Complex polytope

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 ${\mathbb{ℂ}}^1$ 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. However, 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:

• 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, File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4.pngFile:CDel 3.pngFile:CDel node.png

File:ComplexOctagon.svg 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. In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.

File:Complex polygon 4-4-2-perspective-labeled.png 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 ${\mathbb{ℂ}}^2$ (a plane in which each point has two complex numbers as its coordinates, not to be confused with the Argand plane of complex numbers), and the edges are complex lines ${\mathbb{ℂ}}^1$ existing as (affine) subspaces of the plane and intersecting at the vertices. Thus, as a one-dimensional complex space, an edge can be given its own coordinate system, within which the points of the edge are each represented by a single complex number. 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 $x^p-1=0$ 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

File:Complex 1-topes as k-edges.png

Complex 1-polytopes represented in the Argand plane 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.

A real 1-dimensional polytope exists as a closed segment in the real line ${\mathbb{ℝ}}^1$, 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 ${\mathbb{ℂ}}^1$. 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. Unlike points on the real line, points on the complex line have no natural ordering. Thus, unlike real polytopes, no interior can be defined. Despite this, complex 1-polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane.

File:Coxeter node markup real unitary.png

A real edge is generated as the line between a point and its reflective image across a mirror. A unitary reflection order 2 can be seen as a 180 degree rotation around a center. An edge is inactive if the generator point is on the reflective line or at the center.

A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram File:CDel node 1.png. 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 ${\mathbb{ℂ}}^1$ has Coxeter-Dynkin diagram File:CDel pnode 1.png, 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 File:CDel pnode.png, 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. The subgroups of _p[] are any whole divisor d, _d[], where d≥2. A unitary operator generator for File:CDel pnode.png is seen as a rotation by 2π/p radians counter clockwise, and a File:CDel pnode 1.png 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, 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 File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png and the equivalent symmetry group, _p[q]_r, is a ringless diagram File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode.png. 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 File:CDel node 1.pngFile:CDel q.pngFile:CDel node.png. 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 File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png and File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png are ordinary, while File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel node.png is starry.

12 Irreducible Shephard groups

File:Rank2 shephard subgroups.png

12 irreducible Shephard groups with their subgroup index relations. Subgroups index 2 relate by removing a real reflection:
_p[2q]₂ → _p[q]_p, index 2.
_p[4]_q → _p[q]_p, index q.

File:Rank2 shephard subgroups2 series.png

_p[4]₂ subgroups: p=2,3,4...
_p[4]₂ → [p], index p
_p[4]₂ → _p[]×_p[], index 2

Coxeter enumerated this list of regular complex polygons in ${\mathbb{ℂ}}^2$. A regular complex polygon, _p{q}_r or File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png, 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 $g=8/q\cdot (1/p+2/q+1/r-1{)}^{-2}$. 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:

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
	File:CDel node.pngFile:CDel q.pngFile:CDel node.png	File:CDel pnode.pngFile:CDel 4.pngFile:CDel node.png	File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	File:CDel 3node.pngFile:CDel 6.pngFile:CDel node.png	File:CDel 3node.pngFile:CDel 4.pngFile:CDel 3node.png	File:CDel 4node.pngFile:CDel 3.pngFile:CDel 4node.png	File:CDel 3node.pngFile:CDel 8.pngFile:CDel node.png	File:CDel 4node.pngFile:CDel 6.pngFile:CDel node.png	File:CDel 4node.pngFile:CDel 4.pngFile:CDel 3node.png	File:CDel 3node.pngFile:CDel 5.pngFile:CDel 3node.png	File:CDel 5node.pngFile:CDel 3.pngFile:CDel 5node.png	File:CDel 3node.pngFile:CDel 10.pngFile:CDel node.png	File:CDel 5node.pngFile:CDel 6.pngFile:CDel node.png	File:CDel 5node.pngFile:CDel 4.pngFile:CDel 3node.png
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: File:CDel 3node.pngFile:CDel 3.pngFile:CDel node.png, File:CDel 4node.pngFile:CDel 3.pngFile:CDel node.png, File:CDel 5node.pngFile:CDel 3.pngFile:CDel node.png, File:CDel 5node.pngFile:CDel 3.pngFile:CDel 3node.png, File:CDel 3node.pngFile:CDel 5.pngFile:CDel node.png, and File:CDel 5node.pngFile:CDel 5.pngFile:CDel node.png. 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 File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel 3.pngFile:CDel node.png is the same as quasiregular File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel q.pngFile:CDel 3.pngFile:CDel pnode 1.png. As well, regular polygon with the same node orders, File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel q.pngFile:CDel 3.pngFile:CDel pnode.png, have an alternated construction File:CDel node h.pngFile:CDel 3.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel 3.pngFile:CDel pnode.png, 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, File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode.png, can be represented by two matrices:

File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode.png

Name	R1 File:CDel pnode.png	R2 File:CDel rnode.png
Order	p	r
Matrix	$\left[\begin{array}{cc}{e}^{2\pi i/p}& 0\\ (e^{2\pi i/p}-1)k& 1\\ \end{array}\right]$	$\left[\begin{array}{cc}1& (e^{2\pi i/r}-1)k\\ 0& e^{2\pi i/r}\\ \end{array}\right]$

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

File:CDel pnode.pngFile:CDel 2.pngFile:CDel qnode.png Name R1 File:CDel pnode.png R2 File:CDel qnode.png Order p q Matrix $\left[\begin{array}{cc}{e}^{2\pi i/p}& 0\\ 0& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}1& 0\\ 0& e^{2\pi i/q}\\ \end{array}\right]$	File:CDel pnode.pngFile:CDel 4.pngFile:CDel node.png Name R1 File:CDel pnode.png R2 File:CDel node.png Order p 2 Matrix $\left[\begin{array}{cc}{e}^{2\pi i/p}& 0\\ 0& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right]$	File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png Name R1 File:CDel 3node.png R2 File:CDel 3node.png Order 3 3 Matrix $\left[\begin{array}{cc}\frac{-1+\sqrt{3}i}{2}& 0\\ \frac{-3+\sqrt{3}i}{2}& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}1& \frac{-3+\sqrt{3}i}{2}\\ 0& \frac{-1+\sqrt{3}i}{2}\\ \end{array}\right]$
File:CDel 4node.pngFile:CDel 2.pngFile:CDel 4node.png Name R1 File:CDel 4node.png R2 File:CDel 4node.png Order 4 4 Matrix $\left[\begin{array}{cc}i& 0\\ 0& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}1& 0\\ 0& i\\ \end{array}\right]$	File:CDel 4node.pngFile:CDel 4.pngFile:CDel node.png Name R1 File:CDel 4node.png R2 File:CDel node.png Order 4 2 Matrix $\left[\begin{array}{cc}i& 0\\ 0& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}0& 1\\ 1& 0\\ \end{array}\right]$	File:CDel 3node.pngFile:CDel 6.pngFile:CDel node.png Name R1 File:CDel 3node.png R2 File:CDel node.png Order 3 2 Matrix $\left[\begin{array}{cc}\frac{-1+\sqrt{3}i}{2}& 0\\ \frac{-3+\sqrt{3}i}{2}& 1\\ \end{array}\right]$ $\left[\begin{array}{cc}1& -2\\ 0& -1\\ \end{array}\right]$

