Regular complex polygon

<h2 id="regular-complex-polygons">Regular complex polygons</h2>
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.

<h3>Notations</h3>
<h4>Shephard's modified Schläfli notation</h4>
<a href="/facts/Geoffrey_Colin_Shephard/5AtJ6sMW">Shephard</a> originally devised a modified form of <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli's notation</a> 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.

<h4>Coxeter's revised modified Schläfli notation</h4>
A more modern notation p1{q}p2 is due to <a href="/facts/Coxeter/D2l5jWGG">Coxeter</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 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: R1p1 = R2p2 = 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.

<h4>Coxeter–Dynkin diagrams</h4>
Coxeter also generalised the use of <a href="/facts/Coxeter%E2%80%93Dynkin_diagram/Ik6LmXJB">Coxeter–Dynkin diagrams</a> to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram . 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 <a href="/facts/Regular_polygon/ws89Nhpt">regular polygon</a> is 2{q}2 or {q} or .
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 and are ordinary, while is starry.

<h4>12 Irreducible Shephard groups</h4>
<table><tbody><tr><td>12 irreducible Shephard groups with their subgroup index relations.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></td><td>Subgroups from <5,3,2>30, <4,3,2>12 and <3,3,2>6</td></tr><tr><td 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.</td></tr></tbody></table>

Coxeter enumerated this list of regular complex polygons in 
 
 
 
 
 
 C
 
 
 2
 
 
 
 
 {\displaystyle \mathbb {C} ^{2}}
 
. A regular complex polygon, p{q}r or , has p-edges, and r-gonal <a href="/facts/Vertex_figure/3u2ySK0C">vertex figures</a>. p{q}r is a finite polytope if (p + r)q > pr(q − 2).
Its symmetry is written as p[q]r, called a <a href="/facts/Shephard_group/63v5gc1l">Shephard group</a>, analogous to a <a href="/facts/Coxeter_group/bgzHFm1Q">Coxeter group</a>, while also allowing <a href="/facts/Unitary_reflection/63v5gc1l">unitary reflections</a>.
For nonstarry groups, the order of the group p[q]r can be computed as 
 
 
 
 g
 =
 8
 
 /
 
 q
 ⋅
 (
 1
 
 /
 
 p
 +
 2
 
 /
 
 q
 +
 1
 
 /
 
 r
 −
 1
 
 )
 
 −
 2
 
 
 
 
 {\displaystyle g=8/q\cdot (1/p+2/q+1/r-1)^{-2}}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
The <a href="/facts/Coxeter_number/4B80wH8V">Coxeter number</a> for p[q]r is 
 
 
 
 h
 =
 2
 
 /
 
 (
 1
 
 /
 
 p
 +
 2
 
 /
 
 q
 +
 1
 
 /
 
 r
 −
 1
 )
 
 
 {\displaystyle h=2/(1/p+2/q+1/r-1)}
 
, so the group order can also be computed as 
 
 
 
 g
 =
 2
 
 h
 
 2
 
 
 
 /
 
 q
 
 
 {\displaystyle 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: 

<table><tbody><tr><th rowspan="3">Group</th><td>G3 = G(q,1,1)</td><td>G2 = G(p,1,2)</td><td>G4</td><td>G6</td><td>G5</td><td>G8</td><td>G14</td><td>G9</td><td>G10</td><td>G20</td><td>G16</td><td>G21</td><td>G17</td><td>G18</td></tr><tr><td>2[q]2, q = 3,4...</td><td>p[4]2, p = 2,3...</td><td>3[3]3</td><td>3[6]2</td><td>3[4]3</td><td>4[3]4</td><td>3[8]2</td><td>4[6]2</td><td>4[4]3</td><td>3[5]3</td><td>5[3]5</td><td>3[10]2</td><td>5[6]2</td><td>5[4]3</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Order</th><td>2q</td><td>2p2</td><td>24</td><td>48</td><td>72</td><td>96</td><td>144</td><td>192</td><td>288</td><td>360</td><td>600</td><td>720</td><td>1200</td><td>1800</td></tr><tr><th><a href="/facts/Coxeter_number/4B80wH8V">h</a></th><td>q</td><td>2p</td><td>6</td><td colspan="3">12</td><td colspan="3">24</td><td colspan="2">30</td><td colspan="3">60</td></tr></tbody></table>
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: , , , , , and .
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 is the same as quasiregular . As well, regular polygon with the same node orders, , have an <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a> construction , allowing adjacent edges to be two different colors.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
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.

