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Encyclopedia > List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book "Polyhedron Models", by Magnus J. Wenninger. ©1971


It contains the 75 nonprismatic uniform polyhedra, as well as 44 stellated forms of the convex polyhedra. // Uniform polyhedra and tessellations The following list contains ALL 75 uniform polyhedra that are not prisms or antiprisms, 11 uniform tessellations in the plane, and some samplings of the infinite set of prisms and antiprisms. ... Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or in general new polytopes in n dimensions. ...


The polyhedra are grouped below in 5 tables: Regular (1-5), Semiregular (6-18), Nonconvex regular (20-22,41), Stellations and compounds (19-66), and nonconvex uniform (67-119).


The four nonconvex regular polyhedra are listed twice because they belong to both the uniform polyhedra and stellation groupings.

Contents


Platonic solids (Regular) W1 to W5

Index Name Picture Wythoff
Symbol		 Schläfli symbol Symmetry
group		 U# K# V E F Faces by type
1 Tetrahedron 3|2 3
{3,3}		 Td U01 K06 4 6 4 4{3}
2 Octahedron 4|2 3
{3,4}		 Oh U05 K10 6 12 8 8{3}
3 Hexahedron (Cube) 3|2 4
{4,3}		 Oh U06 K11 8 12 6 6{4}
4 Icosahedron 5|2 3
{3,5}		 Ih U22 K27 12 30 20 20{3}
5 Dodecahedron 3|2 5
{5,3}		 Ih U23 K28 20 30 12 12{5}

A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Download high resolution version (643x607, 26 KB)Tetrahedron, made by me using POV-Ray, see image:poly. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Download high resolution version (862x862, 41 KB)Octahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Octahedron_vertfig. ... A hexahedron is a polyhedron with 6 faces. ... Download high resolution version (742x826, 50 KB)Hexahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Cube_vertfig. ... An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... Download high resolution version (819x791, 71 KB)Icosahedron, made by me using POV-Ray, see image:poly. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... Download high resolution version (847x829, 63 KB)Dodecahedron, made by me using POV-Ray, see image:poly. ...

Archimedean solids (Semiregular) W6 to W18

Index Name Picture Wythoff
symbol		 Vertex figure Symmetry
group		 U# K# V E F Faces by type
6 Truncated tetrahedron 2 3|3
3.6.6		 Td U02 K07 12 18 8 4{3}+4{6}
7 Truncated octahedron 2 4|3
4.6.6		 Oh U08 K13 24 36 14 6{4}+8{6}
8 Truncated hexahedron 2 3|4
3.8.8		 Oh U09 K14 24 36 14 8{3}+6{8}
9 Truncated icosahedron 2 5|3
5.6.6		 Ih U25 K30 60 90 32 12{5}+20{6}
10 Truncated dodecahedron 2 3|5
3.10.10		 Ih U26 K31 60 90 32 20{3}+12{10}
11 Cuboctahedron 2|3 4
3.4.3.4		 Oh U07 K12 12 24 14 8{3}+6{4}
12 Icosidodecahedron 2|3 5
3.5.3.5		 Ih U24 K29 30 60 32 20{3}+12{5}
13 Small rhombicuboctahedron 3 4|2
3.4.4.4		 Oh U10 K15 24 48 26 8{3}+(6+12){6}
14 Small rhombicosidodecahedron 3 5|2
3.4.5.4		 Ih U27 K32 60 120 62 20{3}+30{4}+12{5}
15 Great rhombicuboctahedron
(Rhombitruncated cuboctahedron)
(Truncated cuboctahedron)		 2 3 4|
4.6.8		 Oh U11 K16 48 72 26 12{4}+8{6}+6{8}
16 Great rhombicosidodecahedron
(Rhombitruncated icosidodecahedron)
(Truncated icosidodecahedron)		 2 3 5|
4.6.10		 Ih U28 K33 120 180 62 30{4}+20{6}+12{10}
17 Snub cube |2 3 4
3.3.3.3.4		 O U12 K17 24 60 38 (8+24){3}+6{4}
18 Snub dodecahedron |2 3 5
3.3.3.3.5		 I U29 K34 60 150 92 (20+60){3}+12{5}

