User:Tomruen/Uniform polyhedron

From Wikipedia, the free encyclopedia

Contents

[edit] Convex uniform polyhedra

Parent Truncated Rectified Bitruncated
(truncated dual)		 Birectified
(dual)		 Cantellated Cantitruncated
(Omnitruncated)		 Snub
Extended
Schläfli symbol		 \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}
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)		 Image:dynkins-100.png Image:dynkins-110.png Image:dynkins-010.png Image:dynkins-011.png Image:dynkins-001.png Image:dynkins-101.png Image:dynkins-111.png Image:Dynkins-sss.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

[edit] 2D symmetry

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}
(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)

[edit] 3D Nonconvex with right triangles

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}
(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)

[edit] Notes

  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

[edit] 3D Nonconvex with general triangles

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

[edit] Construnction summary chart

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

[edit] Degenerate cases of Wythoff's construction

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