In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

A complex polytope 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.

Precise definitions exist only for the regular complex polytopes, which are configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Coxeter.

Some complex polytopes which are not fully regular have also been described.

Definitions and introduction

The complex line ${\displaystyle \mathbb {C} ^{1))$ has one dimension with real coordinates and another with imaginary coordinates. Applying real coordinates to both dimensions is said to give it two dimensions over the real numbers. A real plane, with the imaginary axis labelled as such, is called an Argand diagram. Because of this it is sometimes called the complex plane. Complex 2-space (also sometimes called the complex plane) is thus a four-dimensional space over the reals, and so on in higher dimensions.

A complex n-polytope in complex n-space is the analogue of a real n-polytope in real n-space.

There is no natural complex analogue of the ordering of points on a real line (or of the associated combinatorial properties). Because of this a complex polytope cannot be seen as a contiguous surface and it does not bound an interior in the way that a real polytope does.

In the case of regular polytopes, a precise definition can be made by using the notion of symmetry. For any regular polytope the symmetry group (here a complex reflection group, called a Shephard group) acts transitively on the flags, that is, on the nested sequences of a point contained in a line contained in a plane and so on.

More fully, say that a collection P of affine subspaces (or flats) of a complex unitary space V of dimension n is a regular complex polytope if it meets the following conditions:^[1][2]

• for every −1 ≤ i < j < k ≤ n, if F is a flat in P of dimension i and H is a flat in P of dimension k such that F ⊂ H then there are at least two flats G in P of dimension j such that F ⊂ G ⊂ H;
• for every i, j such that −1 ≤ i < j − 2, j ≤ n, if F ⊂ G are flats of P of dimensions i, j, then the set of flats between F and G is connected, in the sense that one can get from any member of this set to any other by a sequence of containments; and
• the subset of unitary transformations of V that fix P are transitive on the flags F₀ ⊂ F₁ ⊂ … ⊂F_n of flats of P (with F_i of dimension i for all i).

(Here, a flat of dimension −1 is taken to mean the empty set.) Thus, by definition, regular complex polytopes are configurations in complex unitary space.

The regular complex polytopes were discovered by Shephard (1952), and the theory was further developed by Coxeter (1974).

Three views of regular complex polygon ₄{4}₂,

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.[3] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.

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.

A complex polytope exists in the complex space of equivalent dimension. For example, the vertices of a complex polygon are points in the complex plane ${\displaystyle \mathbb {C} ^{2))$, and the edges are complex lines ${\displaystyle \mathbb {C} ^{1))$ existing as (affine) subspaces of the plane and intersecting at the vertices. Thus, an edge can be given a coordinate system consisting of a single complex number.^{[clarification needed]}

In a regular complex polytope the vertices incident on the edge are arranged symmetrically about their centroid, which is often used as the origin of the edge's coordinate system (in the real case the centroid is just the midpoint of the edge). The symmetry arises from a complex reflection about the centroid; this reflection will leave the magnitude of any vertex unchanged, but change its argument by a fixed amount, moving it to the coordinates of the next vertex in order. So we may assume (after a suitable choice of scale) that the vertices on the edge satisfy the equation ${\displaystyle x^{p}-1=0}$ where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin.

Three real projections of regular complex polygon 4{4}2 are illustrated above, with edges a, b, c, d, e, f, g, h. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices and each vertex lies on two edges, hence each edge meets four other edges. In the first diagram, each edge is represented by a square. The sides of the square are not parts of the polygon but are drawn purely to help visually relate the four vertices. The edges are laid out symmetrically. (Note that the diagram looks similar to the B₄ Coxeter plane projection of the tesseract, but it is structurally different).

The middle diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a real line, and each meeting point of two lines is a vertex. The connectivity between the various edges is clear to see.

The last diagram gives a flavour of the structure projected into three dimensions: the two cubes of vertices are in fact the same size but are seen in perspective at different distances away in the fourth dimension.

Regular complex one-dimensional polytopes

A real 1-dimensional polytope exists as a closed segment in the real line ${\displaystyle \mathbb {R} ^{1))$, defined by its two end points or vertices in the line. Its Schläfli symbol is {} .

