List of uniform polyhedra by Schwarz triangle

There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the acute and obtuse Schwarz triangles. The numbers that can be used for the sides of a non-dihedral acute or obtuse Schwarz triangle that does not necessarily lead to only degenerate uniform polyhedra are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, and 5/4 (but numbers with numerator 4 and those with numerator 5 may not occur together). (4/2 can also be used, but only leads to degenerate uniform polyhedra as 4 and 2 have a common factor.) There are 44 such Schwarz triangles (5 with tetrahedral symmetry, 7 with octahedral symmetry and 32 with icosahedral symmetry), which, together with the infinite family of dihedral Schwarz triangles, can form almost all of the non-degenerate uniform polyhedra. Many degenerate uniform polyhedra, with completely coincident vertices, edges, or faces, may also be generated by the Wythoff construction, and those that arise from Schwarz triangles not using 4/2 are also given in the tables below along with their non-degenerate counterparts. Reflex Schwarz triangles have not been included, as they simply create duplicates or degenerates; however, a few are mentioned outside the tables due to their application to three of the snub polyhedra.

Coxeter's listing of degenerate Wythoffian uniform polyhedra, giving Wythoff symbols, vertex figures, and descriptions using Schläfli symbols. All the uniform polyhedra and all the degenerate Wythoffian uniform polyhedra are listed in this article.

There are a few non-Wythoffian uniform polyhedra, which no Schwarz triangles can generate; however, most of them can be generated using the Wythoff construction as double covers (the non-Wythoffian polyhedron is covered twice instead of once) or with several additional coinciding faces that must be discarded to leave no more than two faces at every edge (see Omnitruncated polyhedron#Other even-sided nonconvex polyhedra). Such polyhedra are marked by an asterisk in this list. The only uniform polyhedra which still fail to be generated by the Wythoff construction are the great dirhombicosidodecahedron and the great disnub dirhombidodecahedron.

Each tiling of Schwarz triangles on a sphere may cover the sphere only once, or it may instead wind round the sphere a whole number of times, crossing itself in the process. The number of times the tiling winds round the sphere is the density of the tiling, and is denoted μ.

Jonathan Bowers' short names for the polyhedra, known as Bowers acronyms, are used instead of the full names for the polyhedra to save space. The Maeder index is also given. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities.

Möbius and Schwarz trianglesEdit

There are 4 spherical triangles with angles π/p, π/q, π/r, where (p q r) are integers: (Coxeter, "Uniform polyhedra", 1954)

  1. (2 2 r) - Dihedral
  2. (2 3 3) - Tetrahedral
  3. (2 3 4) - Octahedral
  4. (2 3 5) - Icosahedral

These are called Möbius triangles.

In addition Schwarz triangles consider (p q r) which are rational numbers. Each of these can be classified in one of the 4 sets above.

Density (μ) Dihedral Tetrahedral Octahedral Icosahedral
d (2 2 n/d)
1 (2 3 3) (2 3 4) (2 3 5)
2 (3/2 3 3) (3/2 4 4) (3/2 5 5), (5/2 3 3)
3 (2 3/2 3) (2 5/2 5)
4 (3 4/3 4) (3 5/3 5)
5 (2 3/2 3/2) (2 3/2 4)
6 (3/2 3/2 3/2) (5/2 5/2 5/2), (3/2 3 5), (5/4 5 5)
7 (2 3 4/3) (2 3 5/2)
8 (3/2 5/2 5)
9 (2 5/3 5)
10 (3 5/3 5/2), (3 5/4 5)
11 (2 3/2 4/3) (2 3/2 5)
13 (2 3 5/3)
14 (3/2 4/3 4/3) (3/2 5/2 5/2), (3 3 5/4)
16 (3 5/4 5/2)
17 (2 3/2 5/2)
18 (3/2 3 5/3), (5/3 5/3 5/2)
19 (2 3 5/4)
21 (2 5/4 5/2)
22 (3/2 3/2 5/2)
23 (2 3/2 5/3)
26 (3/2 5/3 5/3)
27 (2 5/4 5/3)
29 (2 3/2 5/4)
32 (3/2 5/4 5/3)
34 (3/2 3/2 5/4)
38 (3/2 5/4 5/4)
42 (5/4 5/4 5/4)

Although a polyhedron usually has the same density as the Schwarz triangle it is generated from, this is not always the case. Firstly, polyhedra that have faces passing through the centre of the model (including the hemipolyhedra, great dirhombicosidodecahedron, and great disnub dirhombidodecahedron) do not have a well-defined density. Secondly, the distortion necessary to recover uniformity when changing a spherical polyhedron to its planar counterpart can push faces through the centre of the polyhedron and back out the other side, changing the density. This happens in the following cases:

  • The great truncated cuboctahedron, 2 3 4/3 |. While the Schwarz triangle (2 3 4/3) has density 7, recovering uniformity pushes the eight hexagons through the centre, yielding density |7 − 8| = 1, the same as that of the colunar Schwarz triangle (2 3 4) that shares the same great circles.
  • The truncated dodecadodecahedron, 2 5/3 5 |. While the Schwarz triangle (2 5/3 5) has density 9, recovering uniformity pushes the twelve decagons through the centre, yielding density |9 − 12| = 3, the same as that of the colunar Schwarz triangle (2 5/2 5) that shares the same great circles.
  • Three snub polyhedra: the great icosahedron | 2 3/2 3/2, the small retrosnub icosicosidodecahedron | 3/2 3/2 5/2, and the great retrosnub icosidodecahedron | 2 3/2 5/3. Here the vertex figures have been distorted into pentagrams or hexagrams rather than pentagons or hexagons, pushing all the snub triangles through the centre and yielding densities of |5 − 12| = 7, |22 − 60| = 38, and |23 − 60| = 37 respectively. These densities are the same as those of colunar reflex-angled Schwarz triangles that are not included above. Thus the great icosahedron may be considered to come from (2/3 3 3) or (2 3 3/4), the small retrosnub icosicosidodecahedron from (3 3 5/8) or (3 3/4 5/3), and the great retrosnub icosidodecahedron from (2/3 3 5/2), (2 3/4 5/3), or (2 3 5/7). (Coxeter, "Uniform polyhedra", 1954)

Summary tableEdit

 
The eight forms for the Wythoff constructions from a general triangle (p q r). Partial snubs can also be created (not shown in this article).
 
The nine reflexible forms for the Wythoff constructions from a general quadrilateral (p q r s).

There are seven generator points with each set of p,q,r (and a few special forms):

General Right triangle (r=2)
Description Wythoff
symbol
Vertex
configuration
Coxeter
diagram

 
Wythoff
symbol
Vertex
configuration
Schläfli
symbol
Coxeter
diagram
     
regular and
quasiregular
q | p r (p.r)q         q | p 2 pq {p,q}      
p | q r (q.r)p         p | q 2 qp {q,p}      
r | p q (q.p)r         2 | p q (q.p)² t1{p,q}      
truncated and
expanded
q r | p q.2p.r.2p         q 2 | p q.2p.2p t0,1{p,q}      
p r | q p.2q.r.2q         p 2 | q p. 2q.2q t0,1{q,p}      
p q | r 2r.q.2r.p         p q | 2 4.q.4.p t0,2{p,q}      
even-faced p q r | 2r.2q.2p         p q 2 | 4.2q.2p t0,1,2{p,q}      
p q r
s
|
2p.2q.-2p.-2q - p 2 r
s
|
2p.4.-2p.4/3 -
snub | p q r 3.r.3.q.3.p         | p q 2 3.3.q.3.p sr{p,q}      
| p q r s (4.p.4.q.4.r.4.s)/2 - - - -

There are four special cases:

  • p q r
    s
    |
    – This is a mixture of p q r | and p q s |. Both the symbols p q r | and p q s | generate a common base polyhedron with some extra faces. The notation p q r
    s
    |
    then represents the base polyhedron, made up of the faces common to both p q r | and p q s |.
  • | p q r – Snub forms (alternated) are given this otherwise unused symbol.
  • | p q r s – A unique snub form for U75 that isn't Wythoff-constructible using triangular fundamental domains. Four numbers are included in this Wythoff symbol as this polyhedron has a tetragonal spherical fundamental domain.
  • | (p) q (r) s – A unique snub form for Skilling's figure that isn't Wythoff-constructible.

This conversion table from Wythoff symbol to vertex configuration fails for the exceptional five polyhedra listed above whose densities do not match the densities of their generating Schwarz triangle tessellations. In these cases the vertex figure is highly distorted to achieve uniformity with flat faces: in the first two cases it is an obtuse triangle instead of an acute triangle, and in the last three it is a pentagram or hexagram instead of a pentagon or hexagon, winding around the centre twice. This results in some faces being pushed right through the polyhedron when compared with the topologically equivalent forms without the vertex figure distortion and coming out retrograde on the other side.[1]

Dihedral (prismatic)Edit

In dihedral Schwarz triangles, two of the numbers are 2, and the third may be any rational number strictly greater than 1.

  1. (2 2 n/d) – degenerate if gcd(n, d) > 1.

Many of the polyhedra with dihedral symmetry have digon faces that make them degenerate polyhedra (e.g. dihedra and hosohedra). Columns of the table that only give degenerate uniform polyhedra are not included: special degenerate cases (only in the (2 2 2) Schwarz triangle) are marked with a large cross. Uniform crossed antiprisms with a base {p} where p < 3/2 cannot exist as their vertex figures would violate the triangular inequality; these are also marked with a large cross. The 3/2-crossed antiprism (trirp) is degenerate, being flat in Euclidean space, and is also marked with a large cross. The Schwarz triangles (2 2 n/d) are listed here only when gcd(n, d) = 1, as they otherwise result in only degenerate uniform polyhedra.

The list below gives all possible cases where n ≤ 6.