An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone... The truncated tetrahedron is an Archimedean solid. ... Download high resolution version (867x773, 50 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... Download high resolution version (857x789, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Download high resolution version (819x855, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated icosahedron is an Archimedean solid. ... Download high resolution version (878x874, 80 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated dodecahedron is an Archimedean solid. ... Download high resolution version (876x802, 67 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Download high resolution version (818x804, 71 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... Image File history File links Cuboctahedron_vertfig. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Download high resolution version (841x861, 76 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Download high resolution version (823x836, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (858x871, 113 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Download high resolution version (855x868, 83 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Download high resolution version (868x869, 105 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... Download high resolution version (843x833, 97 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, usually regarded as a truncated polyhedron derived by truncating either a dodecahedron or an icosahedron. ... Download high resolution version (872x877, 116 KB)Somethingahedron, made by me using POV-Ray, see image:poly. ...

Kepler-Poinsot solids (Nonconvex regular) W20,W21,W22 and W41

Index Name Picture Wythoff
Symbol		 Schläfli symbol Symmetry
group		 U# K# V E F Faces by type
20 Small stellated dodecahedron 5|25/2
{5/2,5}		 Ih U34 K39 12 30 12 12{5/2}
21 Great dodecahedron 5/2|2 5
{5,5/2}		 Ih U35 K40 12 30 12 12{5}
22 Great stellated dodecahedron 3|25/2
{5/2,3}		 Ih U52 K57 20 30 12 12{5/2}
41 Great icosahedron
(16th stellation of icosahedron)		 5/2|2 3
{3,5/2}		 Ih U53 K58 12 30 20 20{3}

A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... Image File history File links Download high resolution version (639x641, 18 KB) Summary Small stellated dodecahedron, U34 Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ...

Stellations models W19 to W66

Stellation of octahedron

Index Name Symmetry
group		 Picture Facets
2 Octahedron
(regular)		 Oh
19 Stellated octahedron
(Compound of two tetrahedra)		 Oh

An octahedron (plural: octahedra) is a polyhedron with eight faces. ...

Stellations of dodecahedron

Index Name Symmetry
group		 Picture Facets
5 Dodecahedron (regular) Ih
20 Small stellated dodecahedron (regular)
(First stellation of dodecahedron)		 Ih
21 Great dodecahedron (regular)
(Second stellation of dodecahedron)		 Ih
22 Great stellated dodecahedron (regular)
(Third stellation of dodecahedron)		 Ih

A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In geometry, the small stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ...

Stellations of icosahedron

Index Name Symmetry
group		 Picture Facets
4 Icosahedron(regular) Ih
23 Compound of five octahedra
(First compound stellation of icosahedron)		 Ih
24 Compound of five tetrahedra
(Second compound stellation of icosahedron)		 I
25 Compound of ten tetrahedra
(Third compound stellation of icosahedron)		 I
26 Triakis icosahedron
(First stellation of icosahedron)		 Ih
27 Second stellation of icosahedron Ih
28 Third stellation of icosahedron Ih
29 Fourth stellation of icosahedron Ih
30 Fifth stellation of icosahedron Ih
31 Sixth stellation of icosahedron Ih
32 Seventh stellation of icosahedron Ih
33 Eighth stellation of icosahedron Ih
34 Ninth stellation of icosahedron Ih
35 Tenth stellation of icosahedron I
36 Eleventh stellation of icosahedron I
37 Twelfth stellation of icosahedron Ih
38 Thirteenth stellation of icosahedron I
39 Fourteenth stellation of icosahedron I
40 Fifteenth stellation of icosahedron I
41 Great icosahedron(regular)
(Sixteenth stellation of icosahedron)		 Ih
42 Seventeenth stellation of icosahedron Ih

An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... A triakis icosahedron is a catalan solid, the dual polyhedron to the truncated icosahedron. ... The second stellation of icosahedron is a polyhedron created from the icosahedron. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ...