Analogously, a complex 1-polytope exists as a set of p vertex points in the complex line ${\displaystyle \mathbb {C} ^{1))$. These may be represented as a set of points in an Argand diagram (x,y)=x+iy. A regular complex 1-dimensional polytope _p{} has p (p ≥ 2) vertex points arranged to form a convex regular polygon {p} in the Argand plane.^[4]

Unlike points on the real line, points on the complex line have no natural ordering. Thus, unlike real polytopes, no interior can be defined.^[5] Despite this, complex 1-polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane.

A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram . The dot or node of the Coxeter-Dynkin diagram itself represents a reflection generator while the circle around the node means the generator point is not on the reflection, so its reflective image is a distinct point from itself. By extension, a regular complex 1-dimensional polytope in ${\displaystyle \mathbb {C} ^{1))$ has Coxeter-Dynkin diagram , for any positive integer p, 2 or greater, containing p vertices. p can be suppressed if it is 2. It can also be represented by an empty Schläfli symbol _p{}, }_p{, {}_p, or _p{2}₁. The 1 is a notational placeholder, representing a nonexistent reflection, or a period 1 identity generator. (A 0-polytope, real or complex is a point, and is represented as } {, or ₁{2}₁.)

The symmetry is denoted by the Coxeter diagram , and can alternatively be described in Coxeter notation as _p[], []_p or ]_p[, _p[2]₁ or _p[1]_p. The symmetry is isomorphic to the cyclic group, order p.^[6] The subgroups of _p[] are any whole divisor d, _d[], where d≥2.

A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is e^2πi/p = cos(2π/p) + i sin(2π/p). When p = 2, the generator is e^πi = –1, the same as a point reflection in the real plane.

In higher complex polytopes, 1-polytopes form p-edges. A 2-edge is similar to an ordinary real edge, in that it contains two vertices, but need not exist on a real line.

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,^[7] 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 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 regular polygon 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.

12 Irreducible Shephard groups

Coxeter enumerated this list of regular complex polygons in ${\displaystyle \mathbb {C} ^{2))$. A regular complex polygon, _p{q}_r or , 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 ${\displaystyle g=8/q\cdot (1/p+2/q+1/r-1)^{-2))$.^[9]

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

The rank 2 solutions that generate complex polygons are:

Group	G3=G(q,1,1)	G2=G(p,1,2)	G4	G6	G5	G8	G14	G9	G10	G20	G16	G21	G17	G18
	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
Order	2q	2p2	24	48	72	96	144	192	288	360	600	720	1200	1800
h	q	2p	6	12			24			30		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: , , , , , 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 alternated construction , allowing adjacent edges to be two different colors.^[10]

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:^[11]

Name	R1	R2
Order	p	r
Matrix	${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\(e^{2\pi i/p}-1)k&1\\\end{smallmatrix))\right]}$	${\displaystyle \left[{\begin{smallmatrix}1&(e^{2\pi i/r}-1)k\\0&e^{2\pi i/r}\\\end{smallmatrix))\right]}$

With

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

Examples

Name R1 R2 Order p q Matrix ${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&e^{2\pi i/q}\\\end{smallmatrix))\right]}$	Name R1 R2 Order p 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}e^{2\pi i/p}&0\\0&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix))\right]}$	Name R1 R2 Order 3 3 Matrix ${\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3))i}{2))&0\\{\frac {-3+{\sqrt {3))i}{2))&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&{\frac {-3+{\sqrt {3))i}{2))\\0&{\frac {-1+{\sqrt {3))i}{2))\\\end{smallmatrix))\right]}$
Name R1 R2 Order 4 4 Matrix ${\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&0\\0&i\\\end{smallmatrix))\right]}$	Name R1 R2 Order 4 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}i&0\\0&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}0&1\\1&0\\\end{smallmatrix))\right]}$	Name R1 R2 Order 3 2 Matrix ${\displaystyle \left[{\begin{smallmatrix}{\frac {-1+{\sqrt {3))i}{2))&0\\{\frac {-3+{\sqrt {3))i}{2))&1\\\end{smallmatrix))\right]}$ ${\displaystyle \left[{\begin{smallmatrix}1&-2\\0&-1\\\end{smallmatrix))\right]}$

Enumeration of regular complex polygons

Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.^[12]

Group	Order	Coxeter number	Polygon		Vertices	Edges		Notes
G(q,q,2) 2[q]2 = [q] q=2,3,4,...	2q	q	2{q}2		q	q	{}	Real regular polygons Same as Same as if q even