Enumeration of regular complex polygons

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

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	File:CDel node 1.pngFile:CDel q.pngFile:CDel node.png	q	q	{}	Real regular polygons Same as File:CDel node h.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png Same as File:CDel node 1.pngFile:CDel q.pngFile:CDel rat.pngFile:CDel 2x.pngFile:CDel node 1.png 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	File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.png	p2	2p	p{}	same as p{}×p{} or File:CDel pnode 1.pngFile:CDel 2.pngFile:CDel pnode 1.png ${\mathbb{ℝ}}^4$ representation as p-p duoprism
			2(2p2)p	2{4}p	File:CDel node 1.pngFile:CDel 4.pngFile:CDel pnode.png	2p	p2	{}	${\mathbb{ℝ}}^4$ representation as p-p duopyramid
G(2,1,2) 2[4]2 = [4]	8	4		2{4}2 = {4}	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png	4	4	{}	same as {}×{} or File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png Real square
G(3,1,2) 3[4]2	18	6	6(18)2	3{4}2	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png	9	6	3{}	same as 3{}×3{} or File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.png ${\mathbb{ℝ}}^4$ representation as 3-3 duoprism
			2(18)3	2{4}3	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png	6	9	{}	${\mathbb{ℝ}}^4$ representation as 3-3 duopyramid
G(4,1,2) 4[4]2	32	8	8(32)2	4{4}2	File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png	16	8	4{}	same as 4{}×4{} or File:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.png ${\mathbb{ℝ}}^4$ representation as 4-4 duoprism or {4,3,3}
			2(32)4	2{4}4	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png	8	16	{}	${\mathbb{ℝ}}^4$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2) 5[4]2	50	25	5(50)2	5{4}2	File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png	25	10	5{}	same as 5{}×5{} or File:CDel 5node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png ${\mathbb{ℝ}}^4$ representation as 5-5 duoprism
			2(50)5	2{4}5	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 5node.png	10	25	{}	${\mathbb{ℝ}}^4$ representation as 5-5 duopyramid
G(6,1,2) 6[4]2	72	36	6(72)2	6{4}2	File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.png	36	12	6{}	same as 6{}×6{} or File:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.png ${\mathbb{ℝ}}^4$ representation as 6-6 duoprism
			2(72)6	2{4}6	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.png	12	36	{}	${\mathbb{ℝ}}^4$ representation as 6-6 duopyramid
G4=G(1,1,2) 3[3]3 <2,3,3>	24	6	3(24)3	3{3}3	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png	8	8	3{}	Möbius–Kantor configuration self-dual, same as File:CDel node h.pngFile:CDel 6.pngFile:CDel 3node.png ${\mathbb{ℝ}}^4$ representation as {3,3,4}
G6 3[6]2	48	12	3(48)2	3{6}2	File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel node.png	24	16	3{}	same as File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node 1.png
				3{3}2	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel node.png				starry polygon
			2(48)3	2{6}3	File:CDel node 1.pngFile:CDel 6.pngFile:CDel 3node.png	16	24	{}
				2{3}3	File:CDel node 1.pngFile:CDel 3.pngFile:CDel 3node.png				starry polygon
G5 3[4]3	72	12	3(72)3	3{4}3	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png	24	24	3{}	self-dual, same as File:CDel node h.pngFile:CDel 8.pngFile:CDel 3node.png ${\mathbb{ℝ}}^4$ representation as {3,4,3}
G8 4[3]4	96	12	4(96)4	4{3}4	File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.png	24	24	4{}	self-dual, same as File:CDel node h.pngFile:CDel 6.pngFile:CDel 4node.png ${\mathbb{ℝ}}^4$ representation as {3,4,3}
G14 3[8]2	144	24	3(144)2	3{8}2	File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel node.png	72	48	3{}	same as File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png
				3{8/3}2	File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel node.png				starry polygon, same as File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node 1.png
			2(144)3	2{8}3	File:CDel node 1.pngFile:CDel 8.pngFile:CDel 3node.png	48	72	{}
				2{8/3}3	File:CDel node 1.pngFile:CDel 8.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node.png				starry polygon
G9 4[6]2	192	24	4(192)2	4{6}2	File:CDel 4node 1.pngFile:CDel 6.pngFile:CDel node.png	96	48	4{}	same as File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node 1.png
			2(192)4	2{6}4	File:CDel node 1.pngFile:CDel 6.pngFile:CDel 4node.png	48	96	{}
				4{3}2	File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel node.png	96	48	{}	starry polygon
				2{3}4	File:CDel node 1.pngFile:CDel 3.pngFile:CDel 4node.png	48	96	{}	starry polygon
G10 4[4]3	288	24	4(288)3	4{4}3	File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 3node.png	96	72	4{}
		12		4{8/3}3	File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node.png				starry polygon
		24	3(288)4	3{4}4	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 4node.png	72	96	3{}
		12		3{8/3}4	File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 4node.png				starry polygon
G20 3[5]3	360	30	3(360)3	3{5}3	File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel 3node.png	120	120	3{}	self-dual, same as File:CDel node h.pngFile:CDel 10.pngFile:CDel 3node.png ${\mathbb{ℝ}}^4$ representation as {3,3,5}
				3{5/2}3	File:CDel 3node 1.pngFile:CDel 5-2.pngFile:CDel 3node.png				self-dual, starry polygon
G16 5[3]5	600	30	5(600)5	5{3}5	File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel 5node.png	120	120	5{}	self-dual, same as File:CDel node h.pngFile:CDel 6.pngFile:CDel 5node.png ${\mathbb{ℝ}}^4$ representation as {3,3,5}
		10		5{5/2}5	File:CDel 5node 1.pngFile:CDel 5-2.pngFile:CDel 5node.png				self-dual, starry polygon
G21 3[10]2	720	60	3(720)2	3{10}2	File:CDel 3node 1.pngFile:CDel 10.pngFile:CDel node.png	360	240	3{}	same as File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel 3node 1.png
				3{5}2	File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel node.png				starry polygon
				3{10/3}2	File:CDel 3node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel node.png				starry polygon, same as File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node 1.png
				3{5/2}2	File:CDel 3node 1.pngFile:CDel 5-2.pngFile:CDel node.png				starry polygon
			2(720)3	2{10}3	File:CDel node 1.pngFile:CDel 10.pngFile:CDel 3node.png	240	360	{}
				2{5}3	File:CDel node 1.pngFile:CDel 5.pngFile:CDel 3node.png				starry polygon
				2{10/3}3	File:CDel node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node.png				starry polygon
				2{5/2}3	File:CDel node 1.pngFile:CDel 5-2.pngFile:CDel 3node.png				starry polygon
G17 5[6]2	1200	60	5(1200)2	5{6}2	File:CDel 5node 1.pngFile:CDel 6.pngFile:CDel node.png	600	240	5{}	same as File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel 5node 1.png
		20		5{5}2	File:CDel 5node 1.pngFile:CDel 5.pngFile:CDel node.png				starry polygon
		20		5{10/3}2	File:CDel 5node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel node.png				starry polygon
		60		5{3}2	File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel node.png				starry polygon
		60	2(1200)5	2{6}5	File:CDel node 1.pngFile:CDel 6.pngFile:CDel 5node.png	240	600	{}
		20		2{5}5	File:CDel node 1.pngFile:CDel 5.pngFile:CDel 5node.png				starry polygon
		20		2{10/3}5	File:CDel node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 5node.png				starry polygon
		60		2{3}5	File:CDel node 1.pngFile:CDel 3.pngFile:CDel 5node.png				starry polygon
G18 5[4]3	1800	60	5(1800)3	5{4}3	File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel 3node.png	600	360	5{}
		15		5{10/3}3	File:CDel 5node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 3node.png				starry polygon
		30		5{3}3	File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel 3node.png				starry polygon
		30		5{5/2}3	File:CDel 5node 1.pngFile:CDel 5-2.pngFile:CDel 3node.png				starry polygon
		60	3(1800)5	3{4}5	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 5node.png	360	600	3{}
		15		3{10/3}5	File:CDel 3node 1.pngFile:CDel 10.pngFile:CDel rat.pngFile:CDel 3x.pngFile:CDel 5node.png				starry polygon
		30		3{3}5	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 5node.png				starry polygon
		30		3{5/2}5	File:CDel 3node 1.pngFile:CDel 5-2.pngFile:CDel 5node.png				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, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png, with 4 vertices, and 4 edges
  ₂{4}₂, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png, with 4 vertices, and 4 edges
• 2{4}3, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png, with 6 vertices, and 9 edges[13]
  ₂{4}₃, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png, with 6 vertices, and 9 edges
• 2{4}4, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png, with 8 vertices, and 16 edges
  ₂{4}₄, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png, with 8 vertices, and 16 edges
• 2{4}5, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 5node.png, with 10 vertices, and 25 edges
  ₂{4}₅, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 5node.png, with 10 vertices, and 25 edges
• 2{4}6, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.png, with 12 vertices, and 36 edges
  ₂{4}₆, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.png, with 12 vertices, and 36 edges
• 2{4}7, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 7node.png, with 14 vertices, and 49 edges
  ₂{4}₇, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 7node.png, with 14 vertices, and 49 edges
• 2{4}8, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 8node.png, with 16 vertices, and 64 edges
  ₂{4}₈, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 8node.png, with 16 vertices, and 64 edges
• 2{4}9, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 9node.png, with 18 vertices, and 81 edges
  ₂{4}₉, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 9node.png, with 18 vertices, and 81 edges
• 2{4}10, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 10node.png, with 20 vertices, and 100 edges
  ₂{4}₁₀, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 10node.png, 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, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png, with 4 vertices, and 4 2-edges
  ₂{4}₂, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.png, with 4 vertices, and 4 2-edges
• 3{4}2, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.png, with 9 vertices, and 6 (triangular) 3-edges[13]
  ₃{4}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.png, with 9 vertices, and 6 (triangular) 3-edges
• 4{4}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.png, with 16 vertices, and 8 (square) 4-edges
  ₄{4}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.png, with 16 vertices, and 8 (square) 4-edges
• 5{4}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png, with 25 vertices, and 10 (pentagonal) 5-edges
  ₅{4}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png, with 25 vertices, and 10 (pentagonal) 5-edges
• 6{4}2, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.png, with 36 vertices, and 12 (hexagonal) 6-edges
  ₆{4}₂, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.png, with 36 vertices, and 12 (hexagonal) 6-edges
• 7{4}2, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2.pngFile:CDel 7node 1.png, with 49 vertices, and 14 (heptagonal)7-edges
  ₇{4}₂, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2.pngFile:CDel 7node 1.png, with 49 vertices, and 14 (heptagonal)7-edges
• 8{4}2, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2.pngFile:CDel 8node 1.png, with 64 vertices, and 16 (octagonal) 8-edges
  ₈{4}₂, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2.pngFile:CDel 8node 1.png, with 64 vertices, and 16 (octagonal) 8-edges
• 9{4}2, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2.pngFile:CDel 9node 1.png, with 81 vertices, and 18 (enneagonal) 9-edges
  ₉{4}₂, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2.pngFile:CDel 9node 1.png, with 81 vertices, and 18 (enneagonal) 9-edges
• 10{4}2, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2.pngFile:CDel 10node 1.png, with 100 vertices, and 20 (decagonal) 10-edges
  ₁₀{4}₂, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2.pngFile:CDel 10node 1.png, 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, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.png with 9 vertices, 6 3-edges in 2 sets of colors
  ₃{4}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.png with 9 vertices, 6 3-edges in 2 sets of colors
• 2{4}3, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png with 6 vertices, 9 edges in 3 sets
  ₂{4}₃, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png with 6 vertices, 9 edges in 3 sets
• 4{4}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.png with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges
  ₄{4}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.png with 16 vertices, 8 4-edges in 2 sets of colors and filled square 4-edges
• 5{4}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png with 25 vertices, 10 5-edges in 2 sets of colors
  ₅{4}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png with 25 vertices, 10 5-edges in 2 sets of colors

Other Complex polygons _p{r}₂

• 3{6}2, File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node 1.png, with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue[14]
  ₃{6}₂, File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node 1.png, with 24 vertices in black, and 16 3-edges colored in 2 sets of 3-edges in red and blue
• 3{8}2, File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png, with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue[15]
  ₃{8}₂, File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png, with 72 vertices in black, and 48 3-edges colored in 2 sets of 3-edges in red and blue

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, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 3node.png, with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue[16]
  ₃{3}₃, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 3node.png, with 8 vertices in black, and 8 3-edges colored in 2 sets of 3-edges in red and blue
• 3{4}3, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 8.pngFile:CDel 3node.png, with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled[17]
  ₃{4}₃, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 8.pngFile:CDel 3node.png, with 24 vertices and 24 3-edges shown in 3 sets of colors, one set filled
• 4{3}4, File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 4node.png, with 24 vertices and 24 4-edges shown in 4 sets of colors[17]
  ₄{3}₄, File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 4node.png, with 24 vertices and 24 4-edges shown in 4 sets of colors
• 3{5}3, File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 10.pngFile:CDel 3node.png, with 120 vertices and 120 3-edges[18]
  ₃{5}₃, File:CDel 3node 1.pngFile:CDel 5.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 10.pngFile:CDel 3node.png, with 120 vertices and 120 3-edges
• 5{3}5, File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel 5node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 5node.png, with 120 vertices and 120 5-edges[19]
  ₅{3}₅, File:CDel 5node 1.pngFile:CDel 3.pngFile:CDel 5node.png or File:CDel node h.pngFile:CDel 6.pngFile:CDel 5node.png, with 120 vertices and 120 5-edges

Regular complex polytopes

In general, a regular complex polytope is represented by Coxeter as _p{z₁}_q{z₂}_r{z₃}_s... or Coxeter diagram File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 1x.pngFile:CDel 3.pngFile:CDel qnode.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 2x.pngFile:CDel 3.pngFile:CDel rnode.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 3x.pngFile:CDel 3.pngFile:CDel snode.png..., having symmetry _p[z₁]_q[z₂]_r[z₃]_s... or File:CDel pnode.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 1x.pngFile:CDel 3.pngFile:CDel qnode.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 2x.pngFile:CDel 3.pngFile:CDel rnode.pngFile:CDel 3.pngFile:CDel z.pngFile:CDel 3x.pngFile:CDel 3.pngFile:CDel snode.png.... 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 File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.png...File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png. 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 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.png...File:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png. A 1-dimensional regular complex polytope in ${\mathbb{ℂ}}^1$ is represented as File:CDel pnode 1.png, 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 File:CDel pnode.png, a cyclic group of order p. In a higher polytope, _p{} or File:CDel pnode 1.png represents a p-edge element, with a 2-edge, {} or File:CDel node 1.png, representing an ordinary real edge between two vertices. 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. 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

