symbol

Wythoff symbol

conway notation

Polyhedron symbol
(s\*’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}
12 × {5}
12 ×

{

5
2

}

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

## Schwarz triangle

5/35/3
33
55

## Dual

Name
Medial hexagonal hexecontahedron
Image

Composite image

<= Great inverted snub icosidodecahedron | Snub polyhedra | Great snub dodecicosidodecahedron =>
Hecatonicosachoron

## Basic details

Name
Hecatonicosachoron
Other names
120-cell
Number
5
Category
1
Bower’s acronym
hi
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

## Dual

Name
Hexacosichoron
## Cross sections

<= Icositetrachoron | Regular polychora | Hexacosichoron =>
Pentagrammic prism

## Basic details

Name
Pentagrammic prism
Other names

Number

Polyhedron number

Coxeter number

Wenninger number

Bower’s acronym
stip
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 prism | Prisms and Antiprisms | Hexagonal prism =>
Small icosicosidodecahedron

Name
Small icosicosidodecahedron
Other names

Number
51
Polyhedron number
U31
Coxeter number
C40
Wenninger number
W71
Bower’s acronym
siid
Schläfli symbol

Wythoff symbol

Conway notation

Polyhedron symbol
(xo\*x)
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
60
Number of edges
120
Number of faces
20 × {3}
20 × {6}
12 ×

{

5
2

}

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

## Schwarz triangle

5/25/2
33
33

## Dual

Name
Small icosacronic hexecontahedron
Image

Composite image

<= Great dodecicosahedron | Trapezivert polyhedra | Small ditrigonary dodecicosidodecahedron =>