(p q r) q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
(2 2 2)
(μ=1)
X
X
 
4.4.4
cube
4-p
 
3.3.3
tet
2-ap
(2 2 3)
(μ=1)
 
4.3.4
trip
3-p
 
4.3.4
trip
3-p
 
6.4.4
hip
6-p
 
3.3.3.3
oct
3-ap
(2 2 3/2)
(μ=2)
 
4.3.4
trip
3-p
 
4.3.4
trip
3-p
 
6/2.4.4
2trip
6/2-p
X
(2 2 4)
(μ=1)
 
4.4.4
cube
4-p
 
4.4.4
cube
4-p
 
8.4.4
op
8-p
 
3.4.3.3
squap
4-ap
(2 2 4/3)
(μ=3)
 
4.4.4
cube
4-p
 
4.4.4
cube
4-p
 
8/3.4.4
stop
8/3-p
X
(2 2 5)
(μ=1)
 
4.5.4
pip
5-p
 
4.5.4
pip
5-p
 
10.4.4
dip
10-p
 
3.5.3.3
pap
5-ap
(2 2 5/2)
(μ=2)
 
4.5/2.4
stip
5/2-p
 
4.5/2.4
stip
5/2-p
 
10/2.4.4
2pip
10/2-p
 
3.5/2.3.3
stap
5/2-ap
(2 2 5/3)
(μ=3)
 
4.5/2.4
stip
5/2-p
 
4.5/2.4
stip
5/2-p
 
10/3.4.4
stiddip
10/3-p
 
3.5/3.3.3
starp
5/3-ap
(2 2 5/4)
(μ=4)
 
4.5.4
pip
5-p
 
4.5.4
pip
5-p
 
10/4.4.4

10/4-p
X
(2 2 6)
(μ=1)
 
4.6.4
hip
6-p
 
4.6.4
hip
6-p
 
12.4.4
twip
12-p
 
3.6.3.3
hap
6-ap
(2 2 6/5)
(μ=5)
 
4.6.4
hip
6-p
 
4.6.4
hip
6-p
 
12/5.4.4
stwip
12/5-p
X
(2 2 n)
(μ=1)
4.n.4
n-p
4.n.4
n-p
2n.4.4
2n-p
3.n.3.3
n-ap
(2 2 n/d)
(μ=d)
4.n/d.4
n/d-p
4.n/d.4
n/d-p
2n/d.4.4
2n/d-p
3.n/d.3.3
n/d-ap

TetrahedralEdit

In tetrahedral Schwarz triangles, the maximum numerator allowed is 3.

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (3 3 2)
(µ=1)
 
3.3.3
tet
U1
 
3.3.3
tet
U1
 
3.3.3.3
oct
U5
 
3.6.6
tut
U2
 
3.6.6
tut
U2
 
4.3.4.3
co
U7
 
4.6.6
toe
U8
 
3.3.3.3.3
ike
U22
2 (3 3 3/2)
(µ=2)
 
(3.3.3.3.3.3)/2
2tet
 
(3.3.3.3.3.3)/2
2tet
 
(3.3.3.3.3.3)/2
2tet
 
3.6.3/2.6
oho
U3
 
3.6.3/2.6
oho
U3
 
2(6/2.3.6/2.3)
2oct
 
2(6/2.6.6)
2tut
 
2(3.3/2.3.3.3.3)
2oct+8{3}
3 (3 2 3/2)
(µ=3)
 
3.3.3.3
oct
U5
 
3.3.3
tet
U1
 
3.3.3
tet
U1
 
3.6.6
tut
U2
 
2(3/2.4.3.4)
2thah
U4*
 
3(3.6/2.6/2)
3tet
 
2(6/2.4.6)
cho+4{6/2}
U15*
 
3(3.3.3)
3tet
4 (2 3/2 3/2)
(µ=5)
 
3.3.3
tet
U1
 
3.3.3.3
oct
U5
 
3.3.3
tet
U1
 
3.4.3.4
co
U7
 
3(6/2.3.6/2)
3tet
 
3(6/2.3.6/2)
3tet
 
4(6/2.6/2.4)
2oct+6{4}
 
(3.3.3.3.3)/2
gike
U53
5 (3/2 3/2 3/2)
(µ=6)
 
(3.3.3.3.3.3)/2
2tet
 
(3.3.3.3.3.3)/2
2tet
 
(3.3.3.3.3.3)/2
2tet
 
2(6/2.3.6/2.3)
2oct
 
2(6/2.3.6/2.3)
2oct
 
2(6/2.3.6/2.3)
2oct
 
6(6/2.6/2.6/2)
6tet
?

OctahedralEdit

In octahedral Schwarz triangles, the maximum numerator allowed is 4. There also exist octahedral Schwarz triangles which use 4/2 as a number, but these only lead to degenerate uniform polyhedra as 4 and 2 have a common factor.

