UNIFORM POLYHEDRON

A 'uniform polyhedron' is a polyhedron which has regular polygons as faces and is transient on its vertices (i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
Uniform polyhedra may be regular, quasi-regular or semi-regular. The faces and vertices need not be convex, so many of the Uniform polyhedra are also star polyhedra.
Excluding the infinite sets there are 75 uniform polyhedra (or 76 if edges are allowed to coincide).
Categories include:

★ Infinite sets of uniform prisms and antiprisms (including star forms)

★ 5 Platonic solids - regular convex polyhedra

★ 4 Kepler-Poinsot polyhedra - regular nonconvex polyhedra

★ 13 Archimedean solids - quasiregular and semiregular convex polyhedra

★ 14 nonconvex polyhedra with convex faces

★ 39 nonconvex polyhedra with nonconvex faces

★ 1 polyhedron found by John Skilling with pairs of edges that coincide.
They can also be grouped by their symmetry group, which is done below.

Contents
History
Indexing
Dihedral symmetry
Convex forms and fundamental vertex arrangements
Definition of operations
Nonconvex forms listed by symmetry groups and vertex arrangements
Tetrahedral symmetry
Octahedral symmetry
Icosahedral symmetry
Skilling's figure
See also
References
External links

History



★ The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid.

★ Johannes Kepler (1571-1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost.

★ Kepler (1619) discovered two of the regular Kepler-Poinsot polyhedra and Louis Poinsot (1809) discovered the other two.

★ Of the remaining 37 were discoved by Badoureau (1881). Edmund Hess (1878) discovered 2 more and Pitsch (1881) independently discovered 18, not all previously discovered.

★ The famous geometer Donald Coxeter discovered the remaining twelve in collaboration with J.C.P. Miller (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these.

★ In 1954 H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller published the list of uniform polyhedra.

★ In 1970 S. P. Sopov proved their conjecture that the list was complete.

★ In 1974, Magnus Wenninger published his book, ''Polyhedron models'', which is the first published list of all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson (mathematician).

★ In 1975, John Skilling independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility.

★ In 1993, Zvi Har'El produced a complete computer construction of the uniform polyhedra and duals via their Kaleidoscopic construction via a comptuter program called 'Kaleido', and summarized in a paper ''Uniform Solution for Uniform Polyhedra.'', counting figures 1-80.

★ Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system.
Indexing

There are four major published indexing efforts from the works above. To distinguish them, they are given by indexing different letters, 'C' for Coxeter 1954 first enumeration figures, 'W' for the 1974 book Polyhedron models by Wenninger, 'K' for the 1993 Kaleido solution, and 'U' for the Maeder solution used by Mathematica and extensively reproduced elsewhere.
# ['C'] 1954: This paper listed the uniform polyhedra by figures in the paper from 15-92. Starting with 15-32 for the convex forms, 33-35 for 3 infinite prismatic sets, and ending with 36-92 for the nonconvex forms.
# ['W'] 1974: Wenninger's book ''Polyhedron model'' numbered figures 1-119: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra.
# ['K'] 1993 Kaleido: The 80 figures given in the Kaleido solution were grouped by symmetry, numbered 1-80: 1-5 as representatives for the infinite families of prismatic forms with dihedral symmetry, 6-9 with tetrahedral symmetry, 10-26 with Octahedral symmetry, 46-80 with icosahedral symmetry.
# ['U'] 1993 Mathematica: This listing followed the Kaleido one, but moved the 5 prismatic forms to last, shifting the nonprismatic forms back 5, and now 1-75.

Dihedral symmetry


There are two infinite sets of uniform polyhedra with dihedral symmetry: the prisms and antiprisms. These sets both include forms with star polygons.
:''Main article: Prismatic uniform polyhedron''

