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Rank 2

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12 irreducible Shephard groups with their subgroup index relations.[1] 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

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Coxeter enumerated this list of regular complex polygons in . Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular .[2]

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

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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 , , square
2[3]2 6 6 6 3 {} 3 {} Same as , , hexagon
2[4]2 6 8 8 4 {} 4 {} Same as , , octagon
2[5]2 5 10 10 5 {} 5 {} Same as , , decagon
2[q]2 2q 2q 2q q {} q {} Same as , quasiregular polygons
3[2]2 6 6 5 3 {} 2 3{} representation as triangular prism
4[2]2 8 8 6 4 {} 2 4{} representation as square prism
5[2]2 10 10 7 5 {} 2 5{} representation as pentagonal prism
p[2]2 2p 2p p+2 p {} 2 p{} representation as p-prism
3[2]3 9 9 6 3 3{} 3 3{} representation as 3-3 duoprism
Same as
3[2]4 12 12 7 3 4{} 4 3{} representation as 3-4 duoprism
4[2]4 16 16 8 4 4{} 4 4{} representation as 4-4 duoprism, tesseract
5[2]5 25 25 10 5 5{} 5 5{} representation as 5-5 duoprism
Same as
p[2]q pq pq p+q p q{} q p{} 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
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

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3[3]3[3]3 family

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3[3]3[3]3 family
Polyhedron Image Vertices Edges Faces Notes

(27)

(72)

(27)
27 72
Hessian polyhedron
representation as 221
Same as
72 216

representation as 122
Same as
216

216


Same as
648


Same as

3[3]3[4]2 family

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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

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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

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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

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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 δp,r
2
where q is constrained to satisfy q = 2/(1 – (p + r)/pr).[3]

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 .[4]

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 rep.[5] Picture Notes
2[∞]2 = [∞] δ2,2
2
= {∞}
       
{} Real apeirogon
Same as
p[q]r δp,r
2
= p{q}r
p{}
3[12]2 δ3,2
2
= 3{12}2
3{} r{3,6} Same as
δ2,3
2
= 2{12}3
{} {6,3}
3[6]3 δ3,3
2
= 3{6}3
3{} {3,6} Same as
4[8]2 δ4,2
2
= 4{8}2
4{} {4,4} Same as
δ2,4
2
= 2{8}4
{} {4,4}
4[4]4 δ4,4
2
= 4{4}4
4{} {4,4} Same as
6[6]2 δ6,2
2
= 6{6}2
6{} r{3,6} Same as
δ2,6
2
= 2{6}6
{} {3,6}
6[4]3 δ6,3
2
= 6{4}3
6{} {6,3}
δ3,6
2
= 3{4}6
3{} {3,6}
6[3]6 δ6,6
2
= 6{3}6
6{} {3,6} Same as

Quasiregular apeirogons

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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

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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

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Some rank 3 Shephard groups and subgroup relations
- 27
- 54
- 64
- 96
- 125
- 150
- 162
- 336
- 384
- 648
- 750
- 1296
- 2160
- 2160

References

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  1. ^ Coxeter, Complex Regular Polytopes, p. 177, Table III
  2. ^ Regular Complex Polytopes, Table IV. The regular polygons. pp. 178-179
  3. ^ Coxeter, Regular Complex Polytopes, Table VI. The regular honeycombs. p. 111, 136.
  4. ^ Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
  5. ^ Coxeter, Regular Complex Polytopes, 11.6 Apeirogons, pp. 111-112