User:Tomruen/Complex polygons

Rank 2

12 irreducible Shephard groups with their subgroup index relations. Subgroups index 2 relate by removing a real reflection: _p[2q]₂ --> _p[q]_p, index 2.
Also _p[4]_q --> _p[q]_p, index q.

p[4]2 subgroups

Some infinite rank 2 subgroup relations

The symmetry of a regular complex polygon is _p[q]_r, called a Shephard group, analogous to a Coxeter group, while also allowing real and unitary reflections.

The rank 2 solutions that generate complex polygons are: ₂[q]₂, _p[4]₂, ₃[3]₃, ₃[6]₂, ₃[4]₃, ₄[3]₄, ₃[8]₂, ₄[6]₂, ₄[4]₃, ₃[5]₃, ₅[3]₅, ₃[10]₂, ₅[6]₂, and ₅[4]₃ or , , , , , , , , , , , , .

Starry groups with whole q are , , , , , .

Enumeration of regular complex polygons

Coxeter enumerated this list of regular complex polygons in $\mathbb {C} ^{2}$. Groups of the form _p[2q]₂ have a half symmetry _p[q]_p, so a regular polygon is the same as quasiregular .

Group	Order	Image	Polygon		Vertices	Edges		Notes
2[3]2	6		2{6}2		3	3	{}	$\mathbb {R} ^{2}$ equilateral triangle
2[q]2	2q		2{q}2		q	q	{}	$\mathbb {R} ^{2}$ regular polygons
3[4]2	18		3{4}2		9	6	3{}	Same as 3{}×3{} or $\mathbb {R} ^{4}$ representation 3-3 duoprism
			2{4}3		6	9	{}	$\mathbb {R} ^{4}$ representation as 3-3 duopyramid
4[4]2	32		4{4}2		16	8	4{}	Same as 4{}×4{} or $\mathbb {R} ^{4}$ representation 4-4 duoprism or tesseract
			2{4}4		8	16	{}	$\mathbb {R} ^{4}$ representation as 4-4 duopyramid, or 16-cell
5[4]2	50		5{4}2		25	10	5{}	Same as 5{}×5{} or $\mathbb {R} ^{4}$ representation 5-5 duoprism
			2{4}5		10	25	{}	$\mathbb {R} ^{4}$ representation as 5-5 duopyramid
p[4]2	2p2		p{4}2		p2	2p	p{}	Same as p{}×p{} or , $\mathbb {R} ^{2}$ square for p=2 $\mathbb {R} ^{4}$ representation p-p duoprisms and {4,3,3} for p=4
			2{4}p		2p	p2	{}	$\mathbb {R} ^{4}$ representation as p-p duopyramids and {3,3,4} with p=4
3[3]3	24		3{3}3		8	8	3{}	$\mathbb {R} ^{4}$ representation as {3,3,4} Same as
3[6]2	48		3{6}2		24	16	3{}	Same as
			2{6}3		16	24	{}	$\mathbb {R} ^{4}$ representation as {4,3,3}
3[4]3	72		3{4}3		24	24	3{}	$\mathbb {R} ^{4}$ representation as {3,4,3} Same as
4[3]4	96		4{3}4		24	24	4{}	$\mathbb {R} ^{4}$ representation as {3,4,3} Same as
3[8]2	144		3{8}2		72	48	3{}	Same as
			2{8}3		48	72	{}
4[6]2	192		4{6}2		96	48	4{}	Same as
			2{6}4		48	96	{}
4[4]3	288		4{4}3		96	72	4{}
			3{4}4		72	96	3{}
3[5]3	360		3{5}3		120	120	3{}	$\mathbb {R} ^{4}$ representation as {3,3,5} Same as
5[3]5	600		5{3}5		120	120	5{}	$\mathbb {R} ^{4}$ representation as {3,3,5} Same as
3[10]2	720		3{10}2		360	240	3{}	Same as
			2{10}3		240	360	{}
5[6]2	1200		5{6}2		600	240	5{}	Same as $\mathbb {R} ^{4}$ representation as {5,3,3}
			2{6}5		240	600	{}
5[4]3	1800		5{4}3		600	360	5{}	$\mathbb {R} ^{4}$ representation as {5,3,3}
			3{4}5		360	600	3{}

