regular complex polygon

{{Short description|Polygons which have an accompanying imaginary dimension for each real dimension}}

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|+ Three views of regular complex polygon ₄{4}₂, {{CDD|4node_1|3|4|3|node}}

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|320px
This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.Coxeter, Regular Complex Polytopes, 11.3 Petrie Polygon, a simple h-gon formed by the orbit of the flag (O₀,O₀O₁) for the product of the two generating reflections of any nonstarry regular complex polygon, p₁{q}p₂. In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.

|160px
A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.

File:Complex 1-topes as k-edges.png as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges (it often is a complex edge) and only contains vertex elements.]]

In geometry, a regular complex polygon is a generalization of a regular polygon in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A regular polygon exists in 2 real dimensions, $\mathbb{R}^2$ , while a complex polygon exists in two complex dimensions, $\mathbb{C}^2$ , which can be given real representations in 4 dimensions, $\mathbb{R}^4$ , which then must be projected down to 2 or 3 real dimensions to be visualized. A complex polygon is generalized as a complex polytope in $\mathbb{C}^n$ .

A complex polygon may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.

The regular complex polygons have been completely characterized, and can be described using a symbolic notation developed by Coxeter.

A regular complex polygon with all 2-edges can be represented by a graph, while forms with k-edges can only be related by hypergraphs. A k-edge can be seen as a set of vertices, with no order implied. They may be drawn with pairwise 2-edges, but this is not structurally accurate.

Regular complex polygons

While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons _p{4}₂, are limited to 5-edge (pentagonal edges) elements, and infinite regular apeirogons also include 6-edge (hexagonal edges) elements.

=Notations=

==Shephard's modified Schläfli notation==

Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p₁-edges, with a p₂-set as vertex figure and overall symmetry group of order g, we denote the polygon as p₁(g)p₂.

The number of vertices V is then g/p₂ and the number of edges E is g/p₁.

The complex polygon illustrated above has eight square edges (p₁=4) and sixteen vertices (p₂=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.

==Coxeter's revised modified Schläfli notation==

A more modern notation _p₁{q}_p₂ is due to Coxeter,Coxeter, Regular Complex Polytopes, p. xiv and is based on group theory. As a symmetry group, its symbol is _p₁[q]_p₂.

The symmetry group _p₁[q]_p₂ is represented by 2 generators R₁, R₂, where: R₁^p₁ = R₂^p₂ = I. If q is even, (R₂R₁)^q/2 = (R₁R₂)^q/2. If q is odd, (R₂R₁)^(q−1)/2R₂ = (R₁R₂)^(q−1)/2R₁. When q is odd, p₁=p₂.

For ₄[4]₂ has R₁⁴ = R₂² = I, (R₂R₁)² = (R₁R₂)².

For ₃[5]₃ has R₁³ = R₂³ = I, (R₂R₁)²R₂ = (R₁R₂)²R₁.

==Coxeter–Dynkin diagrams==

Coxeter also generalised the use of Coxeter–Dynkin diagrams to complex polytopes, for example the complex polygon _p{q}_r is represented by {{CDD|pnode_1|q|rnode}} and the equivalent symmetry group, _p[q]_r, is a ringless diagram {{CDD|pnode|q|rnode}}. The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is ₂{q}₂ or {q} or {{CDD|node_1|q|node}}.

One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So {{CDD|3node_1|4|node}} and {{CDD|3node_1|3|3node}} are ordinary, while {{CDD|4node_1|3|node}} is starry.

==12 Irreducible Shephard groups==

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12 irreducible Shephard groups with their subgroup index relations.Coxeter, Complex Regular Polytopes, p. 177, Table III

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Subgroups from <5,3,2>₃₀, <4,3,2>₁₂ and <3,3,2>₆

colspan=2|Subgroups relate by removing one reflection:
_p[2q]₂ --> _p[q]_p, index 2 and _p[4]_q --> _p[q]_p, index q.

File:Rank2_shephard_subgroups2_series.png

Coxeter enumerated this list of regular complex polygons in $\mathbb{C}^2$ . A regular complex polygon, _p{q}_r or {{CDD|pnode_1|q|rnode}}, has p-edges, and r-gonal vertex figures. _p{q}_r is a finite polytope if (p + r)q > pr(q − 2).

Its symmetry is written as _p[q]_r, called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections.

For nonstarry groups, the order of the group _p[q]_r can be computed as $g = 8/q \cdot (1/p+2/q+1/r-1)^{-2}$ .Lehrer & Taylor 2009, p. 87

The Coxeter number for _p[q]_r is $h = 2/(1/p+2/q+1/r-1)$ , so the group order can also be computed as $g = 2h^2/q$ . A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.

The rank 2 solutions that generate complex polygons are:

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!rowspan=3|Group

|G₃ = G(q,1,1)

G₂ = G(p,1,2)

G₄

G₆

G₅

G₈

G₁₄

G₉

G₁₀

G₂₀

G₁₆

G₂₁

G₁₇

G₁₈

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|₂[q]₂, q = 3,4...

_p[4]₂, p = 2,3...

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

₅[4]₃

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|{{CDD|node|q|node}}

align=center

!Order

|2q

2p²

144

192

288

360

600

720

1200

1800

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colspan=3|12

colspan=3|24

colspan=2|30

colspan=3|60

Excluded solutions with odd q and unequal p and r are: ₆[3]₂, ₆[3]₃, ₉[3]₃, ₁₂[3]₃, ..., ₅[5]₂, ₆[5]₂, ₈[5]₂, ₉[5]₂, ₄[7]₂, ₉[5]₂, ₃[9]₂, and ₃[11]₂.

Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: {{CDD|3node|3|node}}, {{CDD|4node|3|node}}, {{CDD|5node|3|node}}, {{CDD|5node|3|3node}}, {{CDD|3node|5|node}}, and {{CDD|5node|5|node}}.