<h4>Matrix generators</h4>
The group p[q]r, , can be represented by two matrices:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

<table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>p</td><td>r</td></tr><tr><th>Matrix</th><td> [ e 2 π i / p 0 ( e 2 π i / p − 1 ) k 1 ] {\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\(e^{2\pi i/p}-1)k&1\\\end{smallmatrix}}\right]} </td><td> [ 1 ( e 2 π i / r − 1 ) k 0 e 2 π i / r ] {\displaystyle \left[{\begin{smallmatrix}1&(e^{2\pi i/r}-1)k\\0&e^{2\pi i/r}\end{smallmatrix}}\right]} </td></tr></tbody></table>
With

k
        =
        
          
            
              
                cos
                ⁡
                (
                
                  
                    π
                    p
                  
                
                −
                
                  
                    π
                    r
                  
                
                )
                +
                cos
                ⁡
                (
                
                  
                    
                      2
                      π
                    
                    q
                  
                
                )
              
              
                2
                sin
                ⁡
                
                  
                    π
                    p
                  
                
                sin
                ⁡
                
                  
                    π
                    r
                  
                
              
            
          
        
      
    
    {\displaystyle 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
<table><tbody><tr><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>p</td><td>q</td></tr><tr><th>Matrix</th><td> [ e 2 π i / p 0 0 1 ] {\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]} </td><td> [ 1 0 0 e 2 π i / q ] {\displaystyle \left[{\begin{smallmatrix}1&0\\0&e^{2\pi i/q}\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>p</td><td>2</td></tr><tr><th>Matrix</th><td> [ e 2 π i / p 0 0 1 ] {\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix}}\right]} </td><td> [ 0 1 1 0 ] {\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>3</td><td>3</td></tr><tr><th>Matrix</th><td> [ − 1 + 3 i 2 0 − 3 + 3 i 2 1 ] {\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]} </td><td> [ 1 − 3 + 3 i 2 0 − 1 + 3 i 2 ] {\displaystyle \left[{\begin{smallmatrix}1&{\frac {-3+{\sqrt {3}}i}{2}}\\0&{\frac {-1+{\sqrt {3}}i}{2}}\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td></tr><tr><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>4</td><td>4</td></tr><tr><th>Matrix</th><td> [ i 0 0 1 ] {\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]} </td><td> [ 1 0 0 i ] {\displaystyle \left[{\begin{smallmatrix}1&0\\0&i\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>4</td><td>2</td></tr><tr><th>Matrix</th><td> [ i 0 0 1 ] {\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix}}\right]} </td><td> [ 0 1 1 0 ] {\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td><td><table><tbody><tr><th>Name</th><th>R1</th><th>R2</th></tr><tr><th>Order</th><td>3</td><td>2</td></tr><tr><th>Matrix</th><td> [ − 1 + 3 i 2 0 − 3 + 3 i 2 1 ] {\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3}}i}{2}}&0\\{\frac {-3+{\sqrt {3}}i}{2}}&1\\\end{smallmatrix}}\right]} </td><td> [ 1 − 2 0 − 1 ] {\displaystyle \left[{\begin{smallmatrix}1&-2\\0&-1\\\end{smallmatrix}}\right]} </td></tr></tbody></table></td></tr></tbody></table>
