Regular Complex Polygon

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, $\mathbb^2$, while a complex polygon exists in two complex dimensions, $\mathbb^2$, which can be given real representations in 4 dimensions, $\mathbb^4$, 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 $\mathbb^n$.

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''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 ''p''₁-edges, with a ''p''₂-set as vertex figure and overall symmetry group of order ''g'', we denote the polygon as ''p''₁(''g'')''p''₂.

The number of vertices ''V'' is then ''g''/''p''₂ and the number of edges ''E'' is ''g''/''p''₁.

The complex polygon illustrated above has eight square edges (''p''₁=4) and sixteen vertices (''p''₂=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 _''p''1''p''2 is due to Coxeter, and is based on group theory. As a symmetry group, its symbol is _''p''1 'q''sub>''p''₂.

The symmetry group _''p''1 'q''sub>''p''₂ is represented by 2 generators R₁, R₂, where: R₁^''p''₁ = R₂^''p''₂ = I. If ''q'' is even, (R₂R₁)^''q''/2 = (R₁R₂)^''q''/2. If ''q'' is odd, (R₂R₁)^{(''q''−1)/2}R₂ = (R₁R₂)^{(''q''−1)/2}R₁. When ''q'' is odd, ''p''₁=''p''₂.

For ₄ sub>2 has R₁⁴ = R₂² = I, (R₂R₁)² = (R₁R₂)².

For ₃ sub>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''''r'' is represented by and the equivalent symmetry group, _''p'' 'q''sub>''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 regular polygon is ₂₂ or 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.

12 Irreducible Shephard groups

Coxeter enumerated this list of regular complex polygons in $\mathbb^2$. A regular complex polygon, _''p''''r'' or , has ''p''-edges, and ''r''-gonal vertex figures. _''p''''r'' is a finite polytope if (''p'' + ''r'')''q'' > ''pr''(''q'' − 2).

Its symmetry is written as _''p'' 'q''sub>''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''sub>''r'' can be computed as $g = 8/q \cdot (1/p+2/q+1/r-1)^$.

The Coxeter number for _''p'' 'q''sub>''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:

Excluded solutions with odd ''q'' and unequal ''p'' and ''r'' are: ₆ sub>2, ₆ sub>3, ₉ sub>3, ₁₂ sub>3, ..., ₅ sub>2, ₆ sub>2, ₈ sub>2, ₉ sub>2, ₄ sub>2, ₉ sub>2, ₃ sub>2, and ₃ 1sub>2.

Other whole ''q'' with unequal ''p'' and ''r'', create starry groups with overlapping fundamental domains: , , , , , and .

The dual polygon of _''p''''r'' is _''r''''p''. A polygon of the form _''p''''p'' is self-dual. Groups of the form _''p'' ''q''sub>2 have a half symmetry _''p'' 'q''sub>''p'', so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.

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'', , can be represented by two matrices:

With
: $k = \sqrt \frac$

;Examples

Enumeration of regular complex polygons

Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.

Visualizations of regular complex polygons

2D graphs

Polygons of the form _''p''''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''q'':

Polygons of the form _2''q'' are called generalized orthoplexes. They share vertices with the 4D ''q''-''q'' duopyramids, vertices connected by 2-edges.

Complex bipartite graph square.svg, ₂₂, , with 4 vertices, and 4 edges
Complex polygon 2-4-3-bipartite graph.png, ₂₃, , with 6 vertices, and 9 edges
Complex polygon 2-4-4 bipartite graph.png, ₂₄, , with 8 vertices, and 16 edges
Complex polygon 2-4-5-bipartite graph.png, ₂₅, , with 10 vertices, and 25 edges
6-generalized-2-orthoplex.svg, ₂₆, , with 12 vertices, and 36 edges
7-generalized-2-orthoplex.svg, ₂₇, , with 14 vertices, and 49 edges
8-generalized-2-orthoplex.svg, ₂₈, , with 16 vertices, and 64 edges
9-generalized-2-orthoplex.svg, ₂₉, , with 18 vertices, and 81 edges
10-generalized-2-orthoplex.svg, ₂₁₀, , with 20 vertices, and 100 edges

;Complex polygons _''p''2:

Polygons of the form _''p''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, ₂₂, or , with 4 vertices, and 4 2-edges
3-generalized-2-cube_skew.svg, ₃₂, or , with 9 vertices, and 6 (triangular) 3-edges
4-generalized-2-cube.svg, ₄₂, or , with 16 vertices, and 8 (square) 4-edges
5-generalized-2-cube_skew.svg, ₅₂, or , with 25 vertices, and 10 (pentagonal) 5-edges
6-generalized-2-cube.svg, ₆₂, or , with 36 vertices, and 12 (hexagonal) 6-edges
7-generalized-2-cube_skew.svg, ₇₂, or , with 49 vertices, and 14 (heptagonal)7-edges
8-generalized-2-cube.svg, ₈₂, or , with 64 vertices, and 16 (octagonal) 8-edges
9-generalized-2-cube_skew.svg, ₉₂, or , with 81 vertices, and 18 (enneagonal) 9-edges
10-generalized-2-cube.svg, ₁₀₂, or , with 100 vertices, and 20 (decagonal) 10-edges

;Complex polygons _''p''2:

Complex_polygon_3-6-2.png, ₃₂, or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue
Complex_polygon_3-8-2.png, ₃₂, or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue

;Complex polygons, _''p''''p'':

Polygons of the form _''p''''p'' have equal number of vertices and edges. They are also self-dual.

Complex polygon 3-3-3.svg, ₃₃, or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue
Complex_polygon_3-4-3-fill1.png, ₃₃, or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled
Complex polygon 4-3-4.png, ₄₄, or , with 24 vertices and 24 4-edges shown in 4 sets of colors
Complex polygon 3-5-3.png, ₃₃, or , with 120 vertices and 120 3-edges
Complex polygon 5-3-5.png, ₅₅, or , with 120 vertices and 120 5-edgesCoxeter, Regular Complex Polytopes, p. 49

3D perspective

3D perspective projections of complex polygons _''p''2 can show the point-edge structure of a complex polygon, while scale is not preserved.

The duals _2''p'': are seen by adding vertices inside the edges, and adding edges in place of vertices.

Complex polygon 2-4-3-stereographic0.png, ₂₃, with 6 vertices, 9 edges in 3 sets
Complex polygon 3-4-2-stereographic3.png, ₃₂, with 9 vertices, 6 3-edges in 2 sets of colors as
Complex polygon 4-4-2-stereographic3.svg, ₄₂, with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges as
Complex_polygon_5-4-2-stereographic3.png, ₅₂, with 25 vertices, 10 5-edges in 2 sets of colors as

Quasiregular polygons

A quasiregular polygon is a truncation 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.

Notes

References

* Coxeter, H.S.M. and Moser, W. O. J.; ''Generators and Relations for Discrete Groups'' (1965), esp pp 67–80.
*
* 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

Polytopes