Convex forms and fundamental vertex arrangements


The convex uniform polyhedra can be named by Wythoff construction operations upon a parent form.
Note: ''Dihedra'' are members of an infinite set of two-sided polyhedra (2 identical polygons) which generate the prisms as truncated forms.
Each of these convex forms define set of ''vertices'' that can be identified for the nonconvex forms in the next section.
{| class="wikitable"
|-
!
!Parent
!Truncated
!Rectified
!Bitruncated
(truncated dual)
!Birectified
(dual)
!Cantellated
!Omnitruncated
(Cantitruncated)
!Snub
|-
!rowspan=2|Extended
Schläfli symbol
!egin{Bmatrix} p , q end{Bmatrix}
!tegin{Bmatrix} p , q end{Bmatrix}
!egin{Bmatrix} p \ q end{Bmatrix}
!tegin{Bmatrix} q , p end{Bmatrix}
!egin{Bmatrix} q , p end{Bmatrix}
!regin{Bmatrix} p \ q end{Bmatrix}
!tegin{Bmatrix} p \ q end{Bmatrix}
!segin{Bmatrix} p \ q end{Bmatrix}
|-
!t0{p,q}
!t0,1{p,q}
!t1{p,q}
!t1,2{p,q}
!t2{p,q}
!t0,2{p,q}
!t0,1,2{p,q}
!s{p,q}
|-
!Wythoff symbol
p-q-2
! q | p 2
! 2 q | p
! 2 | p q
! 2 p | q
! p | q 2
! p q | 2
! p q 2 |
! | p q 2
|-
!rowspan=5|Coxeter-Dynkin diagram
(variations)
|-
!
CDW ring.png
CDW p.png
CDW dot.png
CDW q.png
CDW dot.png

!
CDW ring.png
CDW p.png
CDW ring.png
CDW q.png
CDW dot.png

!
CDW dot.png
CDW p.png
CDW ring.png
CDW q.png
CDW dot.png

!
CDW dot.png
CDW p.png
CDW ring.png
CDW q.png
CDW ring.png

!
CDW dot.png
CDW p.png
CDW dot.png
CDW q.png
CDW ring.png

!
CDW ring.png
CDW p.png
CDW dot.png
CDW q.png
CDW ring.png

!
CDW ring.png
CDW p.png
CDW ring.png
CDW q.png
CDW ring.png

!
CDW hole.png
CDW p.png
CDW hole.png
CDW q.png
CDW hole.png

|-
!(o)-p-o-q-o
!(o)-p-(o)-q-o
!o-p-(o)-q-o
!o-p-(o)-q-(o)
!o-p-o-q-(o)
!(o)-p-o-q-(o)
!(o)-p-(o)-q-(o)
!( )-p-( )-q-( )
|-
!xPoQo
!xPxQo
!oPxQo
!oPxQx
!oPoQx
!xPoQx
!xPxQx
!sPsQs
|-
![p,q]:001
![p,q]:011
![p,q]:010
![p,q]:110
![p,q]:100
![p,q]:101
![p,q]:111
![p,q]:111s
|-
!Vertex figure
!pq
!(q.2p.2p)
!(p.q.p.q)
!(p.2q.2q)
!qp
!(p.4.q.4)
!(4.2p.2q)
!(3.3.p.3.q)
|-
|Tetrahedral
3-3-2
|

{3,3}
|

(3.6.6)
|

(3.3.3.3)
|

(3.6.6)
|

{3,3}
|

(3.4.3.4)
|

(4.6.6)
|

(3.3.3.3.3)
|-
|Octahedral
4-3-2
|

{4,3}
|

(3.8.8)
|

(3.4.3.4)
|

(4.6.6)
|

{3,4}
|

(3.4.4.4)
|

(4.6.8)
|

(3.3.3.3.4)
|-
|Icosahedral
5-3-2
|

{5,3}
|

(3.10.10)
|

(3.5.3.5)
|

(5.6.6)
|

{3,5}
|

(3.4.5.4)
|

(4.6.10)
|

(3.3.3.3.5)
|-
|Dihedral
p-2-2
Example p=5
| {5,2}
| 2.10.10
| 2.5.2.5
|

4.4.5
| {2,5}
| 2.4.5.4
|

4.4.10
|

3.3.3.5
|}
Definition of operations

{|width=640|
|

|

Example forms from the cube and octahedron
|}
{|width=720 class="wikitable"
!Operation
!colspan=2|Extended
Schläfli
symbols
!Coxeter-
Dynkin
diagram
!Description
|-
! Parent
|width=70| t0{p,q}
| egin{Bmatrix} p , q end{Bmatrix}
|
Dynkins-100.png

| Any regular polyhedron or tiling
|-
! Rectified
| t1{p,q}
| egin{Bmatrix} p \ q end{Bmatrix}
|
Dynkins-010.png