<h3>Enumeration of regular complex polygons</h3>
Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<table><tbody><tr><th><a href="/facts/Complex_reflection_group/63v5gc1l">Group</a></th><th>Order</th><th>Coxeternumber</th><th colspan="2">Polygon</th><th>Vertices</th><th colspan="2">Edges</th><th>Notes</th></tr><tr><td>G(q,q,2)2[q]2 = [q]q = 2,3,4,...</td><td>2q</td><td>q</td><td>2{q}2</td><td></td><td>q</td><td>q</td><td>{}</td><td>Real <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a>Same as Same as if q even</td></tr></tbody></table>
<table><tbody><tr><th><a href="/facts/Complex_reflection_group/63v5gc1l">Group</a></th><th>Order</th><th>Coxeternumber</th><th colspan="3">Polygon</th><th>Vertices</th><th colspan="2">Edges</th><th>Notes</th></tr><tr><td rowspan="2">G(p,1,2) p[4]2p=2,3,4,...</td><td rowspan="2">2p2</td><td rowspan="2">2p</td><td>p(2p2)2</td><td>p{4}2</td><td>         </td><td>p2</td><td>2p</td><td>p{}</td><td>same as p{}×p{} or R 4 {\displaystyle \mathbb {R} ^{4}} representation as p-p <a href="/facts/Duoprism/D73YMRth">duoprism</a></td></tr><tr><td>2(2p2)p</td><td>2{4}p</td><td></td><td>2p</td><td>p2</td><td>{}</td><td> R 4 {\displaystyle \mathbb {R} ^{4}} representation as p-p <a href="/facts/Duopyramid/B400kvjL">duopyramid</a></td></tr><tr><td>G(2,1,2) 2[4]2 = [4]</td><td>8</td><td>4</td><td></td><td>2{4}2 = {4}</td><td></td><td>4</td><td>4</td><td>{}</td><td>same as {}×{} or Real square</td></tr><tr><td rowspan="2">G(3,1,2) 3[4]2</td><td rowspan="2">18</td><td rowspan="2">6</td><td>6(18)2</td><td>3{4}2</td><td></td><td>9</td><td>6</td><td>3{}</td><td>same as 3{}×3{} or R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/3-3_duoprism/B1dhsgrg">3-3 duoprism</a></td></tr><tr><td>2(18)3</td><td>2{4}3</td><td></td><td>6</td><td>9</td><td>{}</td><td> R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/3-3_duopyramid/B1dhsgrg">3-3 duopyramid</a></td></tr><tr><td rowspan="2">G(4,1,2) 4[4]2</td><td rowspan="2">32</td><td rowspan="2">8</td><td>8(32)2</td><td>4{4}2</td><td></td><td>16</td><td>8</td><td>4{}</td><td>same as 4{}×4{} or R 4 {\displaystyle \mathbb {R} ^{4}} representation as 4-4 duoprism or <a href="/facts/Tesseract/tlqouBoU">{4,3,3}</a></td></tr><tr><td>2(32)4</td><td>2{4}4</td><td></td><td>8</td><td>16</td><td>{}</td><td> R 4 {\displaystyle \mathbb {R} ^{4}} representation as 4-4 duopyramid or <a href="/facts/16-cell/X0YAfobw">{3,3,4}</a></td></tr><tr><td rowspan="2">G(5,1,2) 5[4]2</td><td rowspan="2">50</td><td rowspan="2">25</td><td>5(50)2</td><td>5{4}2</td><td></td><td>25</td><td>10</td><td>5{}</td><td>same as 5{}×5{} or R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/5-5_duoprism/J7TrIYEa">5-5 duoprism</a></td></tr><tr><td>2(50)5</td><td>2{4}5</td><td></td><td>10</td><td>25</td><td>{}</td><td> R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/5-5_duopyramid/J7TrIYEa">5-5 duopyramid</a></td></tr><tr><td rowspan="2">G(6,1,2) 6[4]2</td><td rowspan="2">72</td><td rowspan="2">36</td><td>6(72)2</td><td>6{4}2</td><td></td><td>36</td><td>12</td><td>6{}</td><td>same