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (4 3 2)
(µ=1)
 
4.4.4
cube
U6
 
3.3.3.3
oct
U5
 
3.4.3.4
co
U7
 
3.8.8
tic
U9
 
4.6.6
toe
U8
 
4.3.4.4
sirco
U10
 
4.6.8
girco
U11
 
3.3.3.3.4
snic
U12
2 (4 4 3/2)
(µ=2)
 
(3/2.4)4
oct+6{4}
 
(3/2.4)4
oct+6{4}
 
(4.4.4.4.4.4)/2
2cube
 
3/2.8.4.8
socco
U13
 
3/2.8.4.8
socco
U13
 
2(6/2.4.6/2.4)
2co
 
2(6/2.8.8)
2tic
?
3 (4 3 4/3)
(µ=4)
 
(4.4.4.4.4.4)/2
2cube
 
(3/2.4)4
oct+6{4}
 
(3/2.4)4
oct+6{4}
 
3/2.8.4.8
socco
U13
 
2(4/3.6.4.6)
2cho
U15*
 
3.8/3.4.8/3
gocco
U14
 
6.8.8/3
cotco
U16
?
4 (4 2 3/2)
(µ=5)
 
3.4.3.4
co
U7
 
3.3.3.3
oct
U5
 
4.4.4
cube
U6
 
3.8.8
tic
U9
 
4.4.3/2.4
querco
U17
 
4(4.6/2.6/2)
2oct+6{4}
 
2(4.6/2.8)
sroh+8{6/2}
U18*
?
5 (3 2 4/3)
(µ=7)
 
3.4.3.4
co
U7
 
4.4.4
cube
U6
 
3.3.3.3
oct
U5
 
4.6.6
toe
U8
 
4.4.3/2.4
querco
U17
 
3.8/3.8/3
quith
U19
 
4.6/5.8/3
quitco
U20
?
6 (2 3/2 4/3)
(µ=11)
 
4.4.4
cube
U6
 
3.4.3.4
co
U7
 
3.3.3.3
oct
U5
 
4.3.4.4
sirco
U10
 
4(4.6/2.6/2)
2oct+6{4}
 
3.8/3.8/3
quith
U19
 
2(4.6/2.8/3)
groh+8{6/2}
U21*
?
7 (3/2 4/3 4/3)
(µ=14)
 
(3/2.4)4 = (3.4)4/3
oct+6{4}
 
(4.4.4.4.4.4)/2
2cube
 
(3/2.4)4 = (3.4)4/3
oct+6{4}
 
2(6/2.4.6/2.4)
2co
 
3.8/3.4.8/3
gocco
U14
 
3.8/3.4.8/3
gocco
U14
 
2(6/2.8/3.8/3)
2quith
?

IcosahedralEdit

In icosahedral Schwarz triangles, the maximum numerator allowed is 5. Additionally, the numerator 4 cannot be used at all in icosahedral Schwarz triangles, although numerators 2 and 3 are allowed. (If 4 and 5 could occur together in some Schwarz triangle, they would have to do so in some Möbius triangle as well; but this is impossible as (2 4 5) is a hyperbolic triangle, not a spherical one.)

# (p q r) q | p r
(p.r)q
p | q r
(q.r)p
r | p q
(q.p)r
q r | p
q.2p.r.2p
p r | q
p. 2q.r.2q
p q | r
2r.q.2r.p
p q r |
2r.2q.2p
| p q r
3.r.3.q.3.p
1 (5 3 2)
(µ=1)
 
5.5.5
doe
U23
 
3.3.3.3.3
ike
U22
 
3.5.3.5
id
U24
 
3.10.10
tid
U26
 
5.6.6
ti
U25
 
4.3.4.5
srid
U27
 
4.6.10
grid
U28
 
3.3.3.3.5
snid
U29
2 (3 3 5/2)
(µ=2)
 
3.5/2.3.5/2.3.5/2
sidtid
U30
 
3.5/2.3.5/2.3.5/2
sidtid
U30
 
(310)/2
2ike
 
3.6.5/2.6
siid
U31
 
3.6.5/2.6
siid
U31
 
2(10/2.3.10/2.3)
2id
 
2(10/2.6.6)
2ti
 
3.5/2.3.3.3.3
seside
U32
3 (5 5 3/2)
(µ=2)
 
(5.3/2)5
cid
 
(5.3/2)5
cid
 
(5.5.5.5.5.5)/2
2doe
 
5.10.3/2.10
saddid
U33
 
5.10.3/2.10
saddid
U33
 
2(6/2.5.6/2.5)
2id
 
2(6/2.10.10)
2tid
 
2(3.3/2.3.5.3.5)
2id+40{3}
4 (5 5/2 2)
(µ=3)
 
(5.5.5.5.5)/2
gad
U35
 
5/2.5/2.5/2.5/2.5/2
sissid
U34
 
5/2.5.5/2.5
did
U36
 
5/2.10.10
tigid
U37
 
5.10/2.10/2
3doe
 
4.5/2.4.5
raded
U38
 
2(4.10/2.10)
sird+12{10/2}
U39*
 
3.3.5/2.3.5
siddid
U40
5 (5 3 5/3)
(µ=4)
 
5.5/3.5.5/3.5.5/3
ditdid
U41
 
(3.5/3)5
gacid
 
(3.5)5/3
cid
 
3.10.5/3.10
sidditdid
U43
 
5.6.5/3.6
ided
U44
 
10/3.3.10/3.5
gidditdid
U42
 
10/3.6.10
idtid
U45
 
3.5/3.3.3.3.5
sided
U46
6 (5/2 5/2 5/2)
(µ=6)
 