The dual polygon of _p{q}_r is _r{q}_p. A polygon 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 polygon {{CDD|pnode_1|3|2x|q|3|node}} is the same as quasiregular {{CDD|pnode_1|3|q|3|pnode_1}}. As well, regular polygon with the same node orders, {{CDD|pnode_1|3|q|3|pnode}}, have an alternated construction {{CDD|node_h|3|2x|q|3|pnode}}, allowing adjacent edges to be two different colors.Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179

The group order, g, is used to compute the total number of vertices and edges. It will have g/r vertices, and g/p edges. When p=r, the number of vertices and edges are equal. This condition is required when q is odd.

== Matrix generators ==

The group p[q]r, {{CDD|pnode|q|rnode}}, can be represented by two matrices:Complex Polytopes, 8.9 The Two-Dimensional Case, p. 88

class=wikitable \|+ {{CDD\|pnode\|q\|rnode}} !Name	R₁ {{CDD\|pnode}}	R₂ {{CDD\|rnode}}
align=center !Order \|p \|r
Matrix \| $\left [\begin{smallmatrix}$ e^{2\pi i/p} & 0 \\ (e^{2\pi i/p}-1)k & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 1 & (e^{2\pi i/r}-1)k \\ 0 & e^{2\pi i/r} \end{smallmatrix}\right ]

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|+ {{CDD|pnode|q|rnode}}

!Name

R₁
{{CDD|pnode}}

R₂
{{CDD|rnode}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

e^{2\pi i/p} & 0 \\

(e^{2\pi i/p}-1)k & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

1 & (e^{2\pi i/r}-1)k \\

0 & e^{2\pi i/r}

\end{smallmatrix}\right ]

With

: $k = \sqrt \frac{ \cos(\frac{\pi}{p}-\frac{\pi}{r})+\cos(\frac{2\pi}{q}) }{2\sin\frac{\pi}{p}\sin\frac{\pi}{r} }$

;Examples

class=wikitable
valign=top \| {\| class=wikitable \|+ {{CDD\|pnode\|2\|qnode}} !Name	R₁ {{CDD\|pnode}}	R₂ {{CDD\|qnode}}
align=center !Order \|p \|q
Matrix \| $\left [\begin{smallmatrix}$ e^{2\pi i/p} & 0 \\ 0 & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 1 & 0 \\ 0 & e^{2\pi i/q} \\ \end{smallmatrix}\right ]

class=wikitable

valign=top

{| class=wikitable

|+ {{CDD|pnode|2|qnode}}

!Name

R₁
{{CDD|pnode}}

R₂
{{CDD|qnode}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

e^{2\pi i/p} & 0 \\

0 & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

1 & 0 \\

0 & e^{2\pi i/q} \\

\end{smallmatrix}\right ]

class=wikitable \|+ {{CDD\|pnode\|4\|node}} !Name	R₁ {{CDD\|pnode}}	R₂ {{CDD\|node}}
align=center !Order \|p \|2
Matrix \| $\left [\begin{smallmatrix}$ e^{2\pi i/p} & 0 \\ 0 & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 0 & 1 \\ 1 & 0 \\ \end{smallmatrix}\right ]

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|+ {{CDD|pnode|4|node}}

!Name

R₁
{{CDD|pnode}}

R₂
{{CDD|node}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

e^{2\pi i/p} & 0 \\

0 & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

0 & 1 \\

1 & 0 \\

\end{smallmatrix}\right ]

class=wikitable \|+ {{CDD\|3node\|3\|3node}} !Name	R₁ {{CDD\|3node}}	R₂ {{CDD\|3node}}
align=center !Order \|3 \|3
Matrix \| $\left [\begin{smallmatrix}$ \frac{-1+\sqrt3 i}{2} & 0 \\ \frac{-3+\sqrt3 i}{2} & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 1 & \frac{-3+\sqrt3 i}{2} \\ 0 & \frac{-1+\sqrt3 i}{2} \\ \end{smallmatrix}\right ]

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|+ {{CDD|3node|3|3node}}

!Name

R₁
{{CDD|3node}}

R₂
{{CDD|3node}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

\frac{-1+\sqrt3 i}{2} & 0 \\

\frac{-3+\sqrt3 i}{2} & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

1 & \frac{-3+\sqrt3 i}{2} \\

0 & \frac{-1+\sqrt3 i}{2} \\

\end{smallmatrix}\right ]

|- valign=top

class=wikitable \|+ {{CDD\|4node\|2\|4node}} !Name	R₁ {{CDD\|4node}}	R₂ {{CDD\|4node}}
align=center !Order \|4 \|4
Matrix \| $\left [\begin{smallmatrix}$ i & 0 \\ 0 & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 1 & 0 \\ 0 & i \\ \end{smallmatrix}\right ]

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|+ {{CDD|4node|2|4node}}

!Name

R₁
{{CDD|4node}}

R₂
{{CDD|4node}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

i & 0 \\

0 & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

1 & 0 \\

0 & i \\

\end{smallmatrix}\right ]

class=wikitable \|+ {{CDD\|4node\|4\|node}} !Name	R₁ {{CDD\|4node}}	R₂ {{CDD\|node}}
align=center !Order \|4 \|2
Matrix \| $\left [\begin{smallmatrix}$ i & 0 \\ 0 & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 0 & 1 \\ 1 & 0 \\ \end{smallmatrix}\right ]

class=wikitable

|+ {{CDD|4node|4|node}}

!Name

R₁
{{CDD|4node}}

R₂
{{CDD|node}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

i & 0 \\

0 & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

0 & 1 \\

1 & 0 \\

\end{smallmatrix}\right ]

class=wikitable \|+ {{CDD\|3node\|6\|node}} !Name	R₁ {{CDD\|3node}}	R₂ {{CDD\|node}}
align=center !Order \|3 \|2
Matrix \| $\left [\begin{smallmatrix}$ \frac{-1+\sqrt3 i}{2} & 0 \\ \frac{-3+\sqrt3 i}{2} & 1 \\ \end{smallmatrix}\right ] \| $\left [\begin{smallmatrix}$ 1 & -2 \\ 0 & -1 \\ \end{smallmatrix}\right ]

class=wikitable

|+ {{CDD|3node|6|node}}

!Name

R₁
{{CDD|3node}}

R₂
{{CDD|node}}

align=center

!Order

Matrix

$\left [\begin{smallmatrix}$

\frac{-1+\sqrt3 i}{2} & 0 \\

\frac{-3+\sqrt3 i}{2} & 1 \\

\end{smallmatrix}\right ]

$\left [\begin{smallmatrix}$

1 & -2 \\

0 & -1 \\

\end{smallmatrix}\right ]

= Enumeration of regular complex polygons =

Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.Regular Complex Polytopes, Coxeter, pp. 177–179

class="wikitable sortable"

!Group

data-sort-type="number"|Order

data-sort-type="number"|Coxeter
number

colspan=2|Polygon

data-sort-type="number"|Vertices

colspan=2 data-sort-type="number"|Edges

Notes

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|G(q,q,2)
₂[q]₂ = [q]
q = 2,3,4,...