as 6{}×6{} or R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/6-6_duoprism/Rgcg9TrJ">6-6 duoprism</a></td></tr><tr><td>2(72)6</td><td>2{4}6</td><td></td><td>12</td><td>36</td><td>{}</td><td> R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/6-6_duopyramid/Rgcg9TrJ">6-6 duopyramid</a></td></tr><tr><td>G4=G(1,1,2)3[3]3<a href="/facts/Binary_tetrahedral_group/pBBUSr2g"><2,3,3></a></td><td>24</td><td>6</td><td>3(24)3</td><td><a href="/facts/M%C3%B6bius%E2%80%93Kantor_polygon/BHQdNTjM">3{3}3</a></td><td></td><td>8</td><td>8</td><td>3{}</td><td><a href="/facts/M%C3%B6bius%E2%80%93Kantor_configuration/NgQIafId">Möbius–Kantor configuration</a>self-dual, same as R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/16-cell/X0YAfobw">{3,3,4}</a></td></tr><tr><td rowspan="4">G63[6]2</td><td rowspan="4">48</td><td rowspan="4">12</td><td>3(48)2</td><td>3{6}2</td><td></td><td rowspan="2">24</td><td rowspan="2">16</td><td rowspan="2">3{}</td><td>same as </td></tr><tr><td></td><td>3{3}2</td><td></td><td>starry polygon</td></tr><tr><td>2(48)3</td><td>2{6}3</td><td></td><td rowspan="2">16</td><td rowspan="2">24</td><td rowspan="2">{}</td><td></td></tr><tr><td></td><td>2{3}3</td><td></td><td>starry polygon</td></tr><tr><td>G53[4]3</td><td>72</td><td>12</td><td>3(72)3</td><td>3{4}3</td><td></td><td>24</td><td>24</td><td>3{}</td><td>self-dual, same as R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/24-cell/tfoGnlAD">{3,4,3}</a></td></tr><tr><td>G84[3]4</td><td>96</td><td>12</td><td>4(96)4</td><td>4{3}4</td><td></td><td>24</td><td>24</td><td>4{}</td><td>self-dual, same as R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/24-cell/tfoGnlAD">{3,4,3}</a></td></tr><tr><td rowspan="4">G143[8]2</td><td rowspan="4">144</td><td rowspan="4">24</td><td>3(144)2</td><td>3{8}2</td><td></td><td rowspan="2">72</td><td rowspan="2">48</td><td rowspan="2">3{}</td><td>same as </td></tr><tr><td></td><td>3{8/3}2</td><td></td><td>starry polygon, same as </td></tr><tr><td>2(144)3</td><td>2{8}3</td><td></td><td rowspan="2">48</td><td rowspan="2">72</td><td rowspan="2">{}</td><td></td></tr><tr><td></td><td>2{8/3}3</td><td></td><td>starry polygon</td></tr><tr><td rowspan="4">G94[6]2</td><td rowspan="4">192</td><td rowspan="4">24</td><td>4(192)2</td><td>4{6}2</td><td></td><td>96</td><td>48</td><td>4{}</td><td>same as </td></tr><tr><td>2(192)4</td><td>2{6}4</td><td></td><td>48</td><td>96</td><td>{}</td><td></td></tr><tr><td></td><td>4{3}2</td><td></td><td>96</td><td>48</td><td>{}</td><td>starry polygon</td></tr><tr><td></td><td>2{3}4</td><td></td><td>48</td><td>96</td><td>{}</td><td>starry polygon</td></tr><tr><td rowspan="4">G104[4]3</td><td rowspan="4">288</td><td>24</td><td>4(288)3</td><td>4{4}3</td><td></td><td rowspan="2">96</td><td rowspan="2">72</td><td rowspan="2">4{}</td><td></td></tr><tr><td>12</td><td></td><td>4{8/3}3</td><td></td><td>starry polygon</td></tr><tr><td>24</td><td>3(288)4</td><td>3{4}4</td><td></td><td rowspan="2">72</td><td rowspan="2">96</td><td rowspan="2">3{}</td><td></td></tr><tr><td>12</td><td></td><td>3{8/3}4</td><td></td><td>starry polygon</td></tr><tr><td rowspan="2">G203[5]3</td><td rowspan="2">360</td><td rowspan="2">30</td><td>3(360)3</td><td>3{5}3</td><td></td><td rowspan="2">120</td><td