Group	Order	Coxeter number	Polygon			Vertices	Edges		Notes
G(p,1,2) p[4]2 p=2,3,4,...	2p2	2p	p(2p2)2	p{4}2		p2	2p	p{}	same as p{}×p{} or ${\displaystyle \mathbb {R} ^{4))$ representation as p-p duoprism
			2(2p2)p	2{4}p		2p	p2	{}	${\displaystyle \mathbb {R} ^{4))$ representation as p-p duopyramid
G(2,1,2) 2[4]2 = [4]	8	4		2{4}2 = {4}		4	4	{}	same as {}×{} or Real square
G(3,1,2) 3[4]2	18	6	6(18)2	3{4}2		9	6	3{}	same as 3{}×3{} or ${\displaystyle \mathbb {R} ^{4))$ representation as 3-3 duoprism
			2(18)3	2{4}3		6	9	{}	${\displaystyle \mathbb {R} ^{4))$ representation as 3-3 duopyramid
G(4,1,2) 4[4]2	32	8	8(32)2	4{4}2		16	8	4{}	same as 4{}×4{} or ${\displaystyle \mathbb {R} ^{4))$ representation as 4-4 duoprism or {4,3,3}
			2(32)4	2{4}4		8	16	{}	${\displaystyle \mathbb {R} ^{4))$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2) 5[4]2	50	25	5(50)2	5{4}2		25	10	5{}	same as 5{}×5{} or ${\displaystyle \mathbb {R} ^{4))$ representation as 5-5 duoprism
			2(50)5	2{4}5		10	25	{}	${\displaystyle \mathbb {R} ^{4))$ representation as 5-5 duopyramid
G(6,1,2) 6[4]2	72	36	6(72)2	6{4}2		36	12	6{}	same as 6{}×6{} or ${\displaystyle \mathbb {R} ^{4))$ representation as 6-6 duoprism
			2(72)6	2{4}6		12	36	{}	${\displaystyle \mathbb {R} ^{4))$ representation as 6-6 duopyramid
G4=G(1,1,2) 3[3]3 <2,3,3>	24	6	3(24)3	3{3}3		8	8	3{}	Möbius–Kantor configuration self-dual, same as ${\displaystyle \mathbb {R} ^{4))$ representation as {3,3,4}
G6 3[6]2	48	12	3(48)2	3{6}2		24	16	3{}	same as
				3{3}2					starry polygon
			2(48)3	2{6}3		16	24	{}
				2{3}3					starry polygon
G5 3[4]3	72	12	3(72)3	3{4}3		24	24	3{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4))$ representation as {3,4,3}
G8 4[3]4	96	12	4(96)4	4{3}4		24	24	4{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4))$ representation as {3,4,3}
G14 3[8]2	144	24	3(144)2	3{8}2		72	48	3{}	same as
				3{8/3}2					starry polygon, same as
			2(144)3	2{8}3		48	72	{}
				2{8/3}3					starry polygon
G9 4[6]2	192	24	4(192)2	4{6}2		96	48	4{}	same as
			2(192)4	2{6}4		48	96	{}
				4{3}2		96	48	{}	starry polygon
				2{3}4		48	96	{}	starry polygon
G10 4[4]3	288	24	4(288)3	4{4}3		96	72	4{}
		12		4{8/3}3					starry polygon
		24	3(288)4	3{4}4		72	96	3{}
		12		3{8/3}4					starry polygon
G20 3[5]3	360	30	3(360)3	3{5}3		120	120	3{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4))$ representation as {3,3,5}
				3{5/2}3					self-dual, starry polygon
G16 5[3]5	600	30	5(600)5	5{3}5		120	120	5{}	self-dual, same as ${\displaystyle \mathbb {R} ^{4))$ representation as {3,3,5}
		10		5{5/2}5					self-dual, starry polygon
G21 3[10]2	720	60	3(720)2	3{10}2		360	240	3{}	same as
				3{5}2					starry polygon
				3{10/3}2					starry polygon, same as
				3{5/2}2					starry polygon
			2(720)3	2{10}3		240	360	{}
				2{5}3					starry polygon
				2{10/3}3					starry polygon
				2{5/2}3					starry polygon
G17 5[6]2	1200	60	5(1200)2	5{6}2		600	240	5{}	same as
		20		5{5}2					starry polygon
		20		5{10/3}2					starry polygon
		60		5{3}2					starry polygon
		60	2(1200)5	2{6}5		240	600	{}
		20		2{5}5					starry polygon
		20		2{10/3}5					starry polygon
		60		2{3}5					starry polygon
G18 5[4]3	1800	60	5(1800)3	5{4}3		600	360	5{}
		15		5{10/3}3					starry polygon
		30		5{3}3					starry polygon
		30		5{5/2}3					starry polygon
		60	3(1800)5	3{4}5		360	600	3{}
		15		3{10/3}5					starry polygon
		30		3{3}5					starry polygon
		30		3{5/2}5					starry polygon

Visualizations of regular complex polygons

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.

