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 |
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) |
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) |
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 |
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.