rowspan="2">120</td><td rowspan="2">3{}</td><td>self-dual, same as R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/600-cell/tNBtDUe9">{3,3,5}</a></td></tr><tr><td></td><td>3{5/2}3</td><td></td><td>self-dual, starry polygon</td></tr><tr><td rowspan="2">G165[3]5</td><td rowspan="2">600</td><td>30</td><td>5(600)5</td><td>5{3}5</td><td></td><td rowspan="2">120</td><td rowspan="2">120</td><td rowspan="2">5{}</td><td>self-dual, same as R 4 {\displaystyle \mathbb {R} ^{4}} representation as <a href="/facts/600-cell/tNBtDUe9">{3,3,5}</a></td></tr><tr><td>10</td><td></td><td>5{5/2}5</td><td></td><td>self-dual, starry polygon</td></tr><tr><td rowspan="8">G213[10]2</td><td rowspan="8">720</td><td rowspan="8">60</td><td>3(720)2</td><td>3{10}2</td><td></td><td rowspan="4">360</td><td rowspan="4">240</td><td rowspan="4">3{}</td><td>same as </td></tr><tr><td></td><td>3{5}2</td><td></td><td>starry polygon</td></tr><tr><td></td><td>3{10/3}2</td><td></td><td>starry polygon, same as </td></tr><tr><td></td><td>3{5/2}2</td><td></td><td>starry polygon</td></tr><tr><td>2(720)3</td><td>2{10}3</td><td></td><td rowspan="4">240</td><td rowspan="4">360</td><td rowspan="4">{}</td><td></td></tr><tr><td></td><td>2{5}3</td><td></td><td>starry polygon</td></tr><tr><td></td><td>2{10/3}3</td><td></td><td>starry polygon</td></tr><tr><td></td><td>2{5/2}3</td><td></td><td>starry polygon</td></tr><tr><td rowspan="8">G175[6]2</td><td rowspan="8">1200</td><td>60</td><td>5(1200)2</td><td>5{6}2</td><td></td><td rowspan="4">600</td><td rowspan="4">240</td><td rowspan="4">5{}</td><td>same as </td></tr><tr><td>20</td><td></td><td>5{5}2</td><td></td><td>starry polygon</td></tr><tr><td>20</td><td></td><td>5{10/3}2</td><td></td><td>starry polygon</td></tr><tr><td>60</td><td></td><td>5{3}2</td><td></td><td>starry polygon</td></tr><tr><td>60</td><td>2(1200)5</td><td>2{6}5</td><td></td><td rowspan="4">240</td><td rowspan="4">600</td><td rowspan="4">{}</td><td></td></tr><tr><td>20</td><td></td><td>2{5}5</td><td></td><td>starry polygon</td></tr><tr><td>20</td><td></td><td>2{10/3}5</td><td></td><td>starry polygon</td></tr><tr><td>60</td><td></td><td>2{3}5</td><td></td><td>starry polygon</td></tr><tr><td rowspan="8">G185[4]3</td><td rowspan="8">1800</td><td>60</td><td>5(1800)3</td><td>5{4}3</td><td></td><td rowspan="4">600</td><td rowspan="4">360</td><td rowspan="4">5{}</td><td></td></tr><tr><td>15</td><td></td><td>5{10/3}3</td><td></td><td>starry polygon</td></tr><tr><td>30</td><td></td><td>5{3}3</td><td></td><td>starry polygon</td></tr><tr><td>30</td><td></td><td>5{5/2}3</td><td></td><td>starry polygon</td></tr><tr><td>60</td><td>3(1800)5</td><td>3{4}5</td><td></td><td rowspan="4">360</td><td rowspan="4">600</td><td rowspan="4">3{}</td><td></td></tr><tr><td>15</td><td></td><td>3{10/3}5</td><td></td><td>starry polygon</td></tr><tr><td>30</td><td></td><td>3{3}5</td><td></td><td>starry polygon</td></tr><tr><td>30</td><td></td><td>3{5/2}5</td><td></td><td>starry polygon</td></tr></tbody></table>
<h3>Visualizations of regular complex polygons</h3>
<h4>2D graphs</h4>
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 <a href="/facts/Orthoplex/erk0byF6">orthoplexes</a>. They share vertices with the 4D q-q <a href="/facts/Duopyramid/B400kvjL">duopyramids</a>, vertices connected by 2-edges.