₂{q}₂

{}

align=left| Real regular polygons
Same as {{CDD|node_h|2x|q|node}}
Same as {{CDD|node_1|q|rat|2x|node_1}} if q even

class="wikitable sortable" !Group	data-sort-type="number"\|Order	data-sort-type="number"\|Coxeter number	colspan=3\|Polygon	data-sort-type="number"\|Vertices	colspan=2 data-sort-type="number"\|Edges	Notes
align=center BGCOLOR="#ffffe0" \| rowspan=2\|G(p,1,2) _p[4]₂ p=2,3,4,...	rowspan=2\|2p²	rowspan=2\|2p	p(2p²)2	_p{4}₂	{{CDD\|pnode_1\|4\|node}}	p²	2p	_p{}	align=left\|same as _p{}×_p{} or {{CDD\|pnode_1\|2\|pnode_1}} $\mathbb{R}^4$ representation as p-p duoprism
align=center BGCOLOR="#ffffe0" \|2(2p²)p	₂{4}_p	{{CDD\|node_1\|4\|pnode}}	2p	p²	{}	align=left\| $\mathbb{R}^4$ representation as p-p duopyramid
align=center BGCOLOR="#ffe0e0" \| G(2,1,2) ₂[4]₂ = [4]	8	4		₂{4}₂ = {4}	{{CDD\|node_1\|4\|node}}	4	4	{}	align=left\|same as {}×{} or {{CDD\|node_1\|2\|node_1}} Real square
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(3,1,2) ₃[4]₂	rowspan=2\|18	rowspan=2\|6	6(18)2	₃{4}₂	{{CDD\|3node_1\|4\|node}}	9	6	₃{}	align=left\|same as ₃{}×₃{} or {{CDD\|3node_1\|2\|3node_1}} $\mathbb{R}^4$ representation as 3-3 duoprism
align=center BGCOLOR="#e0ffff" \|2(18)3	₂{4}₃	{{CDD\|node_1\|4\|3node}}	6	9	{}	align=left\| $\mathbb{R}^4$ representation as 3-3 duopyramid
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(4,1,2) ₄[4]₂	rowspan=2\|32	rowspan=2\|8	8(32)2	₄{4}₂	{{CDD\|4node_1\|4\|node}}	16	8	₄{}	align=left\|same as ₄{}×₄{} or {{CDD\|4node_1\|2\|4node_1}} $\mathbb{R}^4$ representation as 4-4 duoprism or {4,3,3}
align=center BGCOLOR="#e0ffff" \|2(32)4	₂{4}₄	{{CDD\|node_1\|4\|4node}}	8	16	{}	align=left\| $\mathbb{R}^4$ representation as 4-4 duopyramid or {3,3,4}
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(5,1,2) ₅[4]₂	rowspan=2\|50	rowspan=2\|25	5(50)2	₅{4}₂	{{CDD\|5node_1\|4\|node}}	25	10	₅{}	align=left\|same as ₅{}×₅{} or {{CDD\|5node_1\|2\|5node_1}} $\mathbb{R}^4$ representation as 5-5 duoprism
align=center BGCOLOR="#e0ffff" \|2(50)5	₂{4}₅	{{CDD\|node_1\|4\|5node}}	10	25	{}	align=left\| $\mathbb{R}^4$ representation as 5-5 duopyramid
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(6,1,2) ₆[4]₂	rowspan=2\|72	rowspan=2\|36	6(72)2	₆{4}₂	{{CDD\|6node_1\|4\|node}}	36	12	₆{}	align=left\|same as ₆{}×₆{} or {{CDD\|6node_1\|2\|6node_1}} $\mathbb{R}^4$ representation as 6-6 duoprism
align=center BGCOLOR="#e0ffff" \|2(72)6	₂{4}₆	{{CDD\|node_1\|4\|6node}}	12	36	{}	align=left\| $\mathbb{R}^4$ representation as 6-6 duopyramid
align=center BGCOLOR="#e0f0ff" \|G₄=G(1,1,2) ₃[3]₃ <2,3,3>	24	6	3(24)3	₃{3}₃	{{CDD\|3node_1\|3\|3node}}	8	8	₃{}	align=left\|Möbius–Kantor configuration self-dual, same as {{CDD\|node_h\|6\|3node}} $\mathbb{R}^4$ representation as {3,3,4}
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₆ ₃[6]₂	rowspan=4\|48	rowspan=4\|12	3(48)2	₃{6}₂	{{CDD\|3node_1\|6\|node}}	rowspan=2\|24	rowspan=2\| 16	rowspan=2\|₃{}	align=left\|same as {{CDD\|3node_1\|3\|3node_1}}
align=center BGCOLOR="#e0ffe0"	\|₃{3}₂	{{CDD\|3node_1\|3\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|2(48)3	₂{6}₃	{{CDD\|node_1\|6\|3node}}	rowspan=2\| 16	rowspan=2\| 24	rowspan=2\|{}
align=center BGCOLOR="#e0ffe0"	\|₂{3}₃	{{CDD\|node_1\|3\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|G₅ ₃[4]₃	72	12	3(72)3	₃{4}₃	{{CDD\|3node_1\|4\|3node}}	24	24	₃{}	align=left \|self-dual, same as {{CDD\|node_h\|8\|3node}} $\mathbb{R}^4$ representation as {3,4,3}
align=center BGCOLOR="#e0f0ff" \|G₈ ₄[3]₄	96	12	4(96)4	₄{3}₄	{{CDD\|4node_1\|3\|4node}}	24	24	₄{}	align=left\|self-dual, same as {{CDD\|node_h\|6\|4node}} $\mathbb{R}^4$ representation as {3,4,3}
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₁₄ ₃[8]₂	rowspan=4\|144	rowspan=4\|24	3(144)2	₃{8}₂	{{CDD\|3node_1\|8\|node}}	rowspan=2\| 72	rowspan=2\| 48	rowspan=2\|₃{}	align=left\| same as {{CDD\|3node_1\|4\|3node_1}}
align=center BGCOLOR="#e0ffe0" \|	₃{8/3}₂	{{CDD\|3node_1\|8\|rat\|3x\|node}}	align=left\| starry polygon, same as {{CDD\|3node_1\|4\|rat\|3x\|3node_1}}
align=center BGCOLOR="#e0f0ff" \| 2(144)3	₂{8}₃	{{CDD\|node_1\|8\|3node}}	rowspan=2\| 48	rowspan=2\| 72	rowspan=2\|{}
