List of uniform polyhedra by vertex figure

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Polyhedron
Class Number and properties
Platonic solids
(5, convex, regular)
Archimedean solids
(13, convex, uniform)
Kepler–Poinsot polyhedra
(4, regular, non-convex)
Uniform polyhedra
(75, uniform)
Prismatoid:
prisms, antiprisms etc.
(4 infinite uniform classes)
Polyhedra tilings (11 regular, in the plane)
Quasi-regular polyhedra
(8)
Johnson solids (92, convex, non-uniform)
Pyramids and Bipyramids (infinite)
Stellations Stellations
Polyhedral compounds (5 regular)
Deltahedra (Deltahedra,
equalatial triangle faces)
Snub polyhedra
(12 uniform, not mirror image)
Zonohedron (Zonohedra,
faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron (infinite)
Catalan solid (13, Archimedean dual)

There are many relations among the uniform polyhedron. Some are obtained by truncating the vertices of the regular or quasi-regular polyhedron. Others share the same vertices and edges as other polyhedron. The grouping below exhibit some of these relations.

The vertex figure of a polyhedron[edit]

The relations can be made apparent by examining the vertex figures. obtained by listing the faces adjacent to each vertex (remember that for uniform polyhedra all vertices are the same, that is vertex-transitive). For example the cube has vertex figure 4.4.4 that is three adjacent square faces. The possible faces are

  • 3 - equilateral triangle
  • 4 - square
  • 5 - regular pentagon
  • 6 - regular hexagon
  • 8 - regular octagon
  • 10 - regular decagon
  • 5/2 - pentagram
  • 8/3 - octagram
  • 10/3 - decagram

Some faces will appear with reverse orientation which is written here as

  • -3 - a triangle with reverse orientation (often written as 3/2)

Others pass through the origin which we write as

  • 6* - hexagon passing through the origin

The Wythoff symbol relates the polyhedron to spherical triangles. Wythoff symbols are written p|q r, p q|r, p q r| where the spherical triangle has angles π/p,π/q,π/r, the bar indicates the position of the vertices in relation to the triangle.

Example vertex figures

Johnson (2000) classified uniform polyhedra according to the following:

  1. Regular (regular polygonal vertex figures): pq, Wythoff symbol q|p 2
  2. Quasi-regular (rectangular or ditrigonal vertex figures): p.q.p.q 2|p q, or p.q.p.q.p.q, Wythoff symbol 3|p q
  3. Versi-regular (orthodiagonal vertex figures), p.q*.-p.q*, Wythoff symbol q q|p
  4. Truncated regular (isosceles triangular vertex figures): p.p.q, Wythoff symbol q 2|p
  5. Versi-quasi-regular (dipteroidal vertex figures), p.q.p.r Wythoff symbol q r|p
  6. Quasi-quasi-regular (trapezoidal vertex figures): p*.q.p*.-r q.r|p or p.q*.-p.q* p q r|
  7. Truncated quasi-regular (scalene triangular vertex figures), p.q.r Wythoff symbol p q r|
  8. Snub quasi-regular (pentagonal, hexagonal, or octagonal vertex figures), Wythoff symbol p q r|
  9. Prisms (truncated hosohedra),
  10. Antiprisms and crossed antiprisms (snub dihedra)

The format of each figure follows the same basic pattern

  1. image of polyhedron
  2. name of polyhedron
  3. alternate names (in brackets)
  4. Wythoff symbol
  5. Numbering systems: W - number used by Wenninger in polyhedra models, U - uniform indexing, K - Kaleido indexing, C - numbering used in Coxeter et al. 'Uniform Polyhedra'.
  6. Number of vertices V, edges E, Faces F and number of faces by type.
  7. Euler characteristic χ = V - E + F

The vertex figures are on the left, followed by the Point groups in three dimensions#The seven remaining point groups, either tetrahedral Td, octahedral Oh or icosahedral Ih.