(5/2)10/2
2sissid
 
(5/2)10/2
2sissid
 
(5/2)10/2
2sissid
 
2(5/2.10/2)2
2did
 
2(5/2.10/2)2
2did
 
2(5/2.10/2)2
2did
 
6(10/2.10/2.10/2)
6doe
 
3(3.5/2.3.5/2.3.5/2)
3sidtid
7 (5 3 3/2)
(µ=6)
 
(3.5.3.5.3.5)/2
gidtid
U47
 
(310)/4
2gike
 
(3.5.3.5.3.5)/2
gidtid
U47
 
2(3.10.3/2.10)
2seihid
U49*
 
5.6.3/2.6
giid
U48
 
5(6/2.3.6/2.5)
3ike+gad
 
2(6.6/2.10)
siddy+20{6/2}
U50*
 
5(3.3.3.3.3.5)/2
5ike+gad
8 (5 5 5/4)
(µ=6)
 
(510)/4
2gad
 
(510)/4
2gad
 
(510)/4
2gad
 
2(5.10.5/4.10)
2sidhid
U51*
 
2(5.10.5/4.10)
2sidhid
U51*
 
10/4.5.10/4.5
2did
 
2(10/4.10.10)
2tigid
 
3(3.5.3.5.3.5)
3cid
9 (3 5/2 2)
(µ=7)
 
(3.3.3.3.3)/2
gike
U53
 
5/2.5/2.5/2
gissid
U52
 
5/2.3.5/2.3
gid
U54
 
5/2.6.6
tiggy
U55
 
3.10/2.10/2
2gad+ike
 
3(4.5/2.4.3)
sicdatrid
 
4.10/2.6
ri+12{10/2}
U56*
 
3.3.5/2.3.3
gosid
U57
10 (5 5/2 3/2)
(µ=8)
 
(5.3/2)5
cid
 
(5/3.3)5
gacid
 
5.5/3.5.5/3.5.5/3
ditdid
U41
 
5/3.10.3.10
sidditdid
U43
 
5(5.10/2.3.10/2)
ike+3gad
 
3(6/2.5/2.6/2.5)
sidtid+gidtid
 
4(6/2.10/2.10)
id+seihid+sidhid
?
(3|3 5/2) + (3/2|3 5)
11 (5 2 5/3)
(µ=9)
 
5.5/2.5.5/2
did
U36
 
5/2.5/2.5/2.5/2.5/2
sissid
U34
 
(5.5.5.5.5)/2
gad
U35
 
5/2.10.10
tigid
U37
 
3(5.4.5/3.4)
cadditradid
 
10/3.5.5
quit sissid
U58
 
10/3.4.10/9
quitdid
U59
 
3.5/3.3.3.5
isdid
U60
12 (3 5/2 5/3)
(µ=10)
 
(3.5/3)5
gacid
 
(5/2)6/2
2gissid
 
(5/2.3)5/3
gacid
 
2(5/2.6.5/3.6)
2sidhei
U62*
 
3(3.10/2.5/3.10/2)
ditdid+gidtid
 
10/3.5/2.10/3.3
gaddid
U61
 
10/3.10/2.6
giddy+12{10/2}
U63*
 
3.5/3.3.5/2.3.3
gisdid
U64
13 (5 3 5/4)
(µ=10)
 
(5.5.5.5.5.5)/2
2doe
 
(3/2.5)5
cid
 
(3.5)5/3
cid
 
3/2.10.5.10
saddid
U33
 
2(5.6.5/4.6)
2gidhei
U65*
 
3(10/4.3.10/4.5)
sidtid+ditdid
 
2(10/4.6.10)
siddy+12{10/4}
U50*
?
14 (5 2 3/2)
(µ=11)
 
5.3.5.3
id
U24
 
3.3.3.3.3
ike
U22
 
5.5.5
doe
U23
 
3.10.10
tid
U26
 
3(5/4.4.3/2.4)
gicdatrid
 
5(5.6/2.6/2)
2ike+gad
 
2(6/2.4.10)
sird+20{6/2}
U39*
 
5(3.3.3.5.3)/2
4ike+gad
15 (3 2 5/3)
(µ=13)
 
3.5/2.3.5/2
gid
U54
 
5/2.5/2.5/2
gissid
U52
 
(3.3.3.3.3)/2
gike
U53
 
5/2.6.6
tiggy
U55
 
3.4.5/3.4
qrid
U67
 
10/3.10/3.3
quit gissid
U66
 
10/3.4.6
gaquatid
U68
 
3.5/3.3.3.3
gisid
U69
16 (5/2 5/2 3/2)
(µ=14)
 
(5/3.3)5
gacid
 
(5/3.3)5
gacid
 
(5/2)6/2
2gissid
 
3(5/3.10/2.3.10/2)
ditdid+gidtid
 
3(5/3.10/2.3.10/2)
ditdid+gidtid
 
2(6/2.5/2.6/2.5/2)
2gid
 
10(6/2.10/2.10/2)
2ike+4gad
?
17 (3 3 5/4)
(µ=14)
 
(3.5.3.5.3.5)/2
gidtid
U47
 
(3.5.3.5.3.5)/2
gidtid
U47
 
(3)10/4
2gike
 
3/2.6.5.6
giid
U48
 
3/2.6.5.6
giid
U48
 
2(10/4.3.10/4.3)
2gid
 
2(10/4.6.6)
2tiggy
?
18 (3 5/2 5/4)
(µ=16)
 