2D orthogonal projections of 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.

• 
  ₂{4}₂, , with 4 vertices, and 4 edges
• 
  ₂{4}₃, , with 6 vertices, and 9 edges^[13]
• 
  ₂{4}₄, , with 8 vertices, and 16 edges
• 
  ₂{4}₅, , with 10 vertices, and 25 edges
• 
  ₂{4}₆, , with 12 vertices, and 36 edges
• 
  ₂{4}₇, , with 14 vertices, and 49 edges
• 
  ₂{4}₈, , with 16 vertices, and 64 edges
• 
  ₂{4}₉, , with 18 vertices, and 81 edges
• 
  ₂{4}₁₀, , 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{4}₂, or , with 4 vertices, and 4 2-edges
• 
  ₃{4}₂, or , with 9 vertices, and 6 (triangular) 3-edges^[13]
• 
  ₄{4}₂, or , with 16 vertices, and 8 (square) 4-edges
• 
  ₅{4}₂, or , with 25 vertices, and 10 (pentagonal) 5-edges
• 
  ₆{4}₂, or , with 36 vertices, and 12 (hexagonal) 6-edges
• 
  ₇{4}₂, or , with 49 vertices, and 14 (heptagonal)7-edges
• 
  ₈{4}₂, or , with 64 vertices, and 16 (octagonal) 8-edges
• 
  ₉{4}₂, or , with 81 vertices, and 18 (enneagonal) 9-edges
• 
  ₁₀{4}₂, or , with 100 vertices, and 20 (decagonal) 10-edges

3D perspective projections of complex polygons _p{4}₂. The duals ₂{4}_p

are seen by adding vertices inside the edges, and adding edges in place of vertices.

• 
  ₃{4}₂, or with 9 vertices, 6 3-edges in 2 sets of colors
• 
  ₂{4}₃, with 6 vertices, 9 edges in 3 sets
• 
  ₄{4}₂, or with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges
• 
  ₅{4}₂, or with 25 vertices, 10 5-edges in 2 sets of colors

Other Complex polygons _p{r}₂

• 
  ₃{6}₂, or , with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue^[14]
• 
  ₃{8}₂, or , with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue^[15]

2D orthogonal projections of complex polygons, _p{r}_p

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

• 
  ₃{3}₃, or , with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue^[16]
• 
  ₃{4}₃, or , with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled^[17]
• 
  ₄{3}₄, or , with 24 vertices and 24 4-edges shown in 4 sets of colors^[17]
• 
  ₃{5}₃, or , with 120 vertices and 120 3-edges^[18]
• 
  ₅{3}₅, or , with 120 vertices and 120 5-edges^[19]

Regular complex polytopes

In general, a regular complex polytope is represented by Coxeter as _p{z₁}_q{z₂}_r{z₃}_s… or Coxeter diagram …, having symmetry _p[z₁]_q[z₂]_r[z₃]_s… or ….^[20]

There are infinite families of regular complex polytopes that occur in all dimensions, generalizing the hypercubes and cross polytopes in real space. Shephard's "generalized orthotope" generalizes the hypercube; it has symbol given by γ^p
_n = _p{4}₂{3}₂…₂{3}₂ and diagram …. Its symmetry group has diagram _p[4]₂[3]₂…₂[3]₂; in the Shephard–Todd classification, this is the group G(p, 1, n) generalizing the signed permutation matrices. Its dual regular polytope, the "generalized cross polytope", is represented by the symbol β^p
_n = ₂{3}₂{3}₂…₂{4}_p and diagram ….^[21]

A 1-dimensional regular complex polytope in ${\displaystyle \mathbb {C} ^{1))$ is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol γ^p
₁ or β^p
₁ as 1-dimensional generalized hypercube or cross polytope. Its symmetry is _p[] or , a cyclic group of order p. In a higher polytope, _p{} or represents a p-edge element, with a 2-edge, {} or , representing an ordinary real edge between two vertices.^[21]

A dual complex polytope is constructed by exchanging k and (n-1-k)-elements of an n-polytope. For example, a dual complex polygon has vertices centered on each edge, and new edges are centered at the old vertices. A v-valence vertex creates a new v-edge, and e-edges become e-valence vertices.^[22] The dual of a regular complex polytope has a reversed symbol. Regular complex polytopes with symmetric symbols, i.e. _p{q}_p, _p{q}_r{q}_p, _p{q}_r{s}_r{q}_p, etc. are self dual.