File:Rank3 shephard subgroups.png

Some rank 3 Shephard groups with their group orders, and the reflective subgroup relations

Coxeter enumerated this list of nonstarry regular complex polyhedra in ${\mathbb{ℂ}}^3$, including the 5 platonic solids in ${\mathbb{ℝ}}^3$. A regular complex polyhedron, _p{n₁}_q{n₂}_r or File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel n.pngFile:CDel 1x.pngFile:CDel 3.pngFile:CDel qnode.pngFile:CDel 3.pngFile:CDel n.pngFile:CDel 2x.pngFile:CDel 3.pngFile:CDel rnode.png, has File:CDel pnode 1.pngFile:CDel 3.pngFile:CDel n.pngFile:CDel 1x.pngFile:CDel 3.pngFile:CDel qnode.png faces, File:CDel pnode 1.png edges, and File:CDel qnode 1.pngFile:CDel 3.pngFile:CDel n.pngFile:CDel 2x.pngFile:CDel 3.pngFile:CDel rnode.png 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
${\mathbb{ℝ}}^3$	G(1,1,3) 2[3]2[3]2 = [3,3]	24	4	α3 = 2{3}2{3}2 = {3,3}	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	4	6	{}	4	{3}	{3}	none	Real tetrahedron Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
${\mathbb{ℝ}}^3$	G23 2[3]2[5]2 = [3,5]	120	10	2{3}2{5}2 = {3,5}	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	12	30	{}	20	{3}	{5}	none	Real icosahedron
				2{5}2{3}2 = {5,3}	File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	20	30	{}	12	{5}	{3}	none	Real dodecahedron
${\mathbb{ℝ}}^3$	G(2,1,3) 2[3]2[4]2 = [3,4]	48	6	β2 3 = β3 = {3,4}	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	6	12	{}	8	{3}	{4}	{4}	Real octahedron Same as {}+{}+{}, order 8 Same as File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, order 24
${\mathbb{ℝ}}^3$				γ2 3 = γ3 = {4,3}	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	8	12	{}	6	{4}	{3}	none	Real cube Same as {}×{}×{} or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png
${\mathbb{ℂ}}^3$	G(p,1,3) 2[3]2[4]p p=2,3,4,...	6p3	3p	βp 3 = 2{3}2{4}p	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png	3p	3p2	{}	p3	{3}	2{4}p	2{4}p	Generalized octahedron Same as p{}+p{}+p{}, order p3 Same as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png, order 6p2
${\mathbb{ℂ}}^3$				γp 3 = p{4}2{3}2	File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	p3	3p2	p{}	3p	p{4}2	{3}	none	Generalized cube Same as p{}×p{}×p{} or File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png
${\mathbb{ℂ}}^3$	G(3,1,3) 2[3]2[4]3	162	9	β3 3 = 2{3}2{4}3	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png	9	27	{}	27	{3}	2{4}3	2{4}3	Same as 3{}+3{}+3{}, order 27 Same as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png, order 54
${\mathbb{ℂ}}^3$				γ3 3 = 3{4}2{3}2	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	27	27	3{}	9	3{4}2	{3}	none	Same as 3{}×3{}×3{} or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png
${\mathbb{ℂ}}^3$	G(4,1,3) 2[3]2[4]4	384	12	β4 3 = 2{3}2{4}4	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png	12	48	{}	64	{3}	2{4}4	2{4}4	Same as 4{}+4{}+4{}, order 64 Same as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, order 96
${\mathbb{ℂ}}^3$				γ4 3 = 4{4}2{3}2	File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	64	48	4{}	12	4{4}2	{3}	none	Same as 4{}×4{}×4{} or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png
${\mathbb{ℂ}}^3$	G(5,1,3) 2[3]2[4]5	750	15	β5 3 = 2{3}2{4}5	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png	15	75	{}	125	{3}	2{4}5	2{4}5	Same as 5{}+5{}+5{}, order 125 Same as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, order 150
${\mathbb{ℂ}}^3$				γ5 3 = 5{4}2{3}2	File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	125	75	5{}	15	5{4}2	{3}	none	Same as 5{}×5{}×5{} or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png
${\mathbb{ℂ}}^3$	G(6,1,3) 2[3]2[4]6	1296	18	β6 3 = 2{3}2{4}6	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png	36	108	{}	216	{3}	2{4}6	2{4}6	Same as 6{}+6{}+6{}, order 216 Same as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, order 216
${\mathbb{ℂ}}^3$				γ6 3 = 6{4}2{3}2	File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	216	108	6{}	18	6{4}2	{3}	none	Same as 6{}×6{}×6{} or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png
${\mathbb{ℂ}}^3$	G25 3[3]3[3]3	648	9	3{3}3{3}3	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	27	72	3{}	27	3{3}3	3{3}3	3{4}2	Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png. ${\mathbb{ℝ}}^6$ representation as 221 Hessian polyhedron
	G26 2[4]3[3]3	1296	18	2{4}3{3}3	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	54	216	{}	72	2{4}3	3{3}3	{6}
				3{3}3{4}2	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png	72	216	3{}	54	3{3}3	3{4}2	3{4}3	Same as File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png ${\mathbb{ℝ}}^6$ representation as 122

Visualizations of regular complex polyhedra

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

• Real {3,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png has 4 vertices, 6 edges, and 4 faces
  Real {3,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png has 4 vertices, 6 edges, and 4 faces
• 3{3}3{3}3, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, has 27 vertices, 72 3-edges, and 27 faces, with one face highlighted blue.[25]
  ₃{3}₃{3}₃, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png or File:CDel node h.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, has 27 vertices, 72 3-edges, and 27 faces, with one face highlighted blue.
• 2{4}3{3}3, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.[26]
  ₂{4}₃{3}₃, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.
• 3{3}3{4}2, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png, has 72 vertices, 216 3-edges, and 54 vertices, with one face highlighted blue.[27]
  ₃{3}₃{4}₂, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png, has 72 vertices, 216 3-edges, and 54 vertices, with one face highlighted blue.

Generalized octahedra

Generalized octahedra have a regular construction as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png and quasiregular form as File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png. All elements are simplexes.

• Real {3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, with 6 vertices, 12 edges, and 8 faces
  Real {3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png, with 6 vertices, 12 edges, and 8 faces
• 2{3}2{4}3, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png, with 9 vertices, 27 edges, and 27 faces
  ₂{3}₂{4}₃, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png, with 9 vertices, 27 edges, and 27 faces
• 2{3}2{4}4, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, with 12 vertices, 48 edges, and 64 faces
  ₂{3}₂{4}₄, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, with 12 vertices, 48 edges, and 64 faces
• 2{3}2{4}5, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, with 15 vertices, 75 edges, and 125 faces
  ₂{3}₂{4}₅, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, with 15 vertices, 75 edges, and 125 faces
• 2{3}2{4}6, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, with 18 vertices, 108 edges, and 216 faces
  ₂{3}₂{4}₆, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, with 18 vertices, 108 edges, and 216 faces
• 2{3}2{4}7, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label7.png, with 21 vertices, 147 edges, and 343 faces
  ₂{3}₂{4}₇, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label7.png, with 21 vertices, 147 edges, and 343 faces
• 2{3}2{4}8, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label8.png, with 24 vertices, 192 edges, and 512 faces
  ₂{3}₂{4}₈, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label8.png, with 24 vertices, 192 edges, and 512 faces
• 2{3}2{4}9, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label9.png, with 27 vertices, 243 edges, and 729 faces
  ₂{3}₂{4}₉, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label9.png, with 27 vertices, 243 edges, and 729 faces
• 2{3}2{4}10, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label1.pngFile:CDel label0.png, with 30 vertices, 300 edges, and 1000 faces
  ₂{3}₂{4}₁₀, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png or File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label1.pngFile:CDel label0.png, with 30 vertices, 300 edges, and 1000 faces

Generalized cubes

Generalized cubes have a regular construction as File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png and prismatic construction as File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, a product of three p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• Real {4,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png has 8 vertices, 12 edges, and 6 faces
  Real {4,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png has 8 vertices, 12 edges, and 6 faces
• 3{4}2{3}2, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png has 27 vertices, 27 3-edges, and 9 faces[25]
  ₃{4}₂{3}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png has 27 vertices, 27 3-edges, and 9 faces
• 4{4}2{3}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, with 64 vertices, 48 edges, and 12 faces
  ₄{4}₂{3}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, with 64 vertices, 48 edges, and 12 faces
• 5{4}2{3}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, with 125 vertices, 75 edges, and 15 faces
  ₅{4}₂{3}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, with 125 vertices, 75 edges, and 15 faces
• 6{4}2{3}2, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, with 216 vertices, 108 edges, and 18 faces
  ₆{4}₂{3}₂, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, with 216 vertices, 108 edges, and 18 faces
• 7{4}2{3}2, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.png, with 343 vertices, 147 edges, and 21 faces
  ₇{4}₂{3}₂, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.png, with 343 vertices, 147 edges, and 21 faces
• 8{4}2{3}2, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.png, with 512 vertices, 192 edges, and 24 faces
  ₈{4}₂{3}₂, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.png, with 512 vertices, 192 edges, and 24 faces
• 9{4}2{3}2, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.png, with 729 vertices, 243 edges, and 27 faces
  ₉{4}₂{3}₂, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.png, with 729 vertices, 243 edges, and 27 faces
• 10{4}2{3}2, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.png, with 1000 vertices, 300 edges, and 30 faces
  ₁₀{4}₂{3}₂, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.png, 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 ${\mathbb{ℂ}}^4$, including the 6 convex regular 4-polytopes in ${\mathbb{ℝ}}^4$.