align=center BGCOLOR="#e0ffe0" \|	₂{8/3}₃	{{CDD\|node_1\|8\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=4\|G₉ ₄[6]₂	rowspan=4\|192	rowspan=4\|24	4(192)2	₄{6}₂	{{CDD\|4node_1\|6\|node}}	96	48	₄{}	align=left\| same as {{CDD\|4node_1\|3\|4node_1}}
align=center BGCOLOR="#e0f0ff" \| 2(192)4	₂{6}₄	{{CDD\|node_1\|6\|4node}}	48	96	{}
align=center BGCOLOR="#e0ffe0" \|	₄{3}₂	{{CDD\|4node_1\|3\|node}}	96	48	{}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{3}₄	{{CDD\|node_1\|3\|4node}}	48	96	{}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₁₀ ₄[4]₃	rowspan=4\|288	24	4(288)3	₄{4}₃	{{CDD\|4node_1\|4\|3node}}	rowspan=2\| 96	rowspan=2\| 72	rowspan=2\|₄{}
align=center BGCOLOR="#e0ffe0" \| 12		₄{8/3}₃	{{CDD\|4node_1\|8\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 24	3(288)4	₃{4}₄	{{CDD\|3node_1\|4\|4node}}	rowspan=2\| 72	rowspan=2\| 96	rowspan=2\|₃{}
align=center BGCOLOR="#e0ffe0" \| 12		₃{8/3}₄	{{CDD\|3node_1\|8\|rat\|3x\|4node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=2\|G₂₀ ₃[5]₃	rowspan=2\|360	rowspan=2\|30	3(360)3	₃{5}₃	{{CDD\|3node_1\|5\|3node}}	rowspan=2\| 120	rowspan=2\| 120	rowspan=2\|₃{}	align=left \|self-dual, same as {{CDD\|node_h\|10\|3node}} $\mathbb{R}^4$ representation as {3,3,5}
align=center BGCOLOR="#e0ffe0" \|	₃{5/2}₃	{{CDD\|3node_1\|5-2\|3node}}	align=left\|self-dual, starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=2\|G₁₆ ₅[3]₅	rowspan=2\|600	30	5(600)5	₅{3}₅	{{CDD\|5node_1\|3\|5node}}	rowspan=2\| 120	rowspan=2\| 120	rowspan=2\|₅{}	align=left \|self-dual, same as {{CDD\|node_h\|6\|5node}} $\mathbb{R}^4$ representation as {3,3,5}
align=center BGCOLOR="#e0ffe0" \| 10		₅{5/2}₅	{{CDD\|5node_1\|5-2\|5node}}	align=left\|self-dual, starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₂₁ ₃[10]₂	rowspan=8\|720	rowspan=8\|60	3(720)2	₃{10}₂	{{CDD\|3node_1\|10\|node}}	rowspan=4\| 360	rowspan=4\| 240	rowspan=4\|₃{}	align=left\| same as {{CDD\|3node_1\|5\|3node_1}}
align=center BGCOLOR="#e0ffe0"		₃{5}₂	{{CDD\|3node_1\|5\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"		₃{10/3}₂	{{CDD\|3node_1\|10\|rat\|3x\|node}}	align=left\| starry polygon, same as {{CDD\|3node_1\|5\|rat\|3x\|3node_1}}
align=center BGCOLOR="#e0ffe0"		₃{5/2}₂	{{CDD\|3node_1\|5-2\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 2(720)3	₂{10}₃	{{CDD\|node_1\|10\|3node}}	rowspan=4\| 240	rowspan=4\| 360	rowspan=4\|{}
align=center BGCOLOR="#e0ffe0" \|	₂{5}₃	{{CDD\|node_1\|5\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{10/3}₃	{{CDD\|node_1\|10\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{5/2}₃	{{CDD\|node_1\|5-2\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₁₇ ₅[6]₂	rowspan=8\|1200	60	5(1200)2	₅{6}₂	{{CDD\|5node_1\|6\|node}}	rowspan=4\| 600	rowspan=4\| 240	rowspan=4\|₅{}	align=left\| same as {{CDD\|5node_1\|3\|5node_1}}
align=center BGCOLOR="#e0ffe0" \| 20		₅{5}₂	{{CDD\|5node_1\|5\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 20		₅{10/3}₂	{{CDD\|5node_1\|10\|rat\|3x\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 60		₅{3}₂	{{CDD\|5node_1\|3\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 60	2(1200)5	₂{6}₅	{{CDD\|node_1\|6\|5node}}	rowspan=4\| 240	rowspan=4\| 600	rowspan=4\|{}
align=center BGCOLOR="#e0ffe0" \| 20		₂{5}₅	{{CDD\|node_1\|5\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 20		₂{10/3}₅	{{CDD\|node_1\|10\|rat\|3x\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 60		₂{3}₅	{{CDD\|node_1\|3\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₁₈ ₅[4]₃	rowspan=8\|1800	60	5(1800)3	₅{4}₃	{{CDD\|5node_1\|4\|3node}}	rowspan=4\| 600	rowspan=4\| 360	rowspan=4\|₅{}	align=left\|
align=center BGCOLOR="#e0ffe0"	15		₅{10/3}₃	{{CDD\|5node_1\|10\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"	30		₅{3}₃	{{CDD\|5node_1\|3\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"	30		₅{5/2}₃	{{CDD\|5node_1\|5-2\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 60	3(1800)5	₃{4}₅	{{CDD\|3node_1\|4\|5node}}	rowspan=4\| 360	rowspan=4\| 600	rowspan=4\|₃{}
align=center BGCOLOR="#e0ffe0" \| 15		₃{10/3}₅	{{CDD\|3node_1\|10\|rat\|3x\|5node}}	align=left\| starry polygon
align=center BGCOLOR="#e0ffe0" \| 30		₃{3}₅	{{CDD\|3node_1\|3\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 30		₃{5/2}₅	{{CDD\|3node_1\|5-2\|5node}}	align=left\|starry polygon