Enumeration of regular complex polyhedra

Coxeter enumerated this list of nonstarry regular complex polyhedra in ${\displaystyle \mathbb {C} ^{3))$, including the 5 platonic solids in ${\displaystyle \mathbb {R} ^{3))$.^[23]

A regular complex polyhedron, _p{n₁}_q{n₂}_r or , has faces, edges, and vertex figures.

A complex regular polyhedron _p{n₁}_q{n₂}_r requires both g₁ = order(_p[n₁]_q) and g₂ = order(_q[n₂]_r) be finite.

Given g = order(_p[n₁]_q[n₂]_r), the number of vertices is g/g₂, and the number of faces is g/g₁. The number of edges is g/pr.

Space	Group	Order	Coxeter number	Polygon		Vertices	Edges		Faces		Vertex figure	Van Oss polygon	Notes
${\displaystyle \mathbb {R} ^{3))$	G(1,1,3) 2[3]2[3]2 = [3,3]	24	4	α3 = 2{3}2{3}2 = {3,3}		4	6	{}	4	{3}	{3}	none	Real tetrahedron Same as
${\displaystyle \mathbb {R} ^{3))$	G23 2[3]2[5]2 = [3,5]	120	10	2{3}2{5}2 = {3,5}		12	30	{}	20	{3}	{5}	none	Real icosahedron
				2{5}2{3}2 = {5,3}		20	30	{}	12	{5}	{3}	none	Real dodecahedron
${\displaystyle \mathbb {R} ^{3))$	G(2,1,3) 2[3]2[4]2 = [3,4]	48	6	β2 3 = β3 = {3,4}		6	12	{}	8	{3}	{4}	{4}	Real octahedron Same as {}+{}+{}, order 8 Same as , order 24
${\displaystyle \mathbb {R} ^{3))$				γ2 3 = γ3 = {4,3}		8	12	{}	6	{4}	{3}	none	Real cube Same as {}×{}×{} or
${\displaystyle \mathbb {C} ^{3))$	G(p,1,3) 2[3]2[4]p p=2,3,4,...	6p3	3p	βp 3 = 2{3}2{4}p		3p	3p2	{}	p3	{3}	2{4}p	2{4}p	Generalized octahedron Same as p{}+p{}+p{}, order p3 Same as , order 6p2
${\displaystyle \mathbb {C} ^{3))$				γp 3 = p{4}2{3}2		p3	3p2	p{}	3p	p{4}2	{3}	none	Generalized cube Same as p{}×p{}×p{} or
${\displaystyle \mathbb {C} ^{3))$	G(3,1,3) 2[3]2[4]3	162	9	β3 3 = 2{3}2{4}3		9	27	{}	27	{3}	2{4}3	2{4}3	Same as 3{}+3{}+3{}, order 27 Same as , order 54
${\displaystyle \mathbb {C} ^{3))$				γ3 3 = 3{4}2{3}2		27	27	3{}	9	3{4}2	{3}	none	Same as 3{}×3{}×3{} or
${\displaystyle \mathbb {C} ^{3))$	G(4,1,3) 2[3]2[4]4	384	12	β4 3 = 2{3}2{4}4		12	48	{}	64	{3}	2{4}4	2{4}4	Same as 4{}+4{}+4{}, order 64 Same as , order 96
${\displaystyle \mathbb {C} ^{3))$				γ4 3 = 4{4}2{3}2		64	48	4{}	12	4{4}2	{3}	none	Same as 4{}×4{}×4{} or
${\displaystyle \mathbb {C} ^{3))$	G(5,1,3) 2[3]2[4]5	750	15	β5 3 = 2{3}2{4}5		15	75	{}	125	{3}	2{4}5	2{4}5	Same as 5{}+5{}+5{}, order 125 Same as , order 150
${\displaystyle \mathbb {C} ^{3))$				γ5 3 = 5{4}2{3}2		125	75	5{}	15	5{4}2	{3}	none	Same as 5{}×5{}×5{} or
${\displaystyle \mathbb {C} ^{3))$	G(6,1,3) 2[3]2[4]6	1296	18	β6 3 = 2{3}2{4}6		36	108	{}	216	{3}	2{4}6	2{4}6	Same as 6{}+6{}+6{}, order 216 Same as , order 216
${\displaystyle \mathbb {C} ^{3))$				γ6 3 = 6{4}2{3}2		216	108	6{}	18	6{4}2	{3}	none	Same as 6{}×6{}×6{} or
${\displaystyle \mathbb {C} ^{3))$	G25 3[3]3[3]3	648	9	3{3}3{3}3		27	72	3{}	27	3{3}3	3{3}3	3{4}2	Same as . ${\displaystyle \mathbb {R} ^{6))$ representation as 221 Hessian polyhedron
	G26 2[4]3[3]3	1296	18	2{4}3{3}3		54	216	{}	72	2{4}3	3{3}3	{6}
				3{3}3{4}2		72	216	3{}	54	3{3}3	3{4}2	3{4}3	Same as [24] ${\displaystyle \mathbb {R} ^{6))$ representation as 122