Truncated forms[edit]

Regular polyhedra and their truncated forms[edit]

Column A lists all the regular polyhedra, column B list their truncated forms. Regular polyhedra all have vertex figures pr: p.p.p etc. and Whycroft symbol p|q r. The truncated forms have vertex figure q.q.r (where q=2p and r) and Whycroft p q|r.

vertex figure group A: regular: p.p.p B: truncated regular: p.p.r

Tetrahedron vertfig.png
3.3.3
Truncated tetrahedron vertfig.png
3.6.6

Td

Tetrahedron.jpg
Tetrahedron
3|2 3
W1, U01, K06, C15
V 4,E 6,F 4=4{3}
χ=2

Truncatedtetrahedron.jpg
Truncated tetrahedron
2 3|3
W6, U02, K07, C16
V 12,E 18,F 8=4{3}+4{6}
χ=2

Octahedron vertfig.png
3.3.3.3

Truncated octahedron vertfig.png
4.6.6

Oh

Octahedron.svg
Octahedron
4|2 3, 34
W2, U05, K10, C17
V 6,E 12,F 8=8{3}
χ=2

Truncatedoctahedron.jpg
Truncated octahedron
2 4|3
W7, U08, K13, C20
V 24,E 36,F 14=6{4}+8{6}
χ=2

Cube vertfig.png
4.4.4

Truncated cube vertfig.png
3.8.8

Oh

Hexahedron.jpg
Hexahedron
(Cube)
3|2 4
W3, U06, K11, C18
V 8,E 12,F 6=6{4}
χ=2

Truncatedhexahedron.jpg
Truncated hexahedron
2 3|4
W8, U09, K14, C21
V 24,E 36,F 14=8{3}+6{8}
χ=2

Icosahedron vertfig.png
3.3.3.3.3
Truncated icosahedron vertfig.png
5.6.6

Ih

Icosahedron.jpg
Icosahedron
5|2 3
W4, U22, K27, C25
V 12,E 30,F 20=20{3}
χ=2

Truncatedicosahedron.jpg
Truncated icosahedron
2 5|3
W9, U25, K30, C27
E 60,V 90,F 32=12{5}+20{6}
χ=2

Dodecahedron vertfig.png
5.5.5

Truncated dodecahedron vertfig.png
4.10.10

Ih

POV-Ray-Dodecahedron.svg
Dodecahedron
3|2 5
W5, U23, K28, C26
V 20,E 30,F 12=12{5}
χ=2

Truncateddodecahedron.jpg
Truncated dodecahedron
2 3|5
W10, U26, K31, C29
V 60,E 90,F 32=20{3}+12{10}
χ=2

Great dodecahedron vertfig.png
5.5.5.5.5
Truncated great dodecahedron vertfig.png
5/2.10.10

Ih

Great dodecahedron.png
Great dodecahedron
5/2|2 5
W21, U35, K40, C44
V 12,E 30,F 12=12{5}
χ=-6

Great truncated dodecahedron.png
Truncated great dodecahedron
25/2|5
W75, U37, K42, C47
V 60,E 90,F 24=12{5/2}+12{10}
χ=-6

Great icosahedron vertfig.png
3.3.3.3.3

Great truncated icosahedron vertfig.png
5/2.6.6.

Ih

Great icosahedron.png
Great icosahedron
(16th stellation of icosahedron)
5/2|2 3
W41, U53, K58, C69
V 12,E 30,F 20=20{3}
χ=2

Great truncated icosahedron.png
Great truncated icosahedron
25/2|3
W95, U55, K60, C71
V 60,E 90,F 32=12{5/2}+20{6}
χ=2

Small stellated dodecahedron vertfig.png
5/2.5/2.5/2.5/2.5/2

Ih

Small stellated dodecahedron.png
Small stellated dodecahedron
5|25/2
W20, U34, K39, C43
V 12,E 30,F 12=12{5/2}
χ=-6

Great stellated dodecahedron vertfig.png
5/2.5/2.5/2

Ih

Great stellated dodecahedron.png
Great stellated dodecahedron
3|25/2
W22, U52, K57, C68
V 20,E 30,F 12=12{5/2}
χ=2

