User:Tomruen/Uniform polyhedron

Convex uniform polyhedra[edit]

	Parent	Truncated	Rectified	Bitruncated (truncated dual)	Birectified (dual)	Cantellated	Cantitruncated (Omnitruncated)	Snub
Extended Schläfli symbol	${\begin{Bmatrix}p,q\end{Bmatrix}}$	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	${\begin{Bmatrix}q,p\end{Bmatrix}}$	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$s{\begin{Bmatrix}p\\q\end{Bmatrix}}$
Extended Schläfli symbol	t₀{p,q}	t_0,1{p,q}	t₁{p,q}	t_1,2{p,q}	t₂{p,q}	t_0,2{p,q}	t_0,1,2{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	p^q	(q.2p.2p)	(p.q.p.q)	(p.2q.2q)	q^p	(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	${\begin{Bmatrix}p,q\end{Bmatrix}}$	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	${\begin{Bmatrix}q,p\end{Bmatrix}}$	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$s{\begin{Bmatrix}p\\q\end{Bmatrix}}$
(Extended-2) Schläfli symbols	t₀{p,q} t₂{q,p}	t_0,1{p,q} t_1,2{q,p}	t₁{p,q} t₁{q,p}	t_1,2{p,q} t_0,1{q,p}	t₂{p,q} t₀{q,p}	t_0,2{p,q} t_0,2{q,p}	t_0,1,2{p,q} t_0,1,2{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	(p^q)	(q.2p.2p)	(p.q)²	(p.2q.2q)	(q^p)	(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	${\begin{Bmatrix}p,q\end{Bmatrix}}$	$t{\begin{Bmatrix}p,q\end{Bmatrix}}$	${\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}q,p\end{Bmatrix}}$	${\begin{Bmatrix}q,p\end{Bmatrix}}$	$r{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$t{\begin{Bmatrix}p\\q\end{Bmatrix}}$	$s{\begin{Bmatrix}p\\q\end{Bmatrix}}$
(Extended-2) Schläfli symbols	t₀{p,q} t₂{q,p}	t_0,1{p,q} t_1,2{q,p}	t₁{p,q} t₁{q,p}	t_1,2{p,q} t_0,1{q,p}	t₂{p,q} t₀{q,p}	t_0,2{p,q} t_0,2{q,p}	t_0,1,2{p,q} t_0,1,2{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	(p^q)	(q.2p.2p)	(p.q)²	(p.2q.2q)	(q^p)	(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]

Johnson 2-1-0 subscripts	00x	0xx	0x0	xx0	x00	x0x	xxx	---
Mod.Schlafli	t₀[p,r,q]	t_0,1[p,r,q]	t₁[p,r,q]	t_1,2[p,r,q]	t₂[p,r,q]	t_0,2[p,r,q]	t_0,1,2[p,r,q]	s{p,r,q}
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.

No.	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}
D_6a	2 3 \| 3/2		3{3,3}
D_6b	2 5 \| 5/2		3{4,3}
D_6c	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
D_30a	\| 3/2 3 3
D_30b	\| 3/2 4 4
D_30c	\| 3/2 5 5
D_30d	\| 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