Space	Group	Order	Coxeter number	Polytope	Vertices	Edges	Faces	Cells	Van Oss polygon	Notes
${\mathbb{ℝ}}^4$	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} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	5	10 {}	10 {3}	5 {3,3}	none	Real 5-cell (simplex)
${\mathbb{ℝ}}^4$	G28 2[3]2[4]2[3]2 = [3,4,3]	1152	12	2{3}2{4}2{3}2 = {3,4,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	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} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png	120	720 {}	1200 {3}	600 {3,3}	{10}	Real 600-cell
				2{5}2{3}2{3}2 = {5,3,3} File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	600	1200 {}	720 {5}	120 {5,3}		Real 120-cell
${\mathbb{ℝ}}^4$	G(2,1,4) 2[3]2[3]2[4]p =[3,3,4]	384	8	β2 4 = β4 = {3,3,4} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	8	24 {}	32 {3}	16 {3,3}	{4}	Real 16-cell Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, order 192
${\mathbb{ℝ}}^4$				γ2 4 = γ4 = {4,3,3} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	16	32 {}	24 {4}	8 {4,3}	none	Real tesseract Same as {}4 or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png, order 16
${\mathbb{ℂ}}^4$	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 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png	4p	6p2 {}	4p3 {3}	p4 {3,3}	2{4}p	Generalized 4-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png, order 24p3
${\mathbb{ℂ}}^4$				γp 4 = p{4}2{3}2{3}2 File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	p4	4p3 p{}	6p2 p{4}2	4p p{4}2{3}2	none	Generalized tesseract Same as p{}4 or File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, order p4
${\mathbb{ℂ}}^4$	G(3,1,4) 2[3]2[3]2[4]3	1944	12	β3 4 = 2{3}2{3}2{4}3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png	12	54 {}	108 {3}	81 {3,3}	2{4}3	Generalized 4-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png, order 648
${\mathbb{ℂ}}^4$				γ3 4 = 3{4}2{3}2{3}2 File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	81	108 3{}	54 3{4}2	12 3{4}2{3}2	none	Same as 3{}4 or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png, order 81
${\mathbb{ℂ}}^4$	G(4,1,4) 2[3]2[3]2[4]4	6144	16	β4 4 = 2{3}2{3}2{4}4 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png	16	96 {}	256 {3}	64 {3,3}	2{4}4	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, order 1536
${\mathbb{ℂ}}^4$				γ4 4 = 4{4}2{3}2{3}2 File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	256	256 4{}	96 4{4}2	16 4{4}2{3}2	none	Same as 4{}4 or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, order 256
${\mathbb{ℂ}}^4$	G(5,1,4) 2[3]2[3]2[4]5	15000	20	β5 4 = 2{3}2{3}2{4}5 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png	20	150 {}	500 {3}	625 {3,3}	2{4}5	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, order 3000
${\mathbb{ℂ}}^4$				γ5 4 = 5{4}2{3}2{3}2 File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	625	500 5{}	150 5{4}2	20 5{4}2{3}2	none	Same as 5{}4 or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, order 625
${\mathbb{ℂ}}^4$	G(6,1,4) 2[3]2[3]2[4]6	31104	24	β6 4 = 2{3}2{3}2{4}6 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png	24	216 {}	864 {3}	1296 {3,3}	2{4}6	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, order 5184
${\mathbb{ℂ}}^4$				γ6 4 = 6{4}2{3}2{3}2 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	1296	864 6{}	216 6{4}2	24 6{4}2{3}2	none	Same as 6{}4 or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, order 1296
${\mathbb{ℂ}}^4$	G32 3[3]3[3]3[3]3	155520	30	3{3}3{3}3{3}3 File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	240	2160 3{}	2160 3{3}3	240 3{3}3{3}3	3{4}3	Witting polytope ${\mathbb{ℝ}}^8$ representation as 421

Visualizations of regular complex 4-polytopes

• Real {3,3,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 5 vertices, 10 edges, 10 {3} faces, and 5 {3,3} cells
  Real {3,3,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 5 vertices, 10 edges, 10 {3} faces, and 5 {3,3} cells
• Real {3,4,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 24 vertices, 96 edges, 96 {3} faces, and 24 {3,4} cells
  Real {3,4,3}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 24 vertices, 96 edges, 96 {3} faces, and 24 {3,4} cells
• Real {5,3,3}, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 600 vertices, 1200 edges, 720 {5} faces, and 120 {5,3} cells
  Real {5,3,3}, File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, had 600 vertices, 1200 edges, 720 {5} faces, and 120 {5,3} cells
• Real {3,3,5}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, had 120 vertices, 720 edges, 1200 {3} faces, and 600 {3,3} cells
  Real {3,3,5}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node.png, had 120 vertices, 720 edges, 1200 {3} faces, and 600 {3,3} cells
• Witting polytope, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, has 240 vertices, 2160 3-edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells
  Witting polytope, File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, has 240 vertices, 2160 3-edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells

Generalized 4-orthoplexes

Generalized 4-orthoplexes have a regular construction as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png and quasiregular form as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png. All elements are simplexes.

• Real {3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, with 8 vertices, 24 edges, 32 faces, and 16 cells
  Real {3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, with 8 vertices, 24 edges, 32 faces, and 16 cells
• 2{3}2{3}2{4}3, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png, with 12 vertices, 54 edges, 108 faces, and 81 cells
  ₂{3}₂{3}₂{4}₃, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png, with 12 vertices, 54 edges, 108 faces, and 81 cells
• 2{3}2{3}2{4}4, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, with 16 vertices, 96 edges, 256 faces, and 256 cells
  ₂{3}₂{3}₂{4}₄, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, with 16 vertices, 96 edges, 256 faces, and 256 cells
• 2{3}2{3}2{4}5, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, with 20 vertices, 150 edges, 500 faces, and 625 cells
  ₂{3}₂{3}₂{4}₅, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, with 20 vertices, 150 edges, 500 faces, and 625 cells
• 2{3}2{3}2{4}6, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, with 24 vertices, 216 edges, 864 faces, and 1296 cells
  ₂{3}₂{3}₂{4}₆, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, with 24 vertices, 216 edges, 864 faces, and 1296 cells
• 2{3}2{3}2{4}7, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label7.png, with 28 vertices, 294 edges, 1372 faces, and 2401 cells
  ₂{3}₂{3}₂{4}₇, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label7.png, with 28 vertices, 294 edges, 1372 faces, and 2401 cells
• 2{3}2{3}2{4}8, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label8.png, with 32 vertices, 384 edges, 2048 faces, and 4096 cells
  ₂{3}₂{3}₂{4}₈, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label8.png, with 32 vertices, 384 edges, 2048 faces, and 4096 cells
• 2{3}2{3}2{4}9, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label9.png, with 36 vertices, 486 edges, 2916 faces, and 6561 cells
  ₂{3}₂{3}₂{4}₉, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label9.png, with 36 vertices, 486 edges, 2916 faces, and 6561 cells
• 2{3}2{3}2{4}10, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label1.pngFile:CDel label0.png, with 40 vertices, 600 edges, 4000 faces, and 10000 cells
  ₂{3}₂{3}₂{4}₁₀, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png or File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label1.pngFile:CDel label0.png, with 40 vertices, 600 edges, 4000 faces, and 10000 cells

Generalized 4-cubes

Generalized tesseracts have a regular construction as File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png and prismatic construction as File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, a product of four p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• Real {4,3,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png, with 16 vertices, 32 edges, 24 faces, and 8 cells
  Real {4,3,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png, with 16 vertices, 32 edges, 24 faces, and 8 cells
• 3{4}2{3}2{3}2, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png, with 81 vertices, 108 edges, 54 faces, and 12 cells
  ₃{4}₂{3}₂{3}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png, with 81 vertices, 108 edges, 54 faces, and 12 cells
• 4{4}2{3}2{3}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, with 256 vertices, 96 edges, 96 faces, and 16 cells
  ₄{4}₂{3}₂{3}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, with 256 vertices, 96 edges, 96 faces, and 16 cells
• 5{4}2{3}2{3}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, with 625 vertices, 500 edges, 150 faces, and 20 cells
  ₅{4}₂{3}₂{3}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, with 625 vertices, 500 edges, 150 faces, and 20 cells
• 6{4}2{3}2{3}2, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, with 1296 vertices, 864 edges, 216 faces, and 24 cells
  ₆{4}₂{3}₂{3}₂, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, with 1296 vertices, 864 edges, 216 faces, and 24 cells
• 7{4}2{3}2{3}2, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.png, with 2401 vertices, 1372 edges, 294 faces, and 28 cells
  ₇{4}₂{3}₂{3}₂, File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.pngFile:CDel 2c.pngFile:CDel 7node 1.png, with 2401 vertices, 1372 edges, 294 faces, and 28 cells
• 8{4}2{3}2{3}2, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.png, with 4096 vertices, 2048 edges, 384 faces, and 32 cells
  ₈{4}₂{3}₂{3}₂, File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.pngFile:CDel 2c.pngFile:CDel 8node 1.png, with 4096 vertices, 2048 edges, 384 faces, and 32 cells
• 9{4}2{3}2{3}2, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.png, with 6561 vertices, 2916 edges, 486 faces, and 36 cells
  ₉{4}₂{3}₂{3}₂, File:CDel 9node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.pngFile:CDel 2c.pngFile:CDel 9node 1.png, with 6561 vertices, 2916 edges, 486 faces, and 36 cells
• 10{4}2{3}2{3}2, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.png, with 10000 vertices, 4000 edges, 600 faces, and 40 cells
  ₁₀{4}₂{3}₂{3}₂, File:CDel 10node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png or File:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.pngFile:CDel 2c.pngFile:CDel 10node 1.png, with 10000 vertices, 4000 edges, 600 faces, and 40 cells

Enumeration of regular complex 5-polytopes

Regular complex 5-polytopes in ${\mathbb{ℂ}}^5$ or higher exist in three families, the real simplexes and the generalized hypercube, and orthoplex.

Space	Group	Order	Polytope	Vertices	Edges	Faces	Cells	4-faces	Van Oss polygon	Notes
${\mathbb{ℝ}}^5$	G(1,1,5) = [3,3,3,3]	720	α5 = {3,3,3,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	6	15 {}	20 {3}	15 {3,3}	6 {3,3,3}	none	Real 5-simplex
${\mathbb{ℝ}}^5$	G(2,1,5) =[3,3,3,4]	3840	β2 5 = β5 = {3,3,3,4} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	10	40 {}	80 {3}	80 {3,3}	32 {3,3,3}	{4}	Real 5-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, order 1920
${\mathbb{ℝ}}^5$			γ2 5 = γ5 = {4,3,3,3} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	32	80 {}	80 {4}	40 {4,3}	10 {4,3,3}	none	Real 5-cube Same as {}5 or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png, order 32
${\mathbb{ℂ}}^5$	G(p,1,5) 2[3]2[3]2[3]2[4]p	120p5	βp 5 = 2{3}2{3}2{3}2{4}p File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png	5p	10p2 {}	10p3 {3}	5p4 {3,3}	p5 {3,3,3}	2{4}p	Generalized 5-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png, order 120p4
${\mathbb{ℂ}}^5$			γp 5 = p{4}2{3}2{3}2{3}2 File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	p5	5p4 p{}	10p3 p{4}2	10p2 p{4}2{3}2	5p p{4}2{3}2{3}2	none	Generalized 5-cube Same as p{}5 or File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, order p5
${\mathbb{ℂ}}^5$	G(3,1,5) 2[3]2[3]2[3]2[4]3	29160	β3 5 = 2{3}2{3}2{3}2{4}3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png	15	90 {}	270 {3}	405 {3,3}	243 {3,3,3}	2{4}3	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png, order 9720
${\mathbb{ℂ}}^5$			γ3 5 = 3{4}2{3}2{3}2{3}2 File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	243	405 3{}	270 3{4}2	90 3{4}2{3}2	15 3{4}2{3}2{3}2	none	Same as 3{}5 or File:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.pngFile:CDel 2c.pngFile:CDel 3node 1.png, order 243
${\mathbb{ℂ}}^5$	G(4,1,5) 2[3]2[3]2[3]2[4]4	122880	β4 5 = 2{3}2{3}2{3}2{4}4 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png	20	160 {}	640 {3}	1280 {3,3}	1024 {3,3,3}	2{4}4	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png, order 30720
${\mathbb{ℂ}}^5$			γ4 5 = 4{4}2{3}2{3}2{3}2 File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	1024	1280 4{}	640 4{4}2	160 4{4}2{3}2	20 4{4}2{3}2{3}2	none	Same as 4{}5 or File:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.pngFile:CDel 2c.pngFile:CDel 4node 1.png, order 1024
${\mathbb{ℂ}}^5$	G(5,1,5) 2[3]2[3]2[3]2[4]5	375000	β5 5 = 2{3}2{3}2{3}2{5}5 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png	25	250 {}	1250 {3}	3125 {3,3}	3125 {3,3,3}	2{5}5	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png, order 75000
${\mathbb{ℂ}}^5$			γ5 5 = 5{4}2{3}2{3}2{3}2 File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	3125	3125 5{}	1250 5{5}2	250 5{5}2{3}2	25 5{4}2{3}2{3}2	none	Same as 5{}5 or File:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.pngFile:CDel 2c.pngFile:CDel 5node 1.png, order 3125
${\mathbb{ℂ}}^5$	G(6,1,5) 2[3]2[3]2[3]2[4]6	933210	β6 5 = 2{3}2{3}2{3}2{4}6 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png	30	360 {}	2160 {3}	6480 {3,3}	7776 {3,3,3}	2{4}6	Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png, order 155520
${\mathbb{ℂ}}^5$			γ6 5 = 6{4}2{3}2{3}2{3}2 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	7776	6480 6{}	2160 6{4}2	360 6{4}2{3}2	30 6{4}2{3}2{3}2	none	Same as 6{}5 or File:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.pngFile:CDel 2c.pngFile:CDel 6node 1.png, order 7776

