Uniform star polyhedron/Uniform polyhedra by Wythoff construction

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

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

These are called Mobius 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 Triangles
1 (2 3 3) (2 3 4) (2 3 5)
d (2 2 n/d)
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 (3/23/2 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)

Right triangle generators

Operation Parent Truncated Rectified Truncated dual Dual Cantellated Omnitruncated Snub
(Extended-1)
Schläfli symbols
\begin{Bmatrix} p , q \end{Bmatrix} t\begin{Bmatrix} p , q \end{Bmatrix} \begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} q , p \end{Bmatrix} \begin{Bmatrix} q , p \end{Bmatrix} r\begin{Bmatrix} p \\ q \end{Bmatrix} t\begin{Bmatrix} p \\ q \end{Bmatrix} s\begin{Bmatrix} p \\ q \end{Bmatrix}
Wythoff Symbol
p-q-2
q | 2 p 2 q | p 2 | p q 2 p | q p | 2 q p q | 2 2 p q | | 2 p q
Vertex Figure (pq) (q.2p.2p) (p.q)2 (p. 2q.2q) (qp) (p. 4.q.4) (4.2p.2q) (3.3.p. 3.q)
Dihedral symmetry
(2 2 2)
(3 2 2)
(4 2 2)
(5 2 2)
(5/2 2 2)
(5/3 2 2)
Tetrahedral symmetry
(3 3 2)
U1

U2

<U5>

U2

U1

<U7>

<U8>

<U22>
(3 3/2 2) -- -- -- -- --
U04
-- --
Octahedral symmetry
(4 3 2)
U6

U9

U7

U8

U5

U10

U11

U12
(4 3/2 2) -- -- -- -- --
U17

U18
--
(4/3 3 2) -- -- --
U19
-- --
U20
--
(4/3 3/2 2) -- -- -- -- -- --
U21
--
Icosahedral symmetry
(5 3 2)
U23

U26

U24

U25

U22

U27

U28

U29
(5 5/2 2)
U34

U37

U36

--

U35

U38

U39

U40
(5/2 3 2)
U52

U55

U54

--

U53

--

U56

U57
(5 5/3 2) -- -- --
U58
-- --
U59

U60
(5/3 3 2) --
U66
-- -- --
U67

U68

U69
(5/3 3/2 2) -- -- -- -- -- --
U73

U74

OTHER (r>2)

Wythoff Symbol q | r p p | r q r | p q r q | p r p | q p q | r r p q | | r p q
Tetrahedral symmetry
(3 3 3/2) U03...
Octahedral symmetry
(4 4 3/2) U13
(4 4/3 3) U14
Icosahedral Symmetry
(5/2 3 3) U30
(5 5 3/2) U33
(5 5/3 3) U41
(5 3 3/2) U47
(5 5 5/4) U51
(5/2 5/3 3) U61
(5 5/4 3) U65
(5/2 5/3 5/3) U70
(5/3 3 3/2) U71
(5/2 3/2 3/2) U72

References