Name	(p,r).m-gon	p{q}r	Cox	p	q	r	h	g	v	e	m
Trionic square	(3,2).3-gon	3{4}2		3	4	2	6	18	9	6	3
Trionous square	(2,3).3-gon	2{4}3		2	4	3	6	18	6	9	3
Tetronic square	(4,2).4-gon	4{4}2		4	4	2	8	32	16	8	4
Tetronus square	(2,4).4-gon	2{4}4		2	4	4	8	32	8	16	4
Pentonic square	(5,2).5-gon	5{4}2		5	4	2	10	50	25	10	5
Pentonous square	(2,5).5-gon	2{4}5		2	4	5	10	50	10	25	5
Hexonic square	(6,2).6-gon	6{4}2		6	4	2	12	72	36	12	6
Hexonous square	(2,6).6-gon	2{4}6		2	4	6	12	72	12	36	6
Triadic octagon		3{3}3		3	3	3	6	24	8	8	8/3
	(3,2).8-gon	3{6}2		3	6	2	12	48	24	16	8
	(2,3).8-gon	2{6}3		2	6	3	12	48	16	24	8
Triadic 24-gon		3{4}3		3	4	3	12	72	24	24	8
Tetradic 24-gon		4{3}4		4	3	4	12	96	24	24	6
	(3,2).24-gon	3{8}2		3	8	2	24	144	72	48	24
	(2,3).24-gon	2{8}3		2	8	3	24	144	48	72	24
	(4,2).24-gon	4{6}2		4	6	2	24	192	96	48	24
	(2,4).24-gon	2{6}4		2	6	4	24	192	48	96	24
	(4,3).24-gon	4{4}3		4	4	3	24	288	96	72	24
	(3,4).24-gon	3{4}4		3	4	4	24	288	72	96	24
Triadic 120-gon		3{5}3		3	5	3	30	360	120	120	40
Pentadic 120-gon		5{3}5		5	3	5	30	600	120	120	24
	(3,2).120-gon	3{10}2		3	10	2	60	720	360	240	120
	(2,3).120-gon	2{10}3		2	10	3	60	720	240	360	120
	(5,2).120-gon	5{6}2		5	6	2	60	1200	600	240	120
	(2,5).120-gon	2{6}5		2	6	5	60	1200	240	600	120
	(5,3).120-gon	5{4}3		5	4	3	60	1800	600	360	120
	(3,5).120-gon	3{4}5		3	4	5	60	1800	360	600	120

Enumeration of quasiregular complex polygons

The truncation of a regular complex polygon, is . It is quasiregular, alternating two types of edges. A regular polyhedron with v vertices and e edges has qe vertices, and v+e edges of two types: v _q{} edges, and e _p{} edges.

Group	Order	Image	Polygon	Vertices	Edges	Edge types		Notes
2[2]2	4			4	4	2 {}	2 {}	Same as , $\mathbb {R} ^{2}$, square
2[3]2	6			6	6	3 {}	3 {}	Same as , $\mathbb {R} ^{2}$, hexagon
2[4]2	6			8	8	4 {}	4 {}	Same as , $\mathbb {R} ^{2}$, octagon
2[5]2	5			10	10	5 {}	5 {}	Same as , $\mathbb {R} ^{2}$, decagon
2[q]2	2q			2q	2q	q {}	q {}	Same as , $\mathbb {R} ^{2}$ quasiregular polygons
3[2]2	6			6	5	3 {}	2 3{}	$\mathbb {R} ^{3}$ representation as triangular prism
4[2]2	8			8	6	4 {}	2 4{}	$\mathbb {R} ^{3}$ representation as square prism
5[2]2	10			10	7	5 {}	2 5{}	$\mathbb {R} ^{3}$ representation as pentagonal prism
p[2]2	2p			2p	p+2	p {}	2 p{}	$\mathbb {R} ^{3}$ representation as p-prism
3[2]3	9			9	6	3 3{}	3 3{}	$\mathbb {R} ^{4}$ representation as 3-3 duoprism Same as
3[2]4	12			12	7	3 4{}	4 3{}	$\mathbb {R} ^{4}$ representation as 3-4 duoprism
4[2]4	16			16	8	4 4{}	4 4{}	$\mathbb {R} ^{4}$ representation as 4-4 duoprism, tesseract
5[2]5	25			25	10	5 5{}	5 5{}	$\mathbb {R} ^{4}$ representation as 5-5 duoprism Same as
p[2]q	pq			pq	p+q	p q{}	q p{}	$\mathbb {R} ^{4}$ representation as p-q duoprism Same as if p=q
3[4]2	18			18	15	9 {}	6 3{}
4[4]2	32			32	24	16 {}	8 4{}
p[4]2	2p2			2p2	p2+2p	p2 {}	2p p{}
3[3]3	24			24	16	8 3{}	8 3{}	Same as
3[6]2	48			48	40	16 3{}
3[4]3	72			72	48	24 3{}	24 3{}	Same as
4[3]4	96			96	48	24 4{}	24 4{}	Same as
3[8]2	144			144	120	72 {}	48 3{}
4[6]2	192			192	144	96 {}	48 4{}
4[4]3	288			288	168	96 3{}	72 4{}
3[5]3	360			360	240	120 3{}	120 3{}	Same as
5[3]5	600			600	240	120 5{}	120 5{}	Same as $\mathbb {R} ^{4}$ representation as {5,3,3}
3[10]2	720			720	600	360 {}	240 3{}
5[6]2	1200			1200	840	600 {}	240 5{}
5[4]3	1800			1800	960	600 3{}	360 5{}

Enumeration of uniform complex polyhedra

3[3]3[3]3 family

3[3]3[3]3 family

Polyhedron	Image	Vertices	Edges			Faces			Notes
						(27)	(72)	(27)
		27	72						Hessian polyhedron $\mathbb {R} ^{6}$ representation as 221 Same as
		72		216					$\mathbb {R} ^{6}$ representation as 122 Same as
		216
		216							Same as
		648							Same as