(3/2.5)5
cid
 
5/3.5.5/3.5.5/3.5
ditdid
U41
 
(5/2.3)5/3
gacid
 
5/3.6.5.6
ided
U44
 
5(3/2.10/2.5.10/2)
ike+3gad
 
5(10/4.5/2.10/4.3)
3sissid+gike
 
4(10/4.10/2.6)
did+sidhei+gidhei
?
19 (5/2 2 3/2)
(µ=17)
 
3.5/2.3.5/2
gid
U54
 
(3.3.3.3.3)/2
gike
U53
 
5/2.5/2.5/2
gissid
U52
 
5(10/2.3.10/2)
2gad+ike
 
5/3.4.3.4
qrid
U67
 
5(6/2.6/2.5/2)
2gike+sissid
 
6(6/2.4.10/2)
2gidtid+rhom
?
20 (5/2 5/3 5/3)
(µ=18)
 
(5/2)10/2
2sissid
 
(5/2)10/2
2sissid
 
(5/2)10/2
2sissid
 
2(5/2.10/2)2
2did
 
2(5/2.10/3.5/3.10/3)
2gidhid
U70*
 
2(5/2.10/3.5/3.10/3)
2gidhid
U70*
 
2(10/3.10/3.10/2)
2quitsissid
?
21 (3 5/3 3/2)
(µ=18)
 
(310)/2
2ike
 
5/2.3.5/2.3.5/2.3
sidtid
U30
 
5/2.3.5/2.3.5/2.3
sidtid
U30
 
5/2.6.3.6
siid
U31
 
2(3.10/3.3/2.10/3)
2geihid
U71*
 
5(6/2.5/3.6/2.3)
sissid+3gike
 
2(6/2.10/3.6)
giddy+20{6/2}
U63*
?
22 (3 2 5/4)
(µ=19)
 
3.5.3.5
id
U24
 
5.5.5
doe
U23
 
3.3.3.3.3
ike
U22
 
5.6.6
ti
U25
 
3(3/2.4.5/4.4)
gicdatrid
 
5(10/4.10/4.3)
2sissid+gike
 
2(10/4.4.6)
ri+12{10/4}
U56*
?
23 (5/2 2 5/4)
(µ=21)
 
5/2.5.5/2.5
did
U36
 
(5.5.5.5.5)/2
gad
U35
 
5/2.5/2.5/2.5/2.5/2
sissid
U34
 
3(10/2.5.10/2)
3doe
 
3(5/3.4.5.4)
cadditradid
 
3(10/4.5/2.10/4)
3gissid
 
6(10/4.4.10/2)
2ditdid+rhom
?
24 (5/2 3/2 3/2)
(µ=22)
 
5/2.3.5/2.3.5/2.3
sidtid
U30
 
(310)/2
2ike
 
5/2.3.5/2.3.5/2.3
sidtid
U30
 
2(3.10/2.3.10/2)
2id
 
5(5/3.6/2.3.6/2)
sissid+3gike
 
5(5/3.6/2.3.6/2)
sissid+3gike
 
10(6/2.6/2.10/2)
4ike+2gad
 
(3.3.3.3.3.5/2)/2
sirsid
U72
25 (2 5/3 3/2)
(µ=23)
 
(3.3.3.3.3)/2
gike
U53
 
5/2.3.5/2.3
gid
U54
 
5/2.5/2.5/2
gissid
U52
 
3(5/2.4.3.4)
sicdatrid
 
10/3.3.10/3
quit gissid
U66
 
5(6/2.5/2.6/2)
2gike+sissid
 
2(6/2.10/3.4)
gird+20{6/2}
U73*
 
(3.3.3.5/2.3)/2
girsid
U74
26 (5/3 5/3 3/2)
(µ=26)
 
(5/2.3)5/3
gacid
 
(5/2.3)5/3
gacid
 
(5/2)6/2
2gissid
 
5/2.10/3.3.10/3
gaddid
U61
 
5/2.10/3.3.10/3
gaddid
U61
 
2(6/2.5/2.6/2.5/2)
2gid
 
2(6/2.10/3.10/3)
2quitgissid
?
27 (2 5/3 5/4)
(µ=27)
 
(5.5.5.5.5)/2
gad
U35
 
5/2.5.5/2.5
did
U36
 
5/2.5/2.5/2.5/2.5/2
sissid
U34
 
5/2.4.5.4
raded
U38
 
10/3.5.10/3
quit sissid
U58
 
3(10/4.5/2.10/4)
3gissid
 
2(10/4.10/3.4)
gird+12{10/4}
U73*
?
28 (2 3/2 5/4)
(µ=29)
 
5.5.5
doe
U23
 
3.5.3.5
id
U24
 
3.3.3.3.3
ike
U22
 
3.4.5.4
srid
U27
 
2(6/2.5.6/2)
2ike+gad
 
5(10/4.3.10/4)
2sissid+gike
 
6(10/4.6/2.4/3)
2sidtid+rhom
?
29 (5/3 3/2 5/4)
(µ=32)
 