|The edges are fully truncated into single points. The polyhedron now has the combined faces of the parent and dual.
|-
!Birectified
Also Dual
| t2{p,q}
| egin{Bmatrix} q , p end{Bmatrix}
|
Dynkins-001.png

|
The birectified (dual) is a further truncation so that the original faces are reduced to points. New faces are formed under each parent vertex. The number of edges is unchanged and are rotated 90 degrees. The dual of the regular polyhedron {p, q} is also a regular polyhedron {q, p}.
|-
!Truncated
| t0,1{p,q}
| tegin{Bmatrix} p , q end{Bmatrix}
|
Dynkins-110.png

|Each original vertex is cut off, with new faces filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated polyhedron. The polyhedron has its original faces doubled in sides, and contains the faces of the dual.

|-
!'Bitruncated'
| t1,2{p,q}
| tegin{Bmatrix} q , p end{Bmatrix}
|
Dynkins-011.png

|Same as truncated dual.
|-
! Cantellated
(or rhombated)
(Also expanded)
| t0,2{p,q}
| regin{Bmatrix} p \ q end{Bmatrix}
|
Dynkins-101.png

|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place, as well as the original vertices are also truncated. A uniform cantellation is half way between both the parent and dual forms.

|-
!Omnitruncated
(or cantitruncated)
(or rhombitruncated)
| t0,1,2{p,q}
| tegin{Bmatrix} p \ q end{Bmatrix}
|
Dynkins-111.png

|The truncation and cantellation operations are applied together create an omnitruncated form which has the parent's faces doubled in sides, the duals faces doubled in sides, and squares where the original edges existed.
|-
!Snub
| s{p,q}
| segin{Bmatrix} p \ q end{Bmatrix}
|
Dynkins-sss.png

|The snub takes the omnitruncated form and rectifies alternate vertices. (This operation is only possible for polyhedra with all even-sided faces.) All the original faces end up with half as many sides, and the square degenerate into edges. Since the omnitruncated forms have 3 faces/vertex, new triangles are formed. Usually these alternated faceting forms are slightly deformed thereafter in order to end again as uniform polyhedra. The possibility of the latter variation depends on the degree of freedom.

|}

Nonconvex forms listed by symmetry groups and vertex arrangements


All the uniform polyhedra are listed below by their symmetry groups and subgrouped by their vertex arrangements.
Regular polyhedra are labeled by their Schläfli symbol. Other nonregular uniform polyhedra are listed with their vertex configuration or their Uniform polyhedron index U(1-80).
Note: For nonconvex forms below an additional descriptor 'Nonuniform' is used when the convex hull of the vertex arrangement has same topology as one of these, but has nonregular faces. For example an ''nonuniform cantellated'' form may have rectangles created in place of the edges rather than squares.
Tetrahedral symmetry

There are 2 convex uniform polyhedra, the tetrahedron and truncated tetrahedron, and one nonconvex form, the tetrahemihexahedron which have ''tetrahedral symmetry''. The tetrahedron is self dual.
In addition the octahedron, truncated octahedron, cuboctahedron, and icosahedron have tetrahedral symmetry as well as higher symmetry. They are added for completeness below, although their nonconvex forms with octahedral symmetry are not included here.
{| class="wikitable"
!Vertex group
!Convex
!colspan=2|Nonconvex
|-
|(Tetrahedral)
|

{3,3}
|-
|Truncated (
★ )
|

(3.6.6)
|-
|Rectified (
★ )
|

{3,4}
|

(4.3/2.4.3)
|-
|Cantellated (
★ )
|

(3.4.3.4)
|-
|Omnitruncated (
★ )
|

(4.6.6)
|-
|Snub (
★ )
|

{3,5}
|}
Octahedral symmetry

There are 8 convex forms, and 10 nonconvex forms with ''octahedral symmetry''.
{| class="wikitable"
!Vertex group
!Convex
!colspan=3|Nonconvex
|-
|(Octahedral)
|