Visualizations of regular complex 5-polytopes

Generalized 5-orthoplexes

Generalized 5-orthoplexes have a regular construction as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png and quasiregular form as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png. All elements are simplexes.

• Real {3,3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, with 10 vertices, 40 edges, 80 faces, 80 cells, and 32 4-faces
  Real {3,3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, with 10 vertices, 40 edges, 80 faces, 80 cells, and 32 4-faces
• 2{3}2{3}2{3}2{4}3, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png, with 15 vertices, 90 edges, 270 faces, 405 cells, and 243 4-faces
  ₂{3}₂{3}₂{3}₂{4}₃, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png, with 15 vertices, 90 edges, 270 faces, 405 cells, and 243 4-faces
• 2{3}2{3}2{3}2{4}4, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png, with 20 vertices, 160 edges, 640 faces, 1280 cells, and 1024 4-faces
  ₂{3}₂{3}₂{3}₂{4}₄, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png, with 20 vertices, 160 edges, 640 faces, 1280 cells, and 1024 4-faces
• 2{3}2{3}2{3}2{4}5, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png, with 25 vertices, 250 edges, 1250 faces, 3125 cells, and 3125 4-faces
  ₂{3}₂{3}₂{3}₂{4}₅, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png, with 25 vertices, 250 edges, 1250 faces, 3125 cells, and 3125 4-faces
• 2{3}2{3}2{3}2{4}6, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png, with 30 vertices, 360 edges, 2160 faces, 6480 cells, 7776 4-faces
  ₂{3}₂{3}₂{3}₂{4}₆, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png, with 30 vertices, 360 edges, 2160 faces, 6480 cells, 7776 4-faces
• 2{3}2{3}2{3}2{4}7, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png, with 35 vertices, 490 edges, 3430 faces, 12005 cells, 16807 4-faces
  ₂{3}₂{3}₂{3}₂{4}₇, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png, with 35 vertices, 490 edges, 3430 faces, 12005 cells, 16807 4-faces
• 2{3}2{3}2{3}2{4}8, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png, with 40 vertices, 640 edges, 5120 faces, 20480 cells, 32768 4-faces
  ₂{3}₂{3}₂{3}₂{4}₈, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png, with 40 vertices, 640 edges, 5120 faces, 20480 cells, 32768 4-faces
• 2{3}2{3}2{3}2{4}9, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png, with 45 vertices, 810 edges, 7290 faces, 32805 cells, 59049 4-faces
  ₂{3}₂{3}₂{3}₂{4}₉, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png, with 45 vertices, 810 edges, 7290 faces, 32805 cells, 59049 4-faces
• 2{3}2{3}2{3}2{4}10, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png, with 50 vertices, 1000 edges, 10000 faces, 50000 cells, 100000 4-faces
  ₂{3}₂{3}₂{3}₂{4}₁₀, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png, with 50 vertices, 1000 edges, 10000 faces, 50000 cells, 100000 4-faces

Generalized 5-cubes

Generalized 5-cubes have a regular construction as File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png and prismatic construction as File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, a product of five p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• Real {4,3,3,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 32 vertices, 80 edges, 80 faces, 40 cells, and 10 4-faces
  Real {4,3,3,3}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 32 vertices, 80 edges, 80 faces, 40 cells, and 10 4-faces
• 3{4}2{3}2{3}2{3}2, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 243 vertices, 405 edges, 270 faces, 90 cells, and 15 4-faces
  ₃{4}₂{3}₂{3}₂{3}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 243 vertices, 405 edges, 270 faces, 90 cells, and 15 4-faces
• 4{4}2{3}2{3}2{3}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 1024 vertices, 1280 edges, 640 faces, 160 cells, and 20 4-faces
  ₄{4}₂{3}₂{3}₂{3}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 1024 vertices, 1280 edges, 640 faces, 160 cells, and 20 4-faces
• 5{4}2{3}2{3}2{3}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 3125 vertices, 3125 edges, 1250 faces, 250 cells, and 25 4-faces
  ₅{4}₂{3}₂{3}₂{3}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 3125 vertices, 3125 edges, 1250 faces, 250 cells, and 25 4-faces
• 6{4}2{3}2{3}2{3}2, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 7776 vertices, 6480 edges, 2160 faces, 360 cells, and 30 4-faces
  ₆{4}₂{3}₂{3}₂{3}₂, File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 7776 vertices, 6480 edges, 2160 faces, 360 cells, and 30 4-faces

Enumeration of regular complex 6-polytopes

Space	Group	Order	Polytope	Vertices	Edges	Faces	Cells	4-faces	5-faces	Van Oss polygon	Notes
${\mathbb{ℝ}}^6$	G(1,1,6) = [3,3,3,3,3]	720	α6 = {3,3,3,3,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	7	21 {}	35 {3}	35 {3,3}	21 {3,3,3}	7 {3,3,3,3}	none	Real 6-simplex
${\mathbb{ℝ}}^6$	G(2,1,6) [3,3,3,4]	46080	β2 6 = β6 = {3,3,3,4} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	12	60 {}	160 {3}	240 {3,3}	192 {3,3,3}	64 {3,3,3,3}	{4}	Real 6-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png, order 23040
${\mathbb{ℝ}}^6$			γ2 6 = γ6 = {4,3,3,3} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	64	192 {}	240 {4}	160 {4,3}	60 {4,3,3}	12 {4,3,3,3}	none	Real 6-cube Same as {}6 or File:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.pngFile:CDel 2c.pngFile:CDel node 1.png, order 64
${\mathbb{ℂ}}^6$	G(p,1,6) 2[3]2[3]2[3]2[4]p	720p6	βp 6 = 2{3}2{3}2{3}2{4}p File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png	6p	15p2 {}	20p3 {3}	15p4 {3,3}	6p5 {3,3,3}	p6 {3,3,3,3}	2{4}p	Generalized 6-orthoplex Same as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png, order 720p5
${\mathbb{ℂ}}^6$			γp 6 = p{4}2{3}2{3}2{3}2 File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	p6	6p5 p{}	15p4 p{4}2	20p3 p{4}2{3}2	15p2 p{4}2{3}2{3}2	6p p{4}2{3}2{3}2{3}2	none	Generalized 6-cube Same as p{}6 or File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, order p6

Visualizations of regular complex 6-polytopes

Generalized 6-orthoplexes

Generalized 6-orthoplexes have a regular construction as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png and quasiregular form as File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png. All elements are simplexes.

• Real {3,3,3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, with 12 vertices, 60 edges, 160 faces, 240 cells, 192 4-faces, and 64 5-faces
  Real {3,3,3,3,4}, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, with 12 vertices, 60 edges, 160 faces, 240 cells, 192 4-faces, and 64 5-faces
• 2{3}2{3}2{3}2{3}2{4}3, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png, with 18 vertices, 135 edges, 540 faces, 1215 cells, 1458 4-faces, and 729 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₃, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png, with 18 vertices, 135 edges, 540 faces, 1215 cells, 1458 4-faces, and 729 5-faces
• 2{3}2{3}2{3}2{3}2{4}4, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png, with 24 vertices, 240 edges, 1280 faces, 3840 cells, 6144 4-faces, and 4096 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₄, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png, with 24 vertices, 240 edges, 1280 faces, 3840 cells, 6144 4-faces, and 4096 5-faces
• 2{3}2{3}2{3}2{3}2{4}5, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png, with 30 vertices, 375 edges, 2500 faces, 9375 cells, 18750 4-faces, and 15625 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₅, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png, with 30 vertices, 375 edges, 2500 faces, 9375 cells, 18750 4-faces, and 15625 5-faces
• 2{3}2{3}2{3}2{3}2{4}6, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png, with 36 vertices, 540 edges, 4320 faces, 19440 cells, 46656 4-faces, and 46656 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₆, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png, with 36 vertices, 540 edges, 4320 faces, 19440 cells, 46656 4-faces, and 46656 5-faces
• 2{3}2{3}2{3}2{3}2{4}7, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png, with 42 vertices, 735 edges, 6860 faces, 36015 cells, 100842 4-faces, 117649 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₇, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 7node.png, with 42 vertices, 735 edges, 6860 faces, 36015 cells, 100842 4-faces, 117649 5-faces
• 2{3}2{3}2{3}2{3}2{4}8, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png, with 48 vertices, 960 edges, 10240 faces, 61440 cells, 196608 4-faces, 262144 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₈, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 8node.png, with 48 vertices, 960 edges, 10240 faces, 61440 cells, 196608 4-faces, 262144 5-faces
• 2{3}2{3}2{3}2{3}2{4}9, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png, with 54 vertices, 1215 edges, 14580 faces, 98415 cells, 354294 4-faces, 531441 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₉, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 9node.png, with 54 vertices, 1215 edges, 14580 faces, 98415 cells, 354294 4-faces, 531441 5-faces
• 2{3}2{3}2{3}2{3}2{4}10, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png, with 60 vertices, 1500 edges, 20000 faces, 150000 cells, 600000 4-faces, 1000000 5-faces
  ₂{3}₂{3}₂{3}₂{3}₂{4}₁₀, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 10node.png, with 60 vertices, 1500 edges, 20000 faces, 150000 cells, 600000 4-faces, 1000000 5-faces