3[3]3[4]2 family

3[3]3[4]2 family

Polyhedron	Image	Vertices	Edges		Faces			Notes
					(54)	(216)	(72)
		72	216					Same as
		216						Same as Real representation r(221)
		54		216
		648						Same as
		432						Real representation t(221)
		432
		1296

2[3]2[4]3 family

2[3]2[4]3 family

Polyhedron	Image	Vertices	Edges		Faces			Notes
					(27)	(12)	(9)
		9	27		={3}			Same as
		12			={3}
		27		27				Same as
		54			={6}
		81			={3}
		81			={3}
		162			={6}

2[3]2[4]4 family

2[3]2[4]4 family

Polyhedron	Image	Vertices	Edges		Faces			Notes
					(64)	(48)	(12)
		12	48		={3}
		48			={3}
		64		48				Same as
		96			={6}
		192			={3}
		192			={3}
		384			={6}

Regular complex apeirogons

Some subgroups of the apeirogonal shepherd groups

11 complex apeirogons _p{q}_r with edge interiors colored in light blue, and edges around one vertex are colored individually. Vertices are shown as small black squares. Edges are seen as p-sided regular polygons and vertex figures are r-gonal.

A quasiregular apeirogon is a mixture of two regular apeirogons and , seen here with blue and pink edges. has only one color of edges because q is odd, making it a double covering.

Coxeter expresses them as δ^p,r
₂ where q is constrained to satisfy q = 2/(1 – (p + r)/pr).

Among the aperiogons, four are self-dual (when p = r), while eight exist as dual polytope pairs. Only one, {∞}, is real. The 12 pairs (p, r) as corresponding to aperiogons are (2,2), (3,2), (2,3), (3,3), (4,2), (2,4), (4,4), (6,2), (2,6), (6,3), (3,6), and (6,6).

Rank 2 complex apeirogons have symmetry _p[q]_r, where 1/p + 2/q + 1/r = 1. There are 8 solutions: ₂[∞]₂, ₃[12]₂, ₄[8]₂, ₆[6]₂, ₃[6]₃, ₆[4]₃, ₄[4]₄, and ₆[3]₆ or , , , , , , .

Including affine nodes, there are 3 more infinite solutions: , , , the first is an index 2 subgroup of the second, while the last is starry.

A regular complex apeirogon _p{q}_r has p-edges and q-gonal vertex figures. The dual apeirogon of _p{q}_r is _r{q}_p. An apeirogon of the form _p{q}_p is self-dual. Groups of the form _p[2q]₂ have a half symmetry _p[q]_p, so a regular apeirogon is the same as quasiregular .

Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form ₂[q]_r have a vertex arrangement as {q/2,p}. The form _p[q]₂ have vertex arrangement as r{p,q/2}.

Rank 2

Space	Group	Polygon		Edge type	$\mathbb {R} ^{2}$ rep.	Picture	Notes
$\mathbb {R} ^{1}$	2[∞]2 = [∞]	δ2,2 2 = {∞}		{}			Real apeirogon Same as
$\mathbb {C} ^{1}$	p[q]r	δp,r 2 = p{q}r		p{}
$\mathbb {C} ^{1}$	3[12]2	δ3,2 2 = 3{12}2		3{}	r{3,6}		Same as
		δ2,3 2 = 2{12}3		{}	{6,3}
$\mathbb {C} ^{1}$	3[6]3	δ3,3 2 = 3{6}3		3{}	{3,6}		Same as
$\mathbb {C} ^{1}$	4[8]2	δ4,2 2 = 4{8}2		4{}	{4,4}		Same as
		δ2,4 2 = 2{8}4		{}	{4,4}
$\mathbb {C} ^{1}$	4[4]4	δ4,4 2 = 4{4}4		4{}	{4,4}		Same as
$\mathbb {C} ^{1}$	6[6]2	δ6,2 2 = 6{6}2		6{}	r{3,6}		Same as
		δ2,6 2 = 2{6}6		{}	{3,6}
$\mathbb {C} ^{1}$	6[4]3	δ6,3 2 = 6{4}3		6{}	{6,3}
		δ3,6 2 = 3{4}6		3{}	{3,6}
$\mathbb {C} ^{1}$	6[3]6	δ6,6 2 = 6{3}6		6{}	{3,6}		Same as

Quasiregular apeirogons

Quasiregular complex apeirogons

p[q]r	p{q}r	t(p{q}r)	r{q}p
∞[3[3]]∞
∞[2]∞
∞[4]2
4[4]4
4[8]2
6[6]2
6[4]3
3[12]2
3[6]3
6[3]6

Quasiregular

There are 7 quasiregular complex apeirogons which alternate edges between two dual complex apeirogons.

p[q]r	4[4]4	4[8]2	6[6]2	6[4]3	3[12]2	3[6]3	6[3]6
p{q}r (Regular)
(Quasiregular)
r{q}p (Regular dual)

Rank 3

Some rank 3 Shephard groups and subgroup relations

- 27

- 54

- 64

- 96

- 125

- 150

- 162

- 336

- 384

- 648

- 750

- 1296

- 2160

- 2160