5/3.5.5/3.5.5/3.5
ditdid
U41
 
(3.5)5/3
cid
 
(3.5/2)5/3
gacid
 
3.10/3.5.10/3
gidditdid
U42
 
3(5/2.6/2.5.6/2)
sidtid+gidtid
 
5(10/4.3.10/4.5/2)
3sissid+gike
 
4(10/4.6/2.10/3)
gid+geihid+gidhid
?
30 (3/2 3/2 5/4)
(µ=34)
 
(3.5.3.5.3.5)/2
gidtid
U47
 
(3.5.3.5.3.5)/2
gidtid
U47
 
(3)10/4
2gike
 
5(3.6/2.5.6/2)
3ike+gad
 
5(3.6/2.5.6/2)
3ike+gad
 
2(10/4.3.10/4.3)
2gid
 
10(10/4.6/2.6/2)
2sissid+4gike
?
31 (3/2 5/4 5/4)
(µ=38)
 
(3.5)5/3
cid
 
(5.5.5.5.5.5)/2
2doe
 
(3.5)5/3
cid
 
2(5.6/2.5.6/2)
2id
 
3(3.10/4.5/4.10/4)
sidtid+ditdid
 
3(3.10/4.5/4.10/4)
sidtid+ditdid
 
10(10/4.10/4.6/2)
4sissid+2gike
 
5(3.3.3.5/4.3.5/4)
4ike+2gad
32 (5/4 5/4 5/4)
(µ=42)
 
(5)10/4
2gad
 
(5)10/4
2gad
 
(5)10/4
2gad
 
2(5.10/4.5.10/4)
2did
 
2(5.10/4.5.10/4)
2did
 
2(5.10/4.5.10/4)
2did
 
6(10/4.10/4.10/4)
2gissid
 
3(3/2.5.3/2.5.3/2.5)
3cid

Non-WythoffianEdit

Hemi formsEdit

These polyhedra (the hemipolyhedra) are generated as double coverings by the Wythoff construction. If a figure generated by the Wythoff construction is composed of two identical components, the "hemi" operator takes only one. The octahemioctahedron is included in the table for completeness, although it is not generated as a double cover by the Wythoff construction.

 
3/2.4.3.4
thah
U4
hemi(3 3/2 | 2)
 
4/3.6.4.6
cho
U15
hemi(4 4/3 | 3)
 
5/4.10.5.10
sidhid
U51
hemi(5 5/4 | 5)
 
5/2.6.5/3.6
sidhei
U62
hemi(5/2 5/3 | 3)
 
5/2.10/3.5/3.10/3
gidhid
U70
hemi(5/2 5/3 | 5/3)
   
3/2.6.3.6
oho
U3
hemi(?)
 
3/2.10.3.10
seihid
U49
hemi(3 3/2 | 5)
 
5.6.5/4.6
gidhei
U65
hemi(5 5/4 | 3)
 
3.10/3.3/2.10/3
geihid
U71
hemi(3 3/2 | 5/3)

Reduced formsEdit

These polyhedra are generated with extra faces by the Wythoff construction. If a figure is generated by the Wythoff construction as being composed of two or three non-identical components, the "reduced" operator removes extra faces (that must be specified) from the figure, leaving only one component.

Wythoff Polyhedron Extra faces   Wythoff Polyhedron Extra faces   Wythoff Polyhedron Extra faces
3 2 3/2 |  
4.6.4/3.6
cho
U15
4{6/2}   4 2 3/2 |  
4.8.4/3.8/7
sroh
U18
8{6/2}   2 3/2 4/3 |  
4.8/3.4/3.8/5
groh
U21
8{6/2}
5 5/2 2 |  
4.10.4/3.10/9
sird
U39
12{10/2}   5 3 3/2 |  
10.6.10/9.6/5
siddy
U50
20{6/2}   3 5/2 2 |  
6.4.6/5.4/3
ri
U56
12{10/2}
5 5/2 3/2 |  
3/2.10.3.10
seihid
U49
id + sidhid   5 5/2 3/2 |  
5/4.10.5.10
sidhid
U51
id + seihid   5 3 5/4 |  
10.6.10/9.6/5
siddy
U50
12{10/4}
3 5/2 5/3 |  
6.10/3.6/5.10/7
giddy
U63
12{10/2}   5 2 3/2 |  
4.10/3.4/3.10/9
sird
U39
20{6/2}   3 5/2 5/4 |  
5.6.5/4.6
gidhei
U65
did + sidhei
3 5/2 5/4 |  
5/2.6.5/3.6
sidhei
U62
did + gidhei   3 5/3 3/2 |  
6.10/3.6/5.10/7
giddy
U63
20{6/2}   3 2 5/4 |  
6.4.6/5.4/3
ri
U56
12{10/4}
2 5/3 3/2 |  
4.10/3.4/3.10/7
gird
U73
20{6/2}   5/3 3/2 5/4 |  
3.10/3.3/2.10/3
geihid
U71
gid + gidhid   5/3 3/2 5/4 |  
5/2.10/3.5/3.10/3
gidhid
U70
gid + geihid
2 5/3 5/4 |  
4.10/3.4/3.10/7
gird
U73
12{10/4}                

The tetrahemihexahedron (thah, U4) is also a reduced version of the {3/2}-cupola (retrograde triangular cupola, ratricu) by {6/2}. As such it may also be called the crossed triangular cuploid.