Generalized 6-cubes

Generalized 6-cubes have a regular construction as File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png and prismatic construction as File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png, a product of six p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

• Real {3,3,3,3,3,4}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 64 vertices, 192 edges, 240 faces, 160 cells, 60 4-faces, and 12 5-faces
  Real {3,3,3,3,3,4}, File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 64 vertices, 192 edges, 240 faces, 160 cells, 60 4-faces, and 12 5-faces
• 3{4}2{3}2{3}2{3}2{3}2, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 729 vertices, 1458 edges, 1215 faces, 540 cells, 135 4-faces, and 18 5-faces
  ₃{4}₂{3}₂{3}₂{3}₂{3}₂, File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 729 vertices, 1458 edges, 1215 faces, 540 cells, 135 4-faces, and 18 5-faces
• 4{4}2{3}2{3}2{3}2{3}2, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 4096 vertices, 6144 edges, 3840 faces, 1280 cells, 240 4-faces, and 24 5-faces
  ₄{4}₂{3}₂{3}₂{3}₂{3}₂, File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 4096 vertices, 6144 edges, 3840 faces, 1280 cells, 240 4-faces, and 24 5-faces
• 5{4}2{3}2{3}2{3}2{3}2, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 15625 vertices, 18750 edges, 9375 faces, 2500 cells, 375 4-faces, and 30 5-faces
  ₅{4}₂{3}₂{3}₂{3}₂{3}₂, File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with 15625 vertices, 18750 edges, 9375 faces, 2500 cells, 375 4-faces, and 30 5-faces

Enumeration of regular complex apeirotopes

Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs. For each dimension there are 12 apeirotopes symbolized as δ^p,r
_n+1 exists in any dimensions ${\mathbb{ℂ}}^n$, or ${\mathbb{ℝ}}^n$ if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2. Each has proportional element counts given as:

k-faces = $(\genfrac{}{}{0ex}{}{n}{k})p^{n-k}{r}^k$, where $(\genfrac{}{}{0ex}{}{n}{m})=\frac{n!}{m!(n-m)!}$ and n! denotes the factorial of n.

Regular complex 1-polytopes

The only regular complex 1-polytope is _∞{}, or File:CDel infinnode 1.png. Its real representation is an apeirogon, {∞}, or File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png.

Regular complex apeirogons

File:Rank2 infinite shephard subgroups.png

Some subgroups of the apeirogonal Shephard groups

File:Complex apeirogon chart2.png

11 complex apeirogons _p{q}_r with edge interiors colored in light blue, and edges around one vertex are colored individually. Vertices are shown as small black squares. Edges are seen as p-sided regular polygons and vertex figures are r-gonal.

File:7 quasiregular complex apeirogons.png

A quasiregular apeirogon File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode 1.png is a mixture of two regular apeirogons File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png and File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode 1.png, seen here with blue and pink edges. File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.png has only one color of edges because q is odd, making it a double covering.

Rank 2 complex apeirogons have symmetry _p[q]_r, where 1/p + 2/q + 1/r = 1. Coxeter expresses them as δ^p,r
₂ where q is constrained to satisfy q = 2/(1 – (p + r)/pr). There are 8 solutions:

2[∞]2	3[12]2	4[8]2	6[6]2	3[6]3	6[4]3	4[4]4	6[3]6
File:CDel node.pngFile:CDel infin.pngFile:CDel node.png	File:CDel 3node.pngFile:CDel 12.pngFile:CDel node.png	File:CDel 4node.pngFile:CDel 8.pngFile:CDel node.png	File:CDel 6node.pngFile:CDel 6.pngFile:CDel node.png	File:CDel 3node.pngFile:CDel 6.pngFile:CDel 3node.png	File:CDel 6node.pngFile:CDel 4.pngFile:CDel 3node.png	File:CDel 4node.pngFile:CDel 4.pngFile:CDel 4node.png	File:CDel 6node.pngFile:CDel 3.pngFile:CDel 6node.png

There are two excluded solutions odd q and unequal p and r: ₁₀[5]₂ and ₁₂[3]₄, or File:CDel 10node.pngFile:CDel 5.pngFile:CDel node.png and File:CDel 12node.pngFile:CDel 3.pngFile:CDel 4node.png. A regular complex apeirogon _p{q}_r has p-edges and r-gonal vertex figures. The dual apeirogon of _p{q}_r is _r{q}_p. An apeirogon of the form _p{q}_p is self-dual. Groups of the form _p[2q]₂ have a half symmetry _p[q]_p, so a regular apeirogon File:CDel pnode 1.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png is the same as quasiregular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel pnode 1.png. Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form ₂{q}_r have a vertex arrangement as {q/2,p}. The form _p{q}₂ have vertex arrangement as r{p,q/2}. Apeirogons of the form _p{4}_r have vertex arrangements {p,r}. Including affine nodes, and ${\mathbb{ℂ}}^2$, there are 3 more infinite solutions: _∞[2]_∞, _∞[4]₂, _∞[3]₃, and File:CDel infinnode 1.pngFile:CDel 2.pngFile:CDel infinnode 1.png, File:CDel infinnode 1.pngFile:CDel 4.pngFile:CDel node.png, and File:CDel infinnode 1.pngFile:CDel 3.pngFile:CDel 3node.png. The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in ${\mathbb{ℂ}}^1$.

Rank 2

Space	Group	Apeirogon		Edge	${\mathbb{ℝ}}^2$ rep.	Picture	Notes
${\mathbb{ℝ}}^1$	2[∞]2 = [∞]	δ2,2 2 = {∞}	File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png	{}		File:Regular apeirogon.svg	Real apeirogon Same as File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.png
${\mathbb{ℂ}}^2$ / ${\mathbb{ℂ}}^1$	∞[4]2	∞{4}2	File:CDel infinnode 1.pngFile:CDel 4.pngFile:CDel node.png	∞{}	{4,4}	File:Complex polygon i-4-2.png	Same as File:CDel infinnode 1.pngFile:CDel 2.pngFile:CDel infinnode 1.png File:Truncated complex polygon i-2-i.png
${\mathbb{ℂ}}^1$	∞[3]3	∞{3}3	File:CDel infinnode 1.pngFile:CDel 3.pngFile:CDel 3node.png	∞{}	{3,6}	File:Complex apeirogon 2-6-6.png	Same as File:CDel infinnode 1.pngFile:CDel split1.pngFile:CDel branch 11.pngFile:CDel label-ii.png File:Truncated complex polygon i-3-i-3-i-3-.png
${\mathbb{ℂ}}^1$	p[q]r	δp,r 2 = p{q}r	File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png	p{}
${\mathbb{ℂ}}^1$	3[12]2	δ3,2 2 = 3{12}2	File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.png	3{}	r{3,6}	File:Complex apeirogon 3-12-2.png	Same as File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.png File:Truncated complex polygon 3-6-3.png
		δ2,3 2 = 2{12}3	File:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.png	{}	{6,3}	File:Complex apeirogon 2-12-3.png
${\mathbb{ℂ}}^1$	3[6]3	δ3,3 2 = 3{6}3	File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.png	3{}	{3,6}	File:Complex apeirogon 3-6-3.png	Same as File:CDel node h.pngFile:CDel 12.pngFile:CDel 3node.png
${\mathbb{ℂ}}^1$	4[8]2	δ4,2 2 = 4{8}2	File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.png	4{}	{4,4}	File:Complex apeirogon 4-8-2.png	Same as File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.png File:Truncated complex polygon 4-4-4.png
		δ2,4 2 = 2{8}4	File:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.png	{}	{4,4}	File:Complex apeirogon 2-8-4.svg
${\mathbb{ℂ}}^1$	4[4]4	δ4,4 2 = 4{4}4	File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.png	4{}	{4,4}	File:Complex apeirogon 4-4-4.png	Same as File:CDel node h.pngFile:CDel 8.pngFile:CDel 4node.png
${\mathbb{ℂ}}^1$	6[6]2	δ6,2 2 = 6{6}2	File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.png	6{}	r{3,6}	File:Complex apeirogon 6-6-2.png	Same as File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.png
		δ2,6 2 = 2{6}6	File:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.png	{}	{3,6}	File:Complex apeirogon 2-6-6.png
${\mathbb{ℂ}}^1$	6[4]3	δ6,3 2 = 6{4}3	File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.png	6{}	{6,3}	File:Complex apeirogon 6-4-3.png
		δ3,6 2 = 3{4}6	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 6node.png	3{}	{3,6}	File:Complex apeirogon 3-4-6.png
${\mathbb{ℂ}}^1$	6[3]6	δ6,6 2 = 6{3}6	File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.png	6{}	{3,6}	File:Complex apeirogon 6-3-6.png	Same as File:CDel node h.pngFile:CDel 6.pngFile:CDel 6node.png

Regular complex apeirohedra

There are 22 regular complex apeirohedra, of the form _p{a}_q{b}_r. 8 are self-dual (p=r and a=b), while 14 exist as dual polytope pairs. Three are entirely real (p=q=r=2). Coxeter symbolizes 12 of them as δ^p,r
₃ or _p{4}₂{4}_r is the regular form of the product apeirotope δ^p,r
₂ × δ^p,r
₂ or _p{q}_r × _p{q}_r, where q is determined from p and r. File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel qnode.png is the same as File:CDel pnode 1.pngFile:CDel 3split1-44.pngFile:CDel branch.pngFile:CDel labelq.png, as well as File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png, for p,r=2,3,4,6. Also File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel pnode.pngFile:CDel 4.pngFile:CDel node.png = File:CDel pnode.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel pnode.png.