{3,4}
|-
|Truncated (
★ )
|

(4.6.6)
|-
|Rectified (
★ )
|

(3.4.3.4)
|

(6.4/3.6.4)
|

(6.3/2.6.3)
|-
|Truncated dual (
★ )
|

(3.8.8)
|

(4.8/3.4/3.8/5)
|

(8/3.3.8/3.4)
|

(4.3/2.4.4)
|-
|Dual (
★ )
|

{4,3}
|-
|Cantellated (
★ )
|

(3.4.4.4)
|

(4.8.4/3.8)
|

(8.3/2.8.4)
|

(8/3.8/3.3)
|-
|Omnitruncated (
★ )
|

(4.6.8)
|-
| Nonuniform omnitruncated (
★ )
| (4.6.8)
|

(8/3.4.6)
|

(8/3.6.8)
|-
| Snub (
★ )
|

(3.3.3.3.4)
|}
Icosahedral symmetry

There are 8 convex forms and 46 nonconvex forms with ''icosahedral symmetry'' (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have nonuniform chiral symmetry, and some have achiral symmetry.
{| class="wikitable"
!Vertex group
!Convex
!colspan=4|Nonconvex
|-
| (Icosahedral)
|

{3,5}
|

{5/2,5}
|

{5,5/2}
|

{3,5/2}
|-
| Truncated (
★ )
|

(5.6.6)
|-
| Nonuniform truncated (
★ )
| (5.6.6)
|

U37
|

U61
|

U67
|

U73
|

U38
|

U44
|

U56
|

U32
|-
| Rectified (
★ )
|

(3.5.3.5)
|

U49
|

U51
|

U54
|

U70
|

U71
|

U36
|

U62
|

U65
|-
| Truncated dual (
★ )
|

(3.10.10)
|

U42
|

U48
|

U63
|-
| Nonuniform truncated dual (
★ )
| (3.10.10)
|

U72
|-
| Dual (
★ )
|

{5,3}
|

{5/2,3}
|

U30
|

U41
|

U47
|-
| Cantellated (
★ )
|

(3.4.5.4)
|

U33
|

U39
|

U58
|-
| Nonuniform Cantellated (
★ )
| (3.4.5.4)
|

U31
|

U43
|

U50
|

U66
|

U55
|

U75
|

U64
|-
| Omnitruncated (
★ )
|

(4.6.10)
|-
| Nonuniform omnitruncated (
★ )
| (4.6.10)
|

U68
|

U59
|

U45
|-
| Snub (
★ )
|

(3.3.3.3.5)
|-
| Nonuniform Snub (
★ )
| (3.3.3.3.5)
|

U40
|

U46
|

U57
|

U69
|

U60
|

U74
|}

Skilling's figure


One further nonconvex polyhedron is the Great disnub dirhombidodecahedron, also known as ''Skilling's figure'', which is vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges.
It is sometimes but not always counted as a uniform polyhedron. It has 'I'h symmetry.
:

See also



★ Polyhedron

★
★ Regular polyhedron

★
★ Quasiregular polyhedron

★
★ Semiregular polyhedron

★ List of uniform polyhedra

★ List of Wenninger polyhedron models

★ Polyhedron model

★ List of uniform polyhedra by vertex figure

★ List of uniform polyhedra by Wythoff symbol

References



★ Brückner, M. ''Vielecke und vielflache. Theorie und geschichte.''. Leipzig, Germany: Teubner, 1900. [1]

★ H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, ''Uniform polyhedra'', 'Phil. Trans.' 1954, 246 A, 401-50 [2]

★ S. P. Sopov ''A proof of the completeness on the list of elementary homogeneous polyhedra.'' (Russian) Ukrain. Geometr. Sb. No. 8, (1970), 139-156

★ Polyhedron Models, , Magnus, Wenninger, Cambridge University Press, 1974, ISBN 0-521-09859-9

★ John Skilling, ''The complete set of uniform polyhedra.'', Philos. Trans. Roy. Soc. London Ser. A 278 (1975), 111-135 [3]

★ Har'El, Z. ''Uniform Solution for Uniform Polyhedra.'', Geometriae Dedicata 47, 57-110, 1993. Zvi Har’El [4], Kaleido software, Images, dual images

★ Mäder, R. E. ''Uniform Polyhedra.'' Mathematica J. 3, 48-57, 1993. [5]

External links



★

★ Uniform Solution for Uniform Polyhedra

★ The Uniform Polyhedra

★ Virtual Polyhedra Uniform Polyhedra

★ Stella: Polyhedron Navigator - Software for generating and printing nets for all uniform polyhedra

★ Paper models:

★
★ Uniform/Dual Polyhedra

★
★ Paper Models of Uniform (and other) Polyhedra

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