Convex uniform polyhedra edit
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 edit
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 edit
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 edit
- 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 ...
- Dihedrals - added as a test, although since they are not "regular", operations applied are polygon-based.
- Tilings - 442, 643, 333 - added, but greater need for consistent colorings/orientatios
3D Nonconvex with general triangles edit
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 edit
Degenerate cases of Wythoff's construction edit
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.