Rank 3

Space	Group	Apeirohedron		Vertex	Edge		Face		van Oss apeirogon	Notes
${\mathbb{ℂ}}^3$	2[3]2[4]∞	∞{4}2{3}2	File:CDel infinnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png			∞{}		∞{4}2		Same as ∞{}×∞{}×∞{} or File:CDel infinnode 1.pngFile:CDel 2c.pngFile:CDel infinnode 1.pngFile:CDel 2c.pngFile:CDel infinnode 1.png Real representation {4,3,4}
${\mathbb{ℂ}}^2$	p[4]2[4]r	p{4}2{4}r	File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png	p2	2pr	p{}	r2	p{4}2	2{q}r	Same as File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png, p,r=2,3,4,6
${\mathbb{ℝ}}^2$	[4,4]	δ2,2 3 = {4,4}	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	4	8	{}	4	{4}	{∞}	Real square tiling Same as File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.png or File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
${\mathbb{ℂ}}^2$	3[4]2[4]2   3[4]2[4]3 4[4]2[4]2   4[4]2[4]4 6[4]2[4]2   6[4]2[4]3   6[4]2[4]6	3{4}2{4}2 2{4}2{4}3 3{4}2{4}3 4{4}2{4}2 2{4}2{4}4 4{4}2{4}4 6{4}2{4}2 2{4}2{4}6 6{4}2{4}3 3{4}2{4}6 6{4}2{4}6	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png	9 4 9 16 4 16 36 4 36 9 36	12 12 18 16 16 32 24 24 36 36 72	3{} {} 3{} 4{} {} 4{} 6{} {} 6{} 3{} 6{}	4 9 9 4 16 16 4 36 9 36 36	3{4}2 {4} 3{4}2 4{4}2 {4} 4{4}2 6{4}2 {4} 6{4}2 3{4}2 6{4}2	p{q}r	Same as File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.png or File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node 1.png Same as File:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.png Same as File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.png Same as File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.png or File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node 1.png Same as File:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.png Same as File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.png Same as File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.png or File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node 1.png Same as File:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.png Same as File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.png Same as File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 6node.png Same as File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.png

Space	Group	Apeirohedron		Vertex	Edge		Face		van Oss apeirogon	Notes
${\mathbb{ℂ}}^2$	2[4]r[4]2	2{4}r{4}2	File:CDel node 1.pngFile:CDel 4.pngFile:CDel rnode.pngFile:CDel 4.pngFile:CDel node.png	2		{}	2	p{4}2'	2{4}r	Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png and File:CDel rnode.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel rnode.png, r=2,3,4,6
${\mathbb{ℝ}}^2$	[4,4]	{4,4}	File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png	2	4	{}	2	{4}	{∞}	Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png and File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png
${\mathbb{ℂ}}^2$	2[4]3[4]2 2[4]4[4]2 2[4]6[4]2	2{4}3{4}2 2{4}4{4}2 2{4}6{4}2	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 4.pngFile:CDel node.png File:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.pngFile:CDel 4.pngFile:CDel node.png	2	9 16 36	{}	2	2{4}3 2{4}4 2{4}6	2{q}r	Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png and File:CDel 3node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png and File:CDel 4node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png Same as File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png and File:CDel 6node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.png

Space	Group	Apeirohedron		Vertex	Edge		Face		van Oss apeirogon	Notes
${\mathbb{ℝ}}^2$	2[6]2[3]2 = [6,3]	{3,6}	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png	1	3	{}	2	{3}	{∞}	Real triangular tiling
		{6,3}	File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	2	3	{}	1	{6}	none	Real hexagonal tiling
${\mathbb{ℂ}}^2$	3[4]3[3]3	3{3}3{4}3	File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel 3node.png	1	8	3{}	3	3{3}3	3{4}6	Same as File:CDel 3node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label-33.png
		3{4}3{3}3	File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	3	8	3{}	1	3{4}3	3{12}2
${\mathbb{ℂ}}^2$	4[3]4[3]4	4{3}4{3}4	File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.pngFile:CDel 3.pngFile:CDel 4node.png	1	6	4{}	1	4{3}4	4{4}4	Self-dual, same as File:CDel node h.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 3.pngFile:CDel 4node.png
${\mathbb{ℂ}}^2$	4[3]4[4]2	4{3}4{4}2	File:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.pngFile:CDel 4.pngFile:CDel node.png	1	12	4{}	3	4{3}4	2{8}4	Same as File:CDel 4node.pngFile:CDel 3.pngFile:CDel 4node 1.pngFile:CDel 3.pngFile:CDel 4node.png
		2{4}4{3}4	File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 3.pngFile:CDel 4node.png	3	12	{}	1	2{4}4	4{4}4

Regular complex 3-apeirotopes

There are 16 regular complex apeirotopes in ${\mathbb{ℂ}}^3$. Coxeter expresses 12 of them by δ^p,r
₃ where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png = File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png. The first case is the ${\mathbb{ℝ}}^3$ cubic honeycomb.

Rank 4

Space	Group	3-apeirotope	Vertex	Edge	Face	Cell	van Oss apeirogon	Notes
${\mathbb{ℂ}}^3$	p[4]2[3]2[4]r	δp,r 3 = p{4}2{3}2{4}r File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png		p{}	p{4}2	p{4}2{3}2	p{q}r	Same as File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png
${\mathbb{ℝ}}^3$	2[4]2[3]2[4]2 =[4,3,4]	δ2,2 3 = 2{4}2{3}2{4}2 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		{}	{4}	{4,3}		Cubic honeycomb Same as File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png or File:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node 1.png or File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node 1.png
${\mathbb{ℂ}}^3$	3[4]2[3]2[4]2	δ3,2 3 = 3{4}2{3}2{4}2 File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		3{}	3{4}2	3{4}2{3}2		Same as File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.png or File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.png or File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node 1.png
		δ2,3 3 = 2{4}2{3}2{4}3 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png		{}	{4}	{4,3}		Same as File:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 12.pngFile:CDel 3node.png
${\mathbb{ℂ}}^3$	3[4]2[3]2[4]3	δ3,3 3 = 3{4}2{3}2{4}3 File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png		3{}	3{4}2	3{4}2{3}2		Same as File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.png
${\mathbb{ℂ}}^3$	4[4]2[3]2[4]2	δ4,2 3 = 4{4}2{3}2{4}2 File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		4{}	4{4}2	4{4}2{3}2		Same as File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.png or File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.png or File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node 1.png
		δ2,4 3 = 2{4}2{3}2{4}4 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png		{}	{4}	{4,3}		Same as File:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 8.pngFile:CDel 4node.png
${\mathbb{ℂ}}^3$	4[4]2[3]2[4]4	δ4,4 3 = 4{4}2{3}2{4}4 File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png		4{}	4{4}2	4{4}2{3}2		Same as File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.pngFile:CDel 2.pngFile:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.png
${\mathbb{ℂ}}^3$	6[4]2[3]2[4]2	δ6,2 3 = 6{4}2{3}2{4}2 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		6{}	6{4}2	6{4}2{3}2		Same as File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.png or File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.png or File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node 1.png
		δ2,6 3 = 2{4}2{3}2{4}6 File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png		{}	{4}	{4,3}		Same as File:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel 6.pngFile:CDel 6node.png
${\mathbb{ℂ}}^3$	6[4]2[3]2[4]3	δ6,3 3 = 6{4}2{3}2{4}3 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png		6{}	6{4}2	6{4}2{3}2		Same as File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.png
		δ3,6 3 = 3{4}2{3}2{4}6 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png		3{}	3{4}2	3{4}2{3}2		Same as File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.png
${\mathbb{ℂ}}^3$	6[4]2[3]2[4]6	δ6,6 3 = 6{4}2{3}2{4}6 File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png		6{}	6{4}2	6{4}2{3}2		Same as File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.pngFile:CDel 2.pngFile:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.png

Rank 4, exceptional cases

Space	Group	3-apeirotope	Vertex	Edge	Face	Cell	van Oss apeirogon	Notes
${\mathbb{ℂ}}^3$	2[4]3[3]3[3]3	3{3}3{3}3{4}2 File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png	1	24 3{}	27 3{3}3	2 3{3}3{3}3	3{4}6	Same as File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel label-33.png
		2{4}3{3}3{3}3 File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	2	27 {}	24 2{4}3	1 2{4}3{3}3	2{12}3
${\mathbb{ℂ}}^3$	2[3]2[4]3[3]3	2{3}2{4}3{3}3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	1	27 {}	72 2{3}2	8 2{3}2{4}3	2{6}6
		3{3}3{4}2{3}2 File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	8	72 3{}	27 3{3}3	1 3{3}3{4}2	3{6}3	Same as File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel split1.pngFile:CDel nodes.pngFile:CDel label-33.png or File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png

Regular complex 4-apeirotopes

There are 15 regular complex apeirotopes in ${\mathbb{ℂ}}^4$. Coxeter expresses 12 of them by δ^p,r
₄ where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png = File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png. The first case is the ${\mathbb{ℝ}}^4$ tesseractic honeycomb. The 16-cell honeycomb and 24-cell honeycomb are real solutions. The last solution is generated has Witting polytope elements.

Rank 5

Space	Group	4-apeirotope	Vertex	Edge	Face	Cell	4-face	van Oss apeirogon	Notes
${\mathbb{ℂ}}^4$	p[4]2[3]2[3]2[4]r	δp,r 4 = p{4}2{3}2{3}2{4}r File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png		p{}	p{4}2	p{4}2{3}2	p{4}2{3}2{3}2	p{q}r	Same as File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png
${\mathbb{ℝ}}^4$	2[4]2[3]2[3]2[4]2	δ2,2 4 = {4,3,3,3} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		{}	{4}	{4,3}	{4,3,3}	{∞}	Tesseractic honeycomb Same as File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png
${\mathbb{ℝ}}^4$	2[3]2[4]2[3]2[3]2 =[3,4,3,3]	{3,3,4,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	1	12 {}	32 {3}	24 {3,3}	3 {3,3,4}		Real 16-cell honeycomb Same as File:CDel nodes 10ru.pngFile:CDel split2.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png
		{3,4,3,3} File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	3	24 {}	32 {3}	12 {3,4}	1 {3,4,3}		Real 24-cell honeycomb Same as File:CDel nodes.pngFile:CDel split2.pngFile:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png or File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png
${\mathbb{ℂ}}^4$	3[3]3[3]3[3]3[3]3	3{3}3{3}3{3}3{3}3 File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png	1	80 3{}	270 3{3}3	80 3{3}3{3}3	1 3{3}3{3}3{3}3	3{4}6	${\mathbb{ℝ}}^8$ representation 521

Regular complex 5-apeirotopes and higher

There are only 12 regular complex apeirotopes in ${\mathbb{ℂ}}^5$ or higher, expressed δ^p,r
_n where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed a product of n apeirogons: File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png ... File:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png = File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png ... File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png. The first case is the real ${\mathbb{ℝ}}^n$ hypercube honeycomb.

Rank 6

Space	Group	5-apeirotopes	Vertices	Edge	Face	Cell	4-face	5-face	van Oss apeirogon	Notes
${\mathbb{ℂ}}^5$	p[4]2[3]2[3]2[3]2[4]r	δp,r 5 = p{4}2{3}2{3}2{3}2{4}r File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel rnode.png		p{}	p{4}2	p{4}2{3}2	p{4}2{3}2{3}2	p{4}2{3}2{3}2{3}2	p{q}r	Same as File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.pngFile:CDel 2.pngFile:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png
${\mathbb{ℝ}}^5$	2[4]2[3]2[3]2[3]2[4]2 =[4,3,3,3,4]	δ2,2 5 = {4,3,3,3,4} File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png		{}	{4}	{4,3}	{4,3,3}	{4,3,3,3}	{∞}	5-cubic honeycomb Same as File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel node 1.pngFile:CDel infin.pngFile:CDel node.png

van Oss polygon

File:Van Oss square hole in octahedron.png

A red square van Oss polygon in the plane of an edge and center of a regular octahedron.