= Visualizations of regular complex polygons =

== 2D graphs==

Polygons of the form _p{2r}_q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.

;Complex polygons ₂{r}_q:

Polygons of the form ₂{4}_q are called generalized orthoplexes. They share vertices with the 4D q-q duopyramids, vertices connected by 2-edges.

Complex bipartite graph square.svg|₂{4}₂, {{CDD|node_1|4|node}}, with 4 vertices, and 4 edges

Complex polygon 2-4-3-bipartite graph.png|₂{4}₃, {{CDD|node_1|4|3node}}, with 6 vertices, and 9 edgesCoxeter, Regular Complex Polytopes, p. 108

Complex polygon 2-4-4 bipartite graph.png|₂{4}₄, {{CDD|node_1|4|4node}}, with 8 vertices, and 16 edges

Complex polygon 2-4-5-bipartite graph.png|₂{4}₅, {{CDD|node_1|4|5node}}, with 10 vertices, and 25 edges

6-generalized-2-orthoplex.svg|₂{4}₆, {{CDD|node_1|4|6node}}, with 12 vertices, and 36 edges

7-generalized-2-orthoplex.svg|₂{4}₇, {{CDD|node_1|4|7node}}, with 14 vertices, and 49 edges

8-generalized-2-orthoplex.svg|₂{4}₈, {{CDD|node_1|4|8node}}, with 16 vertices, and 64 edges

9-generalized-2-orthoplex.svg|₂{4}₉, {{CDD|node_1|4|9node}}, with 18 vertices, and 81 edges

10-generalized-2-orthoplex.svg|₂{4}₁₀, {{CDD|node_1|4|10node}}, with 20 vertices, and 100 edges

;Complex polygons _p{4}₂:

Polygons of the form _p{4}₂ are called generalized hypercubes (squares for polygons). They share vertices with the 4D p-p duoprisms, vertices connected by p-edges. Vertices are drawn in green, and p-edges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.

2-generalized-2-cube.svg|₂{4}₂, {{CDD|node_1|4|node}} or {{CDD|node_1|2|node_1}}, with 4 vertices, and 4 2-edges

3-generalized-2-cube_skew.svg|₃{4}₂, {{CDD|3node_1|4|node}} or {{CDD|3node_1|2|3node_1}}, with 9 vertices, and 6 (triangular) 3-edgesCoxeter, Regular Complex Polytopes, p. 108

4-generalized-2-cube.svg|₄{4}₂, {{CDD|4node_1|4|node}} or {{CDD|4node_1|2|4node_1}}, with 16 vertices, and 8 (square) 4-edges

5-generalized-2-cube_skew.svg|₅{4}₂, {{CDD|5node_1|4|node}} or {{CDD|5node_1|2|5node_1}}, with 25 vertices, and 10 (pentagonal) 5-edges

6-generalized-2-cube.svg|₆{4}₂, {{CDD|6node_1|4|node}} or {{CDD|6node_1|2|6node_1}}, with 36 vertices, and 12 (hexagonal) 6-edges

7-generalized-2-cube_skew.svg|₇{4}₂, {{CDD|7node_1|4|node}} or {{CDD|7node_1|2|7node_1}}, with 49 vertices, and 14 (heptagonal)7-edges

8-generalized-2-cube.svg|₈{4}₂, {{CDD|8node_1|4|node}} or {{CDD|8node_1|2|8node_1}}, with 64 vertices, and 16 (octagonal) 8-edges

9-generalized-2-cube_skew.svg|₉{4}₂, {{CDD|9node_1|4|node}} or {{CDD|9node_1|2|9node_1}}, with 81 vertices, and 18 (enneagonal) 9-edges

10-generalized-2-cube.svg|₁₀{4}₂, {{CDD|10node_1|4|node}} or {{CDD|10node_1|2|10node_1}}, with 100 vertices, and 20 (decagonal) 10-edges

;Complex polygons _p{r}₂:

Complex_polygon_3-6-2.png|₃{6}₂, {{CDD|3node_1|6|node}} or {{CDD|3node_1|3|3node_1}}, with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blueCoxeter, Regular Complex Polytopes, p. 109

Complex_polygon_3-8-2.png|₃{8}₂, {{CDD|3node_1|8|node}} or {{CDD|3node_1|4|3node_1}}, with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blueCoxeter, Regular Complex Polytopes, p. 111

;Complex polygons, _p{r}_p:

Polygons of the form _p{r}_p have equal number of vertices and edges. They are also self-dual.