Visualizations of regular complex polyhedra

2D orthogonal projections of complex polyhedra, _p{s}_t{r}_r

• 
  Real {3,3}, or has 4 vertices, 6 edges, and 4 faces
• 
  ₃{3}₃{3}₃, or , has 27 vertices, 72 3-edges, and 27 faces, with one face highlighted blue.^[25]
• 
  ₂{4}₃{3}₃, has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.^[26]
• 
  ₃{3}₃{4}₂, or , has 72 vertices, 216 3-edges, and 54 vertices, with one face highlighted blue.^[27]

Generalized octahedra

Generalized octahedra have a regular construction as and quasiregular form as . All elements are simplexes.

• 
  Real {3,4}, or , with 6 vertices, 12 edges, and 8 faces
• 
  ₂{3}₂{4}₃, or , with 9 vertices, 27 edges, and 27 faces
• 
  ₂{3}₂{4}₄, or , with 12 vertices, 48 edges, and 64 faces
• 
  ₂{3}₂{4}₅, or , with 15 vertices, 75 edges, and 125 faces
• 
  ₂{3}₂{4}₆, or , with 18 vertices, 108 edges, and 216 faces
• 
  ₂{3}₂{4}₇, or , with 21 vertices, 147 edges, and 343 faces
• 
  ₂{3}₂{4}₈, or , with 24 vertices, 192 edges, and 512 faces
• 
  ₂{3}₂{4}₉, or , with 27 vertices, 243 edges, and 729 faces
• 
  ₂{3}₂{4}₁₀, or , with 30 vertices, 300 edges, and 1000 faces

Generalized cubes

Generalized cubes have a regular construction as and prismatic construction as , a product of three p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• 
  Real {4,3}, or has 8 vertices, 12 edges, and 6 faces
• 
  ₃{4}₂{3}₂, or has 27 vertices, 27 3-edges, and 9 faces^[25]
• 
  ₄{4}₂{3}₂, or , with 64 vertices, 48 edges, and 12 faces
• 
  ₅{4}₂{3}₂, or , with 125 vertices, 75 edges, and 15 faces
• 
  ₆{4}₂{3}₂, or , with 216 vertices, 108 edges, and 18 faces
• 
  ₇{4}₂{3}₂, or , with 343 vertices, 147 edges, and 21 faces
• 
  ₈{4}₂{3}₂, or , with 512 vertices, 192 edges, and 24 faces
• 
  ₉{4}₂{3}₂, or , with 729 vertices, 243 edges, and 27 faces
• 
  ₁₀{4}₂{3}₂, or , with 1000 vertices, 300 edges, and 30 faces

Enumeration of regular complex 4-polytopes

Coxeter enumerated this list of nonstarry regular complex 4-polytopes in ${\displaystyle \mathbb {C} ^{4))$, including the 6 convex regular 4-polytopes in ${\displaystyle \mathbb {R} ^{4))$.^[23]