In addition there are three quasi-truncated forms. These also class as truncated-regular polyhedra.

vertex figures Group Oh Group Ih Group Ih

Stellated truncated hexahedron vertfig.png
3.8/3.8/3
Small stellated truncated dodecahedron vertfig.png
5.10/3.10/3
Great stellated truncated dodecahedron vertfig.png
3.10/3.10/3

Stellated truncated hexahedron.png
Stellated truncated hexahedron
(Quasitruncated hexahedron)
(stellatruncated cube)
2 3|4/3
W92, U19, K24, C66
V 24,E 36,F 14=8{3}+6{8/3}
χ=2

Small stellated truncated dodecahedron.png
Small stellated truncated dodecahedron
(Quasitruncated small stellated dodecahedron)
(Small stellatruncated dodecahedron)
2 5|5/3
W97, U58, K63
V 60,E 90,F 24=12{5}+12{10/3}
χ=-6

Great stellated truncated dodecahedron.png
Great stellated truncated dodecahedron
(Quasitruncated great stellated dodecahedron)
(Great stellatruncated dodecahedron)
2 3|5/3
W104, U66, K71, C83
V 60,E 90,F 32=20{3}+12{10/3}
χ=2

Truncated forms of quasi-regular polyhedra[edit]

Column A lists some quasi-regular polyhedra, column B lists normal truncated forms, column C shows quasi-truncated forms, column D shows a different method of truncation. These truncated forms all have a vertex figure p.q.r and a Wythoff symbol p q r|.

vertex figure group A: quasi-regular: p.q.p.q B: truncated quasi-regular: p.q.r C: truncated quasi-regular: p.q.r D: truncated quasi-regular: p.q.r
Cuboctahedron vertfig.png
3.4.3.4

Great rhombicuboctahedron vertfig.png
4.6.8
Great truncated cuboctahedron vertfig.png
4.6.8/3
Cubitruncated cuboctahedron vertfig.png
8.6.8/3

Oh

Cuboctahedron.jpg
Cuboctahedron
2|3 4
W11, U07, K12, C19
V 12,E 24,F 14=8{3}+6{4}
χ=2

Truncatedcuboctahedron.jpg
Truncated cuboctahedron
(Great rhombicuboctahedron)
2 3 4|
W15, U11, K16, C23
V 48,E 72,F 26=12{4}+8{6}+6{8}
χ=2

Great truncated cuboctahedron.png
Great truncated cuboctahedron
(Quasitruncated cuboctahedron)
2 34/3|
W93, U20, K25, C67
V 48,E 72,F 26=12{4}+8{6}+6{8/3}
χ=2

Cubitruncated cuboctahedron.png
Cubitruncated cuboctahedron
(Cuboctatruncated cuboctahedron)
3 44/3|
W79, U16, K21, C52
V 48,E 72,F 20=8{6}+6{8}+6{8/3}
χ=-4

Icosidodecahedron vertfig.png
3.5.3.5

Great rhombicosidodecahedron vertfig.png
4.6.10
Great truncated icosidodecahedron vertfig.png
4.6.10/3
Icositruncated dodecadodecahedron vertfig.png
10.6.10/3

Ih

Icosidodecahedron.jpg
Icosidodecahedron
2|3 5
W12, U24, K29, C28
V 30,E 60,F 32=20{3}+12{5}
χ=2

Truncatedicosidodecahedron.jpg
Truncated icosidodecahedron
(Great rhombicosidodecahedron)
2 3 5|
W16, U28, K33, C31
V 120,E 180,F 62=30{4}+20{6}+12{10}
χ=2

Great truncated icosidodecahedron.png
Great truncated icosidodecahedron
(Great quasitruncated icosidodecahedron)
2 35/3|
W108, U68, K73, C87
V 120,E 180,F 62=30{4}+20{6}+12{10/3}
χ=2

Icositruncated dodecadodecahedron.png
Icositruncated dodecadodecahedron
(Icosidodecatruncated icosidodecahedron)
3 55/3|
W84, U45, K50, C57
V 120,E 180,F 44=20{6}+12{10}+12{10/3}
χ=-16