Complex polygon 3-3-3.svg|₃{3}₃, {{CDD|3node_1|3|3node}} or {{CDD|node_h|6|3node}}, with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blueCoxeter, Regular Complex Polytopes, p. 30 diagram and p. 47 indices for 8 3-edges

Complex_polygon_3-4-3-fill1.png|₃{4}₃, {{CDD|3node_1|4|3node}} or {{CDD|node_h|8|3node}}, with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filledCoxeter, Regular Complex Polytopes, p. 110

Complex polygon 4-3-4.png|₄{3}₄, {{CDD|4node_1|3|4node}} or {{CDD|node_h|6|4node}}, with 24 vertices and 24 4-edges shown in 4 sets of colorsCoxeter, Regular Complex Polytopes, p. 110

Complex polygon 3-5-3.png|₃{5}₃, {{CDD|3node_1|5|3node}} or {{CDD|node_h|10|3node}}, with 120 vertices and 120 3-edgesCoxeter, Regular Complex Polytopes, p. 48

Complex polygon 5-3-5.png|₅{3}₅, {{CDD|5node_1|3|5node}} or {{CDD|node_h|6|5node}}, with 120 vertices and 120 5-edgesCoxeter, Regular Complex Polytopes, p. 49

== 3D perspective ==

3D perspective projections of complex polygons _p{4}₂ can show the point-edge structure of a complex polygon, while scale is not preserved.

The duals ₂{4}_p: are seen by adding vertices inside the edges, and adding edges in place of vertices.

Complex polygon 2-4-3-stereographic0.png|₂{4}₃, {{CDD|node_1|4|3node}} with 6 vertices, 9 edges in 3 sets

Complex polygon 3-4-2-stereographic3.png|₃{4}₂, {{CDD|3node_1|4|node}} with 9 vertices, 6 3-edges in 2 sets of colors as {{CDD|3node_1|2|3node_1}}

Complex polygon 4-4-2-stereographic3.svg|₄{4}₂, {{CDD|4node_1|4|node}} with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges as {{CDD|4node_1|2|4node_1}}

Complex_polygon_5-4-2-stereographic3.png|₅{4}₂, {{CDD|5node_1|4|node}} with 25 vertices, 10 5-edges in 2 sets of colors as {{CDD|5node_1|2|5node_1}}

Quasiregular polygons

A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon {{CDD|pnode_1|q|rnode_1}} contains alternate edges of the regular polygons {{CDD|pnode_1|q|rnode}} and {{CDD|pnode|q|rnode_1}}. The quasiregular polygon has p vertices on the p-edges of the regular form.

class=wikitable \|+ Example quasiregular polygons !_p[q]_r	₂[4]₂	₃[4]₂	₄[4]₂	₅[4]₂	₆[4]₂	₇[4]₂	₈[4]₂	₃[3]₃	₃[4]₃
align=center !Regular {{CDD\|pnode_1\|q\|rnode}} \|80px {{CDD\|node_1\|4\|node}} 4 2-edges \|80px {{CDD\|3node_1\|4\|node}} 9 3-edges \|80px {{CDD\|4node_1\|4\|node}} 16 4-edges \|80px {{CDD\|5node_1\|4\|node}} 25 5-edges \|80px {{CDD\|6node_1\|4\|node}} 36 6-edges \|80px {{CDD\|7node_1\|4\|node}} 49 7-edges \|80px {{CDD\|8node_1\|4\|node}} 64 8-edges \|80px {{CDD\|3node_1\|3\|3node}} \|80px {{CDD\|3node_1\|4\|3node}}
align=center !Quasiregular {{CDD\|pnode_1\|q\|rnode_1}} \|80px {{CDD\|node_1\|4\|node_1}} = {{CDD\|node_1\|8\|node}} 4+4 2-edges \|80px {{CDD\|3node_1\|4\|node_1}} 6 2-edges 9 3-edges \|80px {{CDD\|4node_1\|4\|node_1}} 8 2-edges 16 4-edges \|80px {{CDD\|5node_1\|4\|node_1}} 10 2-edges 25 5-edges \|80px {{CDD\|6node_1\|4\|node_1}} 12 2-edges 36 6-edges \|80px {{CDD\|7node_1\|4\|node_1}} 14 2-edges 49 7-edges \|80px {{CDD\|8node_1\|4\|node_1}} 16 2-edges 64 8-edges \|80px {{CDD\|3node_1\|3\|3node_1}} = {{CDD\|3node_1\|6\|node}} \|80px {{CDD\|3node_1\|4\|3node_1}} = {{CDD\|3node_1\|8\|node}}
align=center !Regular {{CDD\|pnode_1\|q\|rnode}} \|80px {{CDD\|node_1\|4\|node}} 4 2-edges \|80px {{CDD\|node_1\|4\|3node}} 6 2-edges \|80px {{CDD\|node_1\|4\|4node}} 8 2-edges \|80px {{CDD\|node_1\|4\|5node}} 10 2-edges \|80px {{CDD\|node_1\|4\|6node}} 12 2-edges \|80px {{CDD\|node_1\|4\|7node}} 14 2-edges \|80px {{CDD\|node_1\|4\|8node}} 16 2-edges \|80px {{CDD\|3node_1\|3\|3node}} \|80px {{CDD\|3node_1\|4\|3node}}

Notes

References

Coxeter, H.S.M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
{{citation| last=Coxeter | first=H.S.M. | title=Regular Complex Polytopes | publisher=Cambridge University Press | year=1991 | isbn=0-521-39490-2 | authorlink=Harold Scott MacDonald Coxeter }}
Coxeter, H.S.M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,
Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.
{{citation

| last1 = Shephard | first1 = G. C. | author1-link = Geoffrey Colin Shephard

| last2 = Todd | first2 = J. A.

| doi = 10.4153/cjm-1954-028-3

| journal = Canadian Journal of Mathematics

| mr = 59914

| pages = 274–304

| title = Finite unitary reflection groups

| volume = 6

| year = 1954}}

Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009

Category:Polytopes

class="wikitable sortable" !Group	data-sort-type="number"\|Order	data-sort-type="number"\|Coxeter number	colspan=3\|Polygon	data-sort-type="number"\|Vertices	colspan=2 data-sort-type="number"\|Edges	Notes
align=center BGCOLOR="#ffffe0" \| rowspan=2\|G(p,1,2) _p[4]₂ p=2,3,4,...	rowspan=2\|2p²	rowspan=2\|2p	p(2p²)2	_p{4}₂	{{CDD\|pnode_1\|4\|node}}	p²	2p	_p{}	align=left\|same as _p{}×_p{} or {{CDD\|pnode_1\|2\|pnode_1}} $\mathbb{R}^4$ representation as p-p duoprism
align=center BGCOLOR="#ffffe0" \|2(2p²)p	₂{4}_p	{{CDD\|node_1\|4\|pnode}}	2p	p²	{}	align=left\| $\mathbb{R}^4$ representation as p-p duopyramid
align=center BGCOLOR="#ffe0e0" \| G(2,1,2) ₂[4]₂ = [4]	8	4		₂{4}₂ = {4}	{{CDD\|node_1\|4\|node}}	4	4	{}	align=left\|same as {}×{} or {{CDD\|node_1\|2\|node_1}} Real square
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(3,1,2) ₃[4]₂	rowspan=2\|18	rowspan=2\|6	6(18)2	₃{4}₂	{{CDD\|3node_1\|4\|node}}	9	6	₃{}	align=left\|same as ₃{}×₃{} or {{CDD\|3node_1\|2\|3node_1}} $\mathbb{R}^4$ representation as 3-3 duoprism
align=center BGCOLOR="#e0ffff" \|2(18)3	₂{4}₃	{{CDD\|node_1\|4\|3node}}	6	9	{}	align=left\| $\mathbb{R}^4$ representation as 3-3 duopyramid
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(4,1,2) ₄[4]₂	rowspan=2\|32	rowspan=2\|8	8(32)2	₄{4}₂	{{CDD\|4node_1\|4\|node}}	16	8	₄{}	align=left\|same as ₄{}×₄{} or {{CDD\|4node_1\|2\|4node_1}} $\mathbb{R}^4$ representation as 4-4 duoprism or {4,3,3}
align=center BGCOLOR="#e0ffff" \|2(32)4	₂{4}₄	{{CDD\|node_1\|4\|4node}}	8	16	{}	align=left\| $\mathbb{R}^4$ representation as 4-4 duopyramid or {3,3,4}
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(5,1,2) ₅[4]₂	rowspan=2\|50	rowspan=2\|25	5(50)2	₅{4}₂	{{CDD\|5node_1\|4\|node}}	25	10	₅{}	align=left\|same as ₅{}×₅{} or {{CDD\|5node_1\|2\|5node_1}} $\mathbb{R}^4$ representation as 5-5 duoprism
align=center BGCOLOR="#e0ffff" \|2(50)5	₂{4}₅	{{CDD\|node_1\|4\|5node}}	10	25	{}	align=left\| $\mathbb{R}^4$ representation as 5-5 duopyramid
align=center BGCOLOR="#e0ffff" \| rowspan=2\|G(6,1,2) ₆[4]₂	rowspan=2\|72	rowspan=2\|36	6(72)2	₆{4}₂	{{CDD\|6node_1\|4\|node}}	36	12	₆{}	align=left\|same as ₆{}×₆{} or {{CDD\|6node_1\|2\|6node_1}} $\mathbb{R}^4$ representation as 6-6 duoprism
align=center BGCOLOR="#e0ffff" \|2(72)6	₂{4}₆	{{CDD\|node_1\|4\|6node}}	12	36	{}	align=left\| $\mathbb{R}^4$ representation as 6-6 duopyramid
align=center BGCOLOR="#e0f0ff" \|G₄=G(1,1,2) ₃[3]₃ <2,3,3>	24	6	3(24)3	₃{3}₃	{{CDD\|3node_1\|3\|3node}}	8	8	₃{}	align=left\|Möbius–Kantor configuration self-dual, same as {{CDD\|node_h\|6\|3node}} $\mathbb{R}^4$ representation as {3,3,4}
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₆ ₃[6]₂	rowspan=4\|48	rowspan=4\|12	3(48)2	₃{6}₂	{{CDD\|3node_1\|6\|node}}	rowspan=2\|24	rowspan=2\| 16	rowspan=2\|₃{}	align=left\|same as {{CDD\|3node_1\|3\|3node_1}}
align=center BGCOLOR="#e0ffe0"	\|₃{3}₂	{{CDD\|3node_1\|3\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|2(48)3	₂{6}₃	{{CDD\|node_1\|6\|3node}}	rowspan=2\| 16	rowspan=2\| 24	rowspan=2\|{}
align=center BGCOLOR="#e0ffe0"	\|₂{3}₃	{{CDD\|node_1\|3\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|G₅ ₃[4]₃	72	12	3(72)3	₃{4}₃	{{CDD\|3node_1\|4\|3node}}	24	24	₃{}	align=left \|self-dual, same as {{CDD\|node_h\|8\|3node}} $\mathbb{R}^4$ representation as {3,4,3}
align=center BGCOLOR="#e0f0ff" \|G₈ ₄[3]₄	96	12	4(96)4	₄{3}₄	{{CDD\|4node_1\|3\|4node}}	24	24	₄{}	align=left\|self-dual, same as {{CDD\|node_h\|6\|4node}} $\mathbb{R}^4$ representation as {3,4,3}
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₁₄ ₃[8]₂	rowspan=4\|144	rowspan=4\|24	3(144)2	₃{8}₂	{{CDD\|3node_1\|8\|node}}	rowspan=2\| 72	rowspan=2\| 48	rowspan=2\|₃{}	align=left\| same as {{CDD\|3node_1\|4\|3node_1}}
align=center BGCOLOR="#e0ffe0" \|	₃{8/3}₂	{{CDD\|3node_1\|8\|rat\|3x\|node}}	align=left\| starry polygon, same as {{CDD\|3node_1\|4\|rat\|3x\|3node_1}}
align=center BGCOLOR="#e0f0ff" \| 2(144)3	₂{8}₃	{{CDD\|node_1\|8\|3node}}	rowspan=2\| 48	rowspan=2\| 72	rowspan=2\|{}