Space	Group	Order	Coxeter number	Polytope	Vertices	Edges	Faces	Cells	Van Oss polygon	Notes
${\displaystyle \mathbb {R} ^{4))$	G(1,1,4) 2[3]2[3]2[3]2 = [3,3,3]	120	5	α4 = 2{3}2{3}2{3}2 = {3,3,3}	5	10 {}	10 {3}	5 {3,3}	none	Real 5-cell (simplex)
${\displaystyle \mathbb {R} ^{4))$	G28 2[3]2[4]2[3]2 = [3,4,3]	1152	12	2{3}2{4}2{3}2 = {3,4,3}	24	96 {}	96 {3}	24 {3,4}	{6}	Real 24-cell
	G30 2[3]2[3]2[5]2 = [3,3,5]	14400	30	2{3}2{3}2{5}2 = {3,3,5}	120	720 {}	1200 {3}	600 {3,3}	{10}	Real 600-cell
				2{5}2{3}2{3}2 = {5,3,3}	600	1200 {}	720 {5}	120 {5,3}		Real 120-cell
${\displaystyle \mathbb {R} ^{4))$	G(2,1,4) 2[3]2[3]2[4]p =[3,3,4]	384	8	β2 4 = β4 = {3,3,4}	8	24 {}	32 {3}	16 {3,3}	{4}	Real 16-cell Same as , order 192
${\displaystyle \mathbb {R} ^{4))$				γ2 4 = γ4 = {4,3,3}	16	32 {}	24 {4}	8 {4,3}	none	Real tesseract Same as {}4 or , order 16
${\displaystyle \mathbb {C} ^{4))$	G(p,1,4) 2[3]2[3]2[4]p p=2,3,4,...	24p4	4p	βp 4 = 2{3}2{3}2{4}p	4p	6p2 {}	4p3 {3}	p4 {3,3}	2{4}p	Generalized 4-orthoplex Same as , order 24p3
${\displaystyle \mathbb {C} ^{4))$				γp 4 = p{4}2{3}2{3}2	p4	4p3 p{}	6p2 p{4}2	4p p{4}2{3}2	none	Generalized tesseract Same as p{}4 or , order p4
${\displaystyle \mathbb {C} ^{4))$	G(3,1,4) 2[3]2[3]2[4]3	1944	12	β3 4 = 2{3}2{3}2{4}3	12	54 {}	108 {3}	81 {3,3}	2{4}3	Generalized 4-orthoplex Same as , order 648
${\displaystyle \mathbb {C} ^{4))$				γ3 4 = 3{4}2{3}2{3}2	81	108 3{}	54 3{4}2	12 3{4}2{3}2	none	Same as 3{}4 or , order 81
${\displaystyle \mathbb {C} ^{4))$	G(4,1,4) 2[3]2[3]2[4]4	6144	16	β4 4 = 2{3}2{3}2{4}4	16	96 {}	256 {3}	64 {3,3}	2{4}4	Same as , order 1536
${\displaystyle \mathbb {C} ^{4))$				γ4 4 = 4{4}2{3}2{3}2	256	256 4{}	96 4{4}2	16 4{4}2{3}2	none	Same as 4{}4 or , order 256
${\displaystyle \mathbb {C} ^{4))$	G(5,1,4) 2[3]2[3]2[4]5	15000	20	β5 4 = 2{3}2{3}2{4}5	20	150 {}	500 {3}	625 {3,3}	2{4}5	Same as , order 3000
${\displaystyle \mathbb {C} ^{4))$				γ5 4 = 5{4}2{3}2{3}2	625	500 5{}	150 5{4}2	20 5{4}2{3}2	none	Same as 5{}4 or , order 625
${\displaystyle \mathbb {C} ^{4))$	G(6,1,4) 2[3]2[3]2[4]6	31104	24	β6 4 = 2{3}2{3}2{4}6	24	216 {}	864 {3}	1296 {3,3}	2{4}6	Same as , order 5184
${\displaystyle \mathbb {C} ^{4))$				γ6 4 = 6{4}2{3}2{3}2	1296	864 6{}	216 6{4}2	24 6{4}2{3}2	none	Same as 6{}4 or , order 1296
${\displaystyle \mathbb {C} ^{4))$	G32 3[3]3[3]3[3]3	155520	30	3{3}3{3}3{3}3	240	2160 3{}	2160 3{3}3	240 3{3}3{3}3	3{4}3	Witting polytope ${\displaystyle \mathbb {R} ^{8))$ representation as 421