Dodecadodecahedron vertfig.png
5/2.5.5/2.5
Truncated dodecadodecahedron vertfig.png
4.10.10/3

Ih

Dodecadodecahedron.png
Dodecadodecahedron
2 5|5/2
W73, U36, K41, C45
V 30,E 60, F 24=12{5}+12{5/2}
χ=-6

Truncated dodecadodecahedron.png
Truncated dodecadodecahedron
(Quasitruncated dodecahedron)
2 55/3|
W98, U59, K64, C75
V 120,E 180,F 54=30{4}+12{10}+12{10/3}
χ=-6

Great icosidodecahedron vertfig.png

3.5/2.3.5/2

Ih

Great icosidodecahedron.png
Great icosidodecahedron
2 3|5/2
W94, U54, K59, C70
V 30,E 60, F 32=20{3}+12{5/2}
χ=2

Polyhedra sharing edges and vertices[edit]

Regular[edit]

These are all mentioned elsewhere, but this table shows some relations. They are all regular apart from the tetrahemihexahedron which is versi-regular.

vertex figure V E group regular regular/versi-regular
Octahedron vertfig.png
3.3.3.3

3.4*.-3.4*

6 12 Oh

Octahedron.svg
Octahedron
4|2 3
W2, U05, K10, C17
F 8=8{3}
χ=2

Tetrahemihexahedron.png
Tetrahemihexahedron
3/23|2
W67, U04, K09, C36
F 7=4{3}+3{4}
χ=1

Icosahedron vertfig.png
3.3.3.3.3
Great dodecahedron vertfig.png
5.5.5.5.5

12 30 Ih

Icosahedron.jpg
Icosahedron
5|2 3
W4, U22, K27
F 20=20{3}
χ=2

Great dodecahedron.png
Great dodecahedron
5/2|2 5
W21, U35, K40, C44
F 12=12{5}
χ=-6

Small stellated dodecahedron vertfig.png
5/2.5/2.5/2.5/2.5/2
Great icosahedron vertfig.png
3.3.3.3.3

12 30 Ih

Small stellated dodecahedron.png
Small stellated dodecahedron
5|25/2
W20, U34, K39, C43
F 12=12{5/2}
χ=-6

Great icosahedron.png
Great icosahedron
(16th stellation of icosahedron)
5/2|2 3
W41, U53, K58, C69
F 20=20{3}
χ=2

Quasi-regular and versi-regular[edit]

Rectangular vertex figures, or crossed rectangles first column are quasi-regular second and third columns are hemihedra with faces passing through the origin, called versi-regular by some authors.

vertex figure V E group quasi-regular: p.q.p.q versi-regular: p.s*.-p.s* versi-regular: q.s*.-q.s*
Cuboctahedron vertfig.png

3.4.3.4
3.6*.-3.6*
4.6*.-4.6*

12 24 Oh

Cuboctahedron.jpg
Cuboctahedron
2|3 4
W11, U07, K12, C19
F 14=8{3}+6{4}
χ=2

Octahemioctahedron.png
Octahemioctahedron
3/23|3
W68, U03, K08, C37
F 12=8{3}+4{6}
χ=0

Cubohemioctahedron.png
Cubohemioctahedron
4/34|3
W78, U15, K20, C51
F 10=6{4}+4{6}
χ=-2

Icosidodecahedron vertfig.png

3.5.3.5
3.10*.-3.10*
5.10*.-5.10*

30 60 Ih

Icosidodecahedron.jpg
Icosidodecahedron
2|3 5
W12, U24, K29, C28
F 32=20{3}+12{5}
χ=2

Small icosihemidodecahedron.png
Small icosihemidodecahedron
3/23|5
W89, U49, K54, C63
F 26=20{3}+6{10}
χ=-4

Small dodecahemidodecahedron.png
Small dodecahemidodecahedron
5/45|5
W91, U51, K56, 65
F 18=12{5}+6{10}
χ=-12