Many cases above are derived from degenerate omnitruncated polyhedra p q r |. In these cases, two distinct degenerate cases p q r | and p q s | can be generated from the same p and q; the result has faces {2p}'s, {2q}'s, and coinciding {2r}'s or {2s}'s respectively. These both yield the same nondegenerate uniform polyhedra when the coinciding faces are discarded, which Coxeter symbolised p q r
s
|. These cases are listed below:

 
4.6.4/3.6
cho
U15
2 3 3/2
3/2
|
 
4.8.4/3.8/7
sroh
U18
2 3 3/2
4/2
|
 
4.10.4/3.10/9
sird
U39
2 3 3/2
5/2
|
 
6.10/3.6/5.10/7
giddy
U63
3 5/3 3/2
5/2
|
 
6.4.6/5.4/3
ri
U56
2 3 5/4
5/2
|
 
4.8/3.4/3.8/5
groh
U21
2 4/3 3/2
4/2
|
 
4.10/3.4/3.10/7
gird
U73
2 5/3 3/2
5/4
|
 
10.6.10/9.6/5
siddy
U50
3 5 3/2
5/4
|

In the small and great rhombihexahedra, the fraction 4/2 is used despite it not being in lowest terms. While 2 4 2 | and 2 4/3 2 | represent a single octagonal or octagrammic prism respectively, 2 4 4/2 | and 2 4/3 4/2 | represent three such prisms, which share some of their square faces (precisely those doubled up to produce {8/2}'s). These {8/2}'s appear with fourfold and not twofold rotational symmetry, justifying the use of 4/2 instead of 2.[1]

Other formsEdit

These two uniform polyhedra cannot be generated at all by the Wythoff construction. This is the set of uniform polyhedra commonly described as the "non-Wythoffians". Instead of the triangular fundamental domains of the Wythoffian uniform polyhedra, these two polyhedra have tetragonal fundamental domains.

Skilling's figure is not given an index in Maeder's list due to it being an exotic uniform polyhedron, with ridges (edges in the 3D case) completely coincident. This is also true of some of the degenerate polyhedron included in the above list, such as the small complex icosidodecahedron. This interpretation of edges being coincident allows these figures to have two faces per edge: not doubling the edges would give them 4, 6, 8, 10 or 12 faces meeting at an edge, figures that are usually excluded as uniform polyhedra. Skilling's figure has 4 faces meeting at some edges.

(p q r s) | p q r s
(4.p. 4.q.4.r.4.s)/2
| (p) q (r) s
(p3.4.q.4.r3.4.s.4)/2
(3/2 5/3 3 5/2)  
(4.3/2.4.5/3.4.3.4.5/2)/2
gidrid
U75
 
(3/23.4.5/3.4.33.4.5/2.4)/2
gidisdrid
Skilling
 
Vertex figure of | 3 5/3 5/2
 
Great snub dodecicosidodecahedron
 
Great dirhombicosidodecahedron
 
Vertex figure of | 3/2 5/3 3 5/2
 
Great disnub dirhombidodecahedron
 
Compound of twenty octahedra
 
Compound of twenty tetrahemihexahedra
 
Vertex figure of |(3/2) 5/3 (3) 5/2

Both of these special polyhedra may be derived from the great snub dodecicosidodecahedron, | 3 5/3 5/2 (U64). This is a chiral snub polyhedron, but its pentagrams appear in coplanar pairs. Combining one copy of this polyhedron with its enantiomorph, the pentagrams coincide and may be removed. As the edges of this polyhedron's vertex figure include three sides of a square, with the fourth side being contributed by its enantiomorph, we see that the resulting polyhedron is in fact the compound of twenty octahedra. Each of these octahedra contain one pair of parallel faces that stem from a fully symmetric triangle of | 3 5/3 5/2, while the other three come from the original | 3 5/3 5/2's snub triangles. Additionally, each octahedron can be replaced by the tetrahemihexahedron with the same edges and vertices. Taking the fully symmetric triangles in the octahedra, the original coinciding pentagrams in the great snub dodecicosidodecahedra, and the equatorial squares of the tetrahemihexahedra together yields the great dirhombicosidodecahedron (Miller's monster).[1] Taking the snub triangles of the octahedra instead yields the great disnub dirhombidodecahedron (Skilling's figure).[2]

ReferencesEdit

  1. ^ a b c Coxeter, 1954
  2. ^ Skilling, 1974
  • Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954). "Uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 246 (916): 401–450. doi:10.1098/rsta.1954.0003. ISSN 0080-4614. JSTOR 91532. MR 0062446.CS1 maint: ref=harv (link) [1]
  • Skilling, J. (1974). "The complete set of uniform polyhedra". Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences. The Royal Society. 278 (1278): 111–135. doi:10.1098/rsta.1975.0022. ISSN 1364-503X.CS1 maint: ref=harv (link) [2]

External linksEdit

Richard Klitzing: Polyhedra by

Zvi Har'El: