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User:Tomruen/Uniform polyhedron

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Convex uniform polyhedra

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Parent Truncated Rectified Bitruncated
(truncated dual) 		Birectified
(dual) 		Cantellated Cantitruncated
(Omnitruncated) 		Snub
Extended
Schläfli symbol 		               
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
Coxeter-Dynkin diagram
(variations)
(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

2D symmetry

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Operation Parent Truncated Rectified Truncated dual Dual Cantellated Omnitruncated Snub
(Extended-1)
Schläfli symbols 		               
(Extended-2)
Schläfli symbols 		t0{p,q}
t2{q,p} 		t0,1{p,q}
t1,2{q,p} 		t1{p,q}
t1{q,p} 		t1,2{p,q}
t0,1{q,p} 		t2{p,q}
t0{q,p} 		t0,2{p,q}
t0,2{q,p} 		t0,1,2{p,q}
t0,1,2{q,p} 		s{p,q}
s{q,p}
Wythoff Symbol 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)
Square
4-4-2 		 
{4a,4} 		 
(4b.8a.8a) 		 
(4a.4b.4a.4b) 		 
(4a.8b.8b) 		 
{4b,4} 		 
(4a.4c.4b.4c) 		 
(4c.8a.8b) 		 
(3c.3c.4a.3d.4b)
Pentagonal
(Order 4)
5-4-2
{3,5}
6.5.5
4.5.4.5
3.10.10
{5,4}
4.4.4.5
4.8.10
3.3.4.3.5
Hexagonal
6-3-2 		 
{3,6} 		 
(6.6.6) 		 
(3.6.3.6) 		 
(3.12.12) 		 
{6,3} 		 
(3.4.6.4) 		 
(4.6.12) 		 
(3.3.3.3.6)
Septagonal
(Order 3)
7-3-2 		 
{3,7} 		 
(6.7.7) 		 
(3.7.3.7) 		 
(3.14.14) 		 
{3,7} 		 
(3.4.7.4) 		 
(4.6.14) 		 
(3.3.3.3.7)

3D Nonconvex with right triangles

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Operation Parent Truncated Rectified Truncated dual Dual Cantellated Omnitruncated Snub
(Extended-1)
Schläfli symbols 		               
(Extended-2)
Schläfli symbols 		t0{p,q}
t2{q,p} 		t0,1{p,q}
t1,2{q,p} 		t1{p,q}
t1{q,p} 		t1,2{p,q}
t0,1{q,p} 		t2{p,q}
t0{q,p} 		t0,2{p,q}
t0,2{q,p} 		t0,1,2{p,q}
t0,1,2{q,p} 		s{p,q}
s{q,p}
Johnson
2-1-0 subscripts 		00x 0xx 0x0 xx0 x00 x0x xxx ---
Wythoff Symbol 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)
Icosahedral(2)
5/2-3-2 		 
{3,5/2} 		 
(5/2.6.6) 		 
(3.5/2)2 		 
[3.10/2.10/2] 		 
{5/2,3} 		 
[3.4.5/2.4] 		 
[4.10/2.6] 		 
(3.3.3.3.5/2)
Icosahedral(3)
5-5/2-2 		 
{5,5/2} 		 
(5/2.10.10) 		 
(5/2.5)2 		 
(5.10/2.10/2) 		 
{5/2,5} 		 
(5/2.4.5.4) 		 
[4.10/2.10] 		 
(3.3.5/2.3.5)

Notes

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  1. on nonconvex polyhedra: [a.b...] are degenerate vertex figures - overlapping vertices and edges. Similar uniform polyhedra are displayed, but somewhat different structure.
    • [3.5.5] is a compound {3,5} and {5,5/2}
    • [3.4.5/2.4] is a compound of (5/2.3)3 and 5 {4,3}.
    • [4.5.6] is a compound of ...
    • [4.5.10] is a compound of ...
  2. Dihedrals - added as a test, although since they are not "regular", operations applied are polygon-based.
  3. Tilings - 442, 643, 333 - added, but greater need for consistent colorings/orientatios

3D Nonconvex with general triangles

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Mod.Schlafli t0[p,r,q] t0,1[p,r,q] t1[p,r,q] t1,2[p,r,q] t2[p,r,q] t0,2[p,r,q] t0,1,2[p,r,q] s{p,r,q}
Johnson
2-1-0 subscripts 		00x 0xx 0x0 xx0 x00 x0x xxx ---
Wythoff Symbol q | r p r q | p r | p q r p | q p | r q p q | r r p q | | r p q
Vertex Figure (p.2)q q.2p.2p (p.q)r p.2q.r.2q (q.2)p p.2r.q.2r 2r.2p.2q 3.r.3.p.3.q

Construnction summary chart

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Example operations on octahedron
 
fix
 
fix2
 
Generating triangles
 
(Better) generating triangles
 
fix
 
fix2

Degenerate cases of Wythoff's construction

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See: [1]

This table shows a list of 45 degenerate cases of Wythoff's construction, enumerated by Coxeter in the 1954 paper, Uniform polyhedra. They exist as polyhedral compounds. They are indexed in the order listed in this paper (table 6), with case 6 subindexed in 3 forms: a,b,c.

No. Picture Wythoff Symbol Vertex figure Compounds
D1   4 | 3/2 4 (3.4)4 -{3,4}+3{4,2} (Octahedron with 3 internal central squares)
D2   5 | 3/2 5 (3.5/4)5 -{3,5}+{5,5/2} (icosahedron with internal great dodecahedron)
D3 5 | 3 5/3 (3.5/3)5 {3,5/2}-{5/2,5}
D4 5/2 | 3 5 (3.5)5 {3,5}+{5,5/2}
D5 5/3 | 3 5/2 (3.5/2)5 {3,5/2}+{5/2,5}
D6a 2 3 | 3/2 3{3,3}
D6b 2 5 | 5/2 3{4,3}
D6c 2 5/2 | 5/4 3{5,3}
D7 2 3 | 5/2 {3,5}+2{5,5/2}
D8 2 4 | 3/2 3{4,2}+2{3,4}
D9 2 5 | 3/2 -{5,5/2}+2{3,5}
D10 2 5/2 | 3/2 {5/2,5}+2{3,5}
D11 2 3 | 5/4 -{3,5/2}+2{5/2,5}
D12 3 5/2 | 2 3 5/2)+5{4,3}
D13 5/3 5 | 2 5/3 5)+5{4,3}
D14 3/2 5 | 2 3 5)+5{4,3}
D15 3 5 | 3/2 3{3,5}+{5,5/2}
D16 3/2 5 | 5/2
D17 3 5/2 | 5/4
D18 3 5/3 | 3/2
D19 5/2 5 | 3/2
D20 3 5/3 | 5/2
D21 3 5 | 5/4
D22 3/2 5/2 | 5
D23 3 5/4 5/2 |
D24 2 3/2 5/2 |
D25 2 5/4 5/2 |
D26 2 3/2 5/4 |
D27 3/2 5/4 5/3 |
D28 | 5/2 5/2 5/2
D29 | 5/4 5/4 5/4
D30a | 3/2 3 3
D30b | 3/2 4 4
D30c | 3/2 5 5
D30d | 3/2 5/2 5/2
D31 | 2 2 3/2
D32 | 2 3/2 3
D33 | 2 3/2 4
D34 | 3/2 3 5
D35 | 3/2 5/2 5
D36 | 2 3/2 5
D37 | 2 3/2 5/2
D38 | 3/2 3 5/3
D39 | 3/2 5/3 5/3
D40 | 3/2 5/4 5/4
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