Great icosidodecahedron vertfig.png

3.5/2.3.5/2
3.10*.-3.10*
5/2.10*.-5/2.10*

30 60 Ih

Great icosidodecahedron.png
Great icosidodecahedron
2|5/23
W94, U54, K59, C70
F 32=20{3}+12{5/2}
χ=2

Great icosihemidodecahedron.png
Great icosihemidodecahedron
3 3|5/3
W106, U71, K76, C85
F 26=20{3}+6{10/3}
χ=-4

Great dodecahemidodecahedron.png
Great dodecahemidodecahedron
5/35/2|5/3
W107, U70, K75, C86
F 18=12{5/2}+6{10/3}
χ=-12

Dodecadodecahedron vertfig.png

5.5/2.5.5/2
5.6*.-5.6*
5/2.6*.-5/2.6*

30 60 Ih

Dodecadodecahedron.png
Dodecadodecahedron
2|5/25
W73, U36, K41, C45
F 24=12{5}+12{5/2}
χ=-6

Great dodecahemicosahedron.png
Great dodecahemicosahedron
5/45|3
W102, U65, K70, C81
F 22=12{5}+10{6}
χ=-8

Small dodecahemicosahedron.png
Small dodecahemicosahedron
5/35/2|3
W100, U62, K67, C78
F 22=12{5/2}+10{6}
χ=-8

Ditrigonal regular and versi-regular[edit]

Ditrigonal (that is di(2) -tri(3)-ogonal) vertex figures are the 3-fold analog of a rectangle. These are all quasi-regular as all edges are isomorphic. The compound of 5-cubes shares the same set of edges and vertices. The cross forms have a non-orientable vertex figure so the "-" notation has not been used and the "*" faces pass near rather than through the origin.

vertex figure V E group ditrogonal crossed-ditrogonal crossed-ditrogonal
Small ditrigonal icosidodecahedron vertfig.png

5/2.3.5/2.3.5/2.3
5/2.5*.5/2.5*.5/2.5*
3.5*.3.5*.3.5*

20 60 Ih

Small ditrigonal icosidodecahedron.png
Small ditrigonal icosidodecahedron
3|5/23
W70, U30, K35, C39
F 32=20{3}+12{5/2}
χ=-8

Ditrigonal dodecadodecahedron.png
Ditrigonal dodecadodecahedron
3|5/35
W80, U41, K46, C53
F 24=12{5}+12{5/2}
χ=-16

Great ditrigonal icosidodecahedron.png
Great ditrigonal icosidodecahedron
3/2|3 5
W87, U47, K52, C61
F 32=20{3}+12{5}
χ=-8

versi-quasi-regular and quasi-quasi-regular[edit]

Group III: trapezoid or crossed trapezoid vertex figures. The first column include the convex rhombic polyhedra, created by inserting two squares into the vertex figures of the Cuboctahedron and Icosidodecahedron.

vertex figure V E group trapezoid: p.q.r.q crossed-trapezoid: p.s*.-r.s* crossed-trapezoid: q.s*.-q.s*
Small rhombicuboctahedron vertfig.png

3.4.4.4
3.8*.-4.8*
4.8*.-4.8*

24 48 Oh

Rhombicuboctahedron.jpg
Small rhombicuboctahedron
(rhombicuboctahedron)
3 4|2
W13, U10, K15, C22
F 26=8{3}+(6+12){4}
χ=2

Small cubicuboctahedron.png
Small cubicuboctahedron
3/24|4
W69, U13, K18, C38
F 20=8{3}+6{4}+6{8}
χ=-4

Small rhombihexahedron.png
Small rhombihexahedron
2 3/2 4|
W86, U18, K23, C60
F 18=12{4}+6{8}
χ=-6

Great cubicuboctahedron vertfig.png

3.8/3.4.8/3
3.4*.-4.4*
8/3.4*.-8/3.4*

24 48 Oh

Great cubicuboctahedron.png
Great cubicuboctahedron
3 4|4/3
W77, U14, K19, C50
F 20=8{3}+6{4}+6{8/3}
χ=-4

Uniform great rhombicuboctahedron.png
Nonconvex great rhombicuboctahedron
(Quasirhombicuboctahedron)
3/24|2
W85, U17, K22, C59
F 26=8{3}+(6+12){4}
χ=2

Great rhombihexahedron.png
Great rhombihexahedron
2 4/33/2|
W103, U21, K26, C82
F 18=12{4}+6{8/3}
χ=-6

Small rhombicosidodecahedron vertfig.png

3.4.5.4
3.10*.-5.10*
4.10*.-4.10*

60 120 Ih

Rhombicosidodecahedron.jpg
Small rhombicosidodecahedron
(rhombicosidodecahedron)
3 5|2
W14, U27, K32, C30
F 62=20{3}+30{4}+12{5}
χ=2

Small dodecicosidodecahedron.png
Small dodecicosidodecahedron
3/25|5
W72, U33, K38, C42
F 44=20{3}+12{5}+12{10}
χ=-16

Small rhombidodecahedron.png
Small rhombidodecahedron
25/25|
W74, U39, K44, C46
F 42=30{4}+12{10}
χ=-18

Rhombidodecadodecahedron vertfig.png

5/2.4.5.4
5/2.6*.-5.6*
4.6*.-4.6*

60 120 Ih

Rhombidodecadodecahedron.png
Rhombidodecadodecahedron
5/25|2
W76, U38, K43, C48
F 54=30{4}+12{5}+12{5/2}
χ=-6

Icosidodecadodecahedron.png
Icosidodecadodecahedron
5/35|3
W83, U44, K49, C56
F 44=12{5}+12{5/2}+20{6}
χ=-16

Rhombicosahedron.png
Rhombicosahedron
2 35/2|
W96, U56, K61, C72
F 50=30{4}+20{6}
χ=-10

Great dodecicosidodecahedron vertfig.png

3.10/3.5/2.10/3
3.4*.-5/2.4*
10/3.4*.-10/3.4*

60 120 Ih

Great dodecicosidodecahedron.png
Great dodecicosidodecahedron
5/23|5/3
W99, U61, K66, C77
F 44=20{3}+12{5/2}+12{10/3 }
χ=-16

Uniform great rhombicosidodecahedron.png
Nonconvex great rhombicosidodecahedron
(Quasirhombicosidodecahedron)
5/33|2
W105, U67, K72, C84
F 62=20{3}+30{4}+12{5/2}
χ=2

Great rhombidodecahedron.png
Great rhombidodecahedron
2 3/25/3|
W109, U73, K78, C89
F 42=30{4}+12{10/3}
χ=-18

Small icosicosidodecahedron vertfig.png

3.6.5/2.6
3.10*.-5/2.10*
6.10*.-6.10*

60 120 Ih

Small icosicosidodecahedron.png
Small icosicosidodecahedron
5/23|3
W71, U31, K36, C40
F 52=20{3}+12{5/2}+20{6}
χ=-8

Small ditrigonal dodecicosidodecahedron.png
Small ditrigonal dodecicosidodecahedron
5/33|5
W82, U43, K48, C55
F 44=20{3}+12{5/2}+12{10}
χ=-16

Small dodecicosahedron.png
Small dodecicosahedron
3 3/2 5|
W90, U50, K55, C64
F 32=20{6}+12{10}
χ=-28

Great ditrigonal dodecicosidodecahedron vertfig.png

3.10/3.5.10/3
3.6*.-5.6*
10/3.6*.-10/3.6*

60 120 Ih

Great ditrigonal dodecicosidodecahedron.png
Great ditrigonal dodecicosidodecahedron
3 5|5/3
W81, U42, K47, C54
F 44=20{3}+12{5}+12{10/3}
χ=-16

Great icosicosidodecahedron.png
Great icosicosidodecahedron
3/25|3
W88, U48, K53, C62
F 52=20{3}+12{5}+20{6}
χ=-8

Great dodecicosahedron.png
Great dodecicosahedron
3 5/35/2|
W101, U63, K68, C79
F 32=20{6}+12{10/3}
χ=-28