Stellations of cuboctahedron

Index Name Symmetry
group		 Picture Facets (triangle) Facets (square)
11 Cuboctahedron
(regular)		 Oh
43 Compound of cube and octahedron
(First stellation of cuboctahedron)		 Oh
44 Second stellation of cuboctahedron Oh
45 Third stellation of cuboctahedron Oh
46 Fourth stellation of cuboctahedron Oh

A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...

Stellations of icosidodecahedron

Index Name Symmetry
group		 Picture Facets
Triangle		 Facets
Pentagon
12 Icosidodecahedron
(regular)		 Ih
47 (First stellation of icosidodecahedron)
Compound of dodecahedron and icosahedron		 Ih
48 Second stellation of icosidodecahedron Ih
49 Third stellation of icosidodecahedron Ih
50 Fourth stellation of icosidodecahedron
(Compound of small stellated dodecahedron
and triakis icosahedron)		 Ih
51 Fifth stellation of icosidodecahedron
(Compound of small stellated dodecahedron
and five octahedra)		 Ih
52 Sixth stellation of icosidodecahedron Ih
53 Seventh stellation of icosidodecahedron Ih
54 Eighth stellation of icosidodecahedron
(Compound of five tetrahedra
and great dodecahedron)		 I
55 Ninth stellation of icosidodecahedron Ih
56 Tenth stellation of icosidodecahedron Ih
57 Eleventh stellation of icosidodecahedron Ih
58 Twelfth stellation of icosidodecahedron Ih
59 Thirteenth stellation of icosidodecahedron Ih
60 Fourteenth stellation of icosidodecahedron Ih
61 Compound of
great stellated dodecahedron
and great icosahedron		 Ih
62 Fifteenth stellation of icosidodecahedron Ih
63 Sixteenth stellation of icosidodecahedron Ih
64 Seventeenth stellation of icosidodecahedron Ih
65 Eighteenth stellation of icosidodecahedron Ih
66 Nineteenth stellation of icosidodecahedron Ih

An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... In geometry, the great stellated dodecahedron is a Kepler-Poinsot solid. ... In geometry, the great icosahedron is a Kepler-Poinsot solid. ... Image File history File links Download high resolution version (900x900, 62 KB) Licensing This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ...

Uniform nonconvex solids W67 to W119

Index Name Picture Wythoff
Symbol		 Vertex figure Symmetry
group		 U# K# V E F Faces by type
67 Tetrahemihexahedron 3/23|2
4.3/2.4.3		 Oh U04 K09 6 12 7 4{3}+3{4}
68 Octahemioctahedron 3/23|3
6.3/2.6.3		 Oh U03 K08 12 24 12 8{3}+4{6}
69 Small cubicuboctahedron 3/24|4
8.3/2.8.4		 Oh U13 K18 24 48 20 8{3}+6{4}+6{8}
70 Small ditrigonal icosidodecahedron 3|5/23
(5/2.3)3		 Ih U30 K35 20 60 32 20{3}+12{5/2}
71 Small icosicosidodecahedron 5/23|3
6.5/2.6.3		 Ih U31 K36 60 120 52 20{3}+12{5/2}+20{6}
72 Small dodecicosidodecahedron 3/25|5
10.3/2.10.5		 Ih U33 K38 60 120 44 20{3}+12{5}+12{10}
73 Dodecadodecahedron 2|5/25
(5/2.5)2		 Ih U36 K41 30 60 24 12{5}+12{5/2}
74 Small rhombidodecahedron 25/25|
10.4.10/9.4/3		 Ih U39 K44 60 120 42 30{4}+12{10}
75 Truncated great dodecahedron 25/2|5
10.10.5/2		 Ih U37 K42 60 90 24 12{5/2}+12{10}
76 Rhombidodecadodecahedron 5/25|2
4.5/2.4.5		 Ih U38 K43 6