Complex polygons p{4}2

Polygons of the form p{4}2 are called generalized <a href="/facts/Hypercube/q2mRPbIz">hypercubes</a> (squares for polygons). They share vertices with the 4D p-p <a href="/facts/Duoprism/D73YMRth">duoprisms</a>, 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.

Complex polygons p{r}2

Complex polygons, p{r}p

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

<h4>3D perspective</h4>
3D <a href="/facts/Perspective_(graphical)/zkNx7Wpf">perspective</a> 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.

<h2 id="quasiregular-polygons">Quasiregular polygons</h2>
A <a href="/facts/Quasiregular_polyhedron/1ukrFhec">quasiregular</a> polygon is a <a href="/facts/Truncation_(geometry)/DYb589Je">truncation</a> of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has p vertices on the p-edges of the regular form.

Example quasiregular polygons<table><tbody><tr><th>p[q]r</th><th>2[4]2</th><th>3[4]2</th><th>4[4]2</th><th>5[4]2</th><th>6[4]2</th><th>7[4]2</th><th>8[4]2</th><th>3[3]3</th><th>3[4]3</th></tr><tr><th>Regular</th><td> 4 2-edges</td><td> 9 3-edges</td><td> 16 4-edges</td><td> 25 5-edges</td><td> 36 6-edges</td><td> 49 7-edges</td><td> 64 8-edges</td><td></td><td></td></tr><tr><th>Quasiregular</th><td> = 4+4 2-edges</td><td> 6 2-edges9 3-edges</td><td> 8 2-edges16 4-edges</td><td> 10 2-edges25 5-edges</td><td> 12 2-edges36 6-edges</td><td> 14 2-edges49 7-edges</td><td> 16 2-edges64 8-edges</td><td> = </td><td> = </td></tr><tr><th>Regular</th><td> 4 2-edges</td><td> 6 2-edges</td><td> 8 2-edges</td><td> 10 2-edges</td><td> 12 2-edges</td><td> 14 2-edges</td><td> 16 2-edges</td><td></td><td></td></tr></tbody></table>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.</li>
<li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> (1991), Regular Complex Polytopes, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-39490-2</li>
<li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,</li>
<li>Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.</li>
<li><a href="/facts/Geoffrey_Colin_Shephard/5AtJ6sMW">Shephard, G. C.</a>; Todd, J. A. (1954), "Finite unitary reflection groups", Canadian Journal of Mathematics, 6: 274–304, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4153%2Fcjm-1954-028-3">10.4153/cjm-1954-028-3</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0059914">0059914</a></li>
<li>Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">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. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Coxeter, Regular Complex Polytopes, p. xiv <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Coxeter, Complex Regular Polytopes, p. 177, Table III <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lehrer & Taylor 2009, p. 87 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Complex Polytopes, 8.9 The Two-Dimensional Case, p. 88 <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Regular Complex Polytopes, Coxeter, pp. 177–179 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Regular complex polygon open-in-new

Regular complex polygon