align=center BGCOLOR="#e0ffe0" \|	₂{8/3}₃	{{CDD\|node_1\|8\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=4\|G₉ ₄[6]₂	rowspan=4\|192	rowspan=4\|24	4(192)2	₄{6}₂	{{CDD\|4node_1\|6\|node}}	96	48	₄{}	align=left\| same as {{CDD\|4node_1\|3\|4node_1}}
align=center BGCOLOR="#e0f0ff" \| 2(192)4	₂{6}₄	{{CDD\|node_1\|6\|4node}}	48	96	{}
align=center BGCOLOR="#e0ffe0" \|	₄{3}₂	{{CDD\|4node_1\|3\|node}}	96	48	{}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{3}₄	{{CDD\|node_1\|3\|4node}}	48	96	{}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \|rowspan=4\|G₁₀ ₄[4]₃	rowspan=4\|288	24	4(288)3	₄{4}₃	{{CDD\|4node_1\|4\|3node}}	rowspan=2\| 96	rowspan=2\| 72	rowspan=2\|₄{}
align=center BGCOLOR="#e0ffe0" \| 12		₄{8/3}₃	{{CDD\|4node_1\|8\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 24	3(288)4	₃{4}₄	{{CDD\|3node_1\|4\|4node}}	rowspan=2\| 72	rowspan=2\| 96	rowspan=2\|₃{}
align=center BGCOLOR="#e0ffe0" \| 12		₃{8/3}₄	{{CDD\|3node_1\|8\|rat\|3x\|4node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=2\|G₂₀ ₃[5]₃	rowspan=2\|360	rowspan=2\|30	3(360)3	₃{5}₃	{{CDD\|3node_1\|5\|3node}}	rowspan=2\| 120	rowspan=2\| 120	rowspan=2\|₃{}	align=left \|self-dual, same as {{CDD\|node_h\|10\|3node}} $\mathbb{R}^4$ representation as {3,3,5}
align=center BGCOLOR="#e0ffe0" \|	₃{5/2}₃	{{CDD\|3node_1\|5-2\|3node}}	align=left\|self-dual, starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=2\|G₁₆ ₅[3]₅	rowspan=2\|600	30	5(600)5	₅{3}₅	{{CDD\|5node_1\|3\|5node}}	rowspan=2\| 120	rowspan=2\| 120	rowspan=2\|₅{}	align=left \|self-dual, same as {{CDD\|node_h\|6\|5node}} $\mathbb{R}^4$ representation as {3,3,5}
align=center BGCOLOR="#e0ffe0" \| 10		₅{5/2}₅	{{CDD\|5node_1\|5-2\|5node}}	align=left\|self-dual, starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₂₁ ₃[10]₂	rowspan=8\|720	rowspan=8\|60	3(720)2	₃{10}₂	{{CDD\|3node_1\|10\|node}}	rowspan=4\| 360	rowspan=4\| 240	rowspan=4\|₃{}	align=left\| same as {{CDD\|3node_1\|5\|3node_1}}
align=center BGCOLOR="#e0ffe0"		₃{5}₂	{{CDD\|3node_1\|5\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"		₃{10/3}₂	{{CDD\|3node_1\|10\|rat\|3x\|node}}	align=left\| starry polygon, same as {{CDD\|3node_1\|5\|rat\|3x\|3node_1}}
align=center BGCOLOR="#e0ffe0"		₃{5/2}₂	{{CDD\|3node_1\|5-2\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 2(720)3	₂{10}₃	{{CDD\|node_1\|10\|3node}}	rowspan=4\| 240	rowspan=4\| 360	rowspan=4\|{}
align=center BGCOLOR="#e0ffe0" \|	₂{5}₃	{{CDD\|node_1\|5\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{10/3}₃	{{CDD\|node_1\|10\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \|	₂{5/2}₃	{{CDD\|node_1\|5-2\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₁₇ ₅[6]₂	rowspan=8\|1200	60	5(1200)2	₅{6}₂	{{CDD\|5node_1\|6\|node}}	rowspan=4\| 600	rowspan=4\| 240	rowspan=4\|₅{}	align=left\| same as {{CDD\|5node_1\|3\|5node_1}}
align=center BGCOLOR="#e0ffe0" \| 20		₅{5}₂	{{CDD\|5node_1\|5\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 20		₅{10/3}₂	{{CDD\|5node_1\|10\|rat\|3x\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 60		₅{3}₂	{{CDD\|5node_1\|3\|node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 60	2(1200)5	₂{6}₅	{{CDD\|node_1\|6\|5node}}	rowspan=4\| 240	rowspan=4\| 600	rowspan=4\|{}
align=center BGCOLOR="#e0ffe0" \| 20		₂{5}₅	{{CDD\|node_1\|5\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 20		₂{10/3}₅	{{CDD\|node_1\|10\|rat\|3x\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 60		₂{3}₅	{{CDD\|node_1\|3\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| rowspan=8\|G₁₈ ₅[4]₃	rowspan=8\|1800	60	5(1800)3	₅{4}₃	{{CDD\|5node_1\|4\|3node}}	rowspan=4\| 600	rowspan=4\| 360	rowspan=4\|₅{}	align=left\|
align=center BGCOLOR="#e0ffe0"	15		₅{10/3}₃	{{CDD\|5node_1\|10\|rat\|3x\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"	30		₅{3}₃	{{CDD\|5node_1\|3\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0"	30		₅{5/2}₃	{{CDD\|5node_1\|5-2\|3node}}	align=left\|starry polygon
align=center BGCOLOR="#e0f0ff" \| 60	3(1800)5	₃{4}₅	{{CDD\|3node_1\|4\|5node}}	rowspan=4\| 360	rowspan=4\| 600	rowspan=4\|₃{}
align=center BGCOLOR="#e0ffe0" \| 15		₃{10/3}₅	{{CDD\|3node_1\|10\|rat\|3x\|5node}}	align=left\| starry polygon
align=center BGCOLOR="#e0ffe0" \| 30		₃{3}₅	{{CDD\|3node_1\|3\|5node}}	align=left\|starry polygon
align=center BGCOLOR="#e0ffe0" \| 30		₃{5/2}₅	{{CDD\|3node_1\|5-2\|5node}}	align=left\|starry polygon