A van Oss polygon is a regular polygon in the plane (real plane ${\mathbb{ℝ}}^2$, or unitary plane ${\mathbb{ℂ}}^2$) in which both an edge and the centroid of a regular polytope lie, and formed of elements of the polytope. Not all regular polytopes have Van Oss polygons. For example, the van Oss polygons of a real octahedron are the three squares whose planes pass through its center. In contrast a cube does not have a van Oss polygon because the edge-to-center plane cuts diagonally across two square faces and the two edges of the cube which lie in the plane do not form a polygon. Infinite honeycombs also have van Oss apeirogons. For example, the real square tiling and triangular tiling have apeirogons {∞} van Oss apeirogons. If it exists, the van Oss polygon of regular complex polytope of the form _p{q}_r{s}_t... has p-edges.

Non-regular complex polytopes

Product complex polytopes

Example product complex polytope

File:Complex polygon 2x5 stereographic3.png Complex product polygon File:CDel node 1.pngFile:CDel 2.pngFile:CDel 5node 1.png or {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges, with its real representation as a 3-dimensional pentagonal prism.

File:Dual complex polygon 2x5 perspective.png The dual polygon,{}+5{} has 7 vertices centered on the edges of the original, connected by 10 edges. Its real representation is a pentagonal bipyramid.

Some complex polytopes can be represented as Cartesian products. These product polytopes are not strictly regular since they'll have more than one facet type, but some can represent lower symmetry of regular forms if all the orthogonal polytopes are identical. For example, the product _p{}×_p{} or File:CDel pnode 1.pngFile:CDel 2.pngFile:CDel pnode 1.png of two 1-dimensional polytopes is the same as the regular _p{4}₂ or File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.png. More general products, like _p{}×_q{} have real representations as the 4-dimensional p-q duoprisms. The dual of a product polytope can be written as a sum _p{}+_q{} and have real representations as the 4-dimensional p-q duopyramid. The _p{}+_p{} can have its symmetry doubled as a regular complex polytope ₂{4}_p or File:CDel node 1.pngFile:CDel 4.pngFile:CDel pnode.png. Similarly, a ${\mathbb{ℂ}}^3$ complex polyhedron can be constructed as a triple product: _p{}×_p{}×_p{} or File:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.pngFile:CDel 2c.pngFile:CDel pnode 1.png is the same as the regular generalized cube, _p{4}₂{3}₂ or File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, as well as product _p{4}₂×_p{} or File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 2.pngFile:CDel pnode 1.png.

Quasiregular polygons

A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode 1.png contains alternate edges of the regular polygons File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png and File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode 1.png. The quasiregular polygon has p vertices on the p-edges of the regular form.

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
Regular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png	File:2-generalized-2-cube.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png 4 2-edges	File:3-generalized-2-cube skew.svg File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png 9 3-edges	File:4-generalized-2-cube.svg File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.png 16 4-edges	File:5-generalized-2-cube skew.svg File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.png 25 5-edges	File:6-generalized-2-cube.svg File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.png 36 6-edges	File:7-generalized-2-cube skew.svg File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node.png 49 8-edges	File:8-generalized-2-cube.svg File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node.png 64 8-edges	File:Complex polygon 3-3-3.svg File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png	File:Complex polygon 3-4-3.png File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png
Quasiregular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode 1.png	File:Truncated 2-generalized-square.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png = File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.png 4+4 2-edges	File:Truncated 3-generalized-square skew.svg File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node 1.png 6 2-edges 9 3-edges	File:Truncated 4-generalized-square.svg File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node 1.png 8 2-edges 16 4-edges	File:Truncated 5-generalized-square skew.svg File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node 1.png 10 2-edges 25 5-edges	File:Truncated 6-generalized-square.svg File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node 1.png 12 2-edges 36 6-edges	File:Truncated 7-generalized-square skew.svg File:CDel 7node 1.pngFile:CDel 4.pngFile:CDel node 1.png 14 2-edges 49 7-edges	File:Truncated 8-generalized-square.svg File:CDel 8node 1.pngFile:CDel 4.pngFile:CDel node 1.png 16 2-edges 64 8-edges	File:Complex polygon 3-6-2.png File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node 1.png = File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel node.png	File:Complex polygon 3-8-2.png File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png = File:CDel 3node 1.pngFile:CDel 8.pngFile:CDel node.png
Regular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png	File:2-generalized-2-orthoplex.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png 4 2-edges	File:3-generalized-2-orthoplex skew.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png 6 2-edges	File:3-generalized-2-orthoplex.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png 8 2-edges	File:5-generalized-2-orthoplex skew.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 5node.png 10 2-edges	File:6-generalized-2-orthoplex.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 6node.png 12 2-edges	File:7-generalized-2-orthoplex skew.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 7node.png 14 2-edges	File:8-generalized-2-orthoplex.svg File:CDel node 1.pngFile:CDel 4.pngFile:CDel 8node.png 16 2-edges	File:Complex polygon 3-3-3.svg File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png	File:Complex polygon 3-4-3.png File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel 3node.png

Quasiregular apeirogons

There are 7 quasiregular complex apeirogons which alternate edges of a regular apeirogon and its regular dual. The vertex arrangements of these apeirogon have real representations with the regular and uniform tilings of the Euclidean plane. The last column for the 6{3}6 apeirogon is not only self-dual, but the dual coincides with itself with overlapping hexagonal edges, thus their quasiregular form also has overlapping hexagonal edges, so it can't be drawn with two alternating colors like the others. The symmetry of the self-dual families can be doubled, so creating an identical geometry as the regular forms: File:CDel pnode 1.pngFile:CDel q.pngFile:CDel pnode 1.png = File:CDel pnode 1.pngFile:CDel 2x.pngFile:CDel q.pngFile:CDel node.png

p[q]r	4[8]2	4[4]4	6[6]2	6[4]3	3[12]2	3[6]3	6[3]6
Regular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode.png or p{q}r	File:Complex apeirogon 4-8-2.png File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.png	File:Complex apeirogon 4-4-4.png File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node.png	File:Complex apeirogon 6-6-2.png File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.png	File:Complex apeirogon 6-4-3.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node.png	File:Complex apeirogon 3-12-2.png File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.png	File:Complex apeirogon 3-6-3.png File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node.png	File:Complex apeirogon 6-3-6.png File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node.png
Quasiregular File:CDel pnode 1.pngFile:CDel q.pngFile:CDel rnode 1.png	File:Truncated complex polygon 4-8-2.png File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node 1.png	File:Truncated complex polygon 4-4-4.png File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel 4node 1.png = File:CDel 4node 1.pngFile:CDel 8.pngFile:CDel node.png	File:Truncated complex polygon 6-6-2.png File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node 1.png	File:Truncated complex polygon 6-4-3.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png	File:Truncated complex polygon 3-12-2.png File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node 1.png	File:Truncated complex polygon 3-6-3.png File:CDel 3node 1.pngFile:CDel 6.pngFile:CDel 3node 1.png = File:CDel 3node 1.pngFile:CDel 12.pngFile:CDel node.png	File:Truncated complex polygon 6-3-6.png File:CDel 6node 1.pngFile:CDel 3.pngFile:CDel 6node 1.png = File:CDel 6node 1.pngFile:CDel 6.pngFile:CDel node.png
Regular dual File:CDel pnode.pngFile:CDel q.pngFile:CDel rnode 1.png or r{q}p	File:Complex apeirogon 2-8-4.svg File:CDel 4node.pngFile:CDel 8.pngFile:CDel node 1.png	File:Complex apeirogon 4-4-4b.png File:CDel 4node.pngFile:CDel 4.pngFile:CDel 4node 1.png	File:Complex apeirogon 2-6-6.png File:CDel 6node.pngFile:CDel 6.pngFile:CDel node 1.png	File:Complex apeirogon 3-4-6.png File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel 3node 1.png	File:Complex apeirogon 2-12-3.png File:CDel 3node.pngFile:CDel 12.pngFile:CDel node 1.png	File:Complex apeirogon 3-6-3b.png File:CDel 3node.pngFile:CDel 6.pngFile:CDel 3node 1.png	File:Complex apeirogon 6-3-6b.png File:CDel 6node.pngFile:CDel 3.pngFile:CDel 6node 1.png

Quasiregular polyhedra

File:3-generalized-octahedron truncation sequence.gif

Example truncation of 3-generalized octahedron, ₂{3}₂{4}₃, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png, to its rectified limit, showing outlined-green triangles faces at the start, and blue ₂{4}₃, File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png, vertex figures expanding as new faces.

Like real polytopes, a complex quasiregular polyhedron can be constructed as a rectification (a complete truncation) of a regular polyhedron. Vertices are created mid-edge of the regular polyhedron and faces of the regular polyhedron and its dual are positioned alternating across common edges. For example, a p-generalized cube, File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, has p³ vertices, 3p² edges, and 3p p-generalized square faces, while the p-generalized octahedron, File:CDel pnode.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, has 3p vertices, 3p² edges and p³ triangular faces. The middle quasiregular form p-generalized cuboctahedron, File:CDel pnode.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, has 3p² vertices, 3p³ edges, and 3p+p³ faces. Also the rectification of the Hessian polyhedron File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, is File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png, a quasiregular form sharing the geometry of the regular complex polyhedron File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png.

Quasiregular examples

Generalized cube/octahedra						Hessian polyhedron
	p=2 (real)	p=3	p=4	p=5	p=6
Generalized cubes File:CDel pnode 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png (regular)	File:2-generalized-3-cube.svg Cube File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 8 vertices, 12 2-edges, and 6 faces.	File:3-generalized-3-cube redblueface.svg File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 27 vertices, 27 3-edges, and 9 faces, with one File:CDel 3node 1.pngFile:CDel 4.pngFile:CDel node.png face blue and red	File:4-generalized-3-cube.svg File:CDel 4node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 64 vertices, 48 4-edges, and 12 faces.	File:5-generalized-3-cube.svg File:CDel 5node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 125 vertices, 75 5-edges, and 15 faces.	File:6-generalized-3-cube.svg File:CDel 6node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, 216 vertices, 108 6-edges, and 18 faces.	File:Complex polyhedron 3-3-3-3-3.png File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.png, 27 vertices, 72 6-edges, and 27 faces.
Generalized cuboctahedra File:CDel pnode.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png (quasiregular)	File:Rectified 2-generalized-3-cube.svg Cuboctahedron File:CDel node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 12 vertices, 24 2-edges, and 6+8 faces.	File:Rectified 3-generalized-3-cube blueface.svg File:CDel 3node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 27 vertices, 81 2-edges, and 9+27 faces, with one File:CDel node 1.pngFile:CDel 4.pngFile:CDel 3node.png face blue	File:Rectified 4-generalized-3-cube blueface.svg File:CDel 4node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 48 vertices, 192 2-edges, and 12+64 faces, with one File:CDel node 1.pngFile:CDel 4.pngFile:CDel 4node.png face blue	File:Rectified 5-generalized-3-cube.svg File:CDel 5node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 75 vertices, 375 2-edges, and 15+125 faces.	File:Rectified 6-generalized-3-cube.svg File:CDel 6node.pngFile:CDel 4.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png, 108 vertices, 648 2-edges, and 18+216 faces.	File:Complex polyhedron 3-3-3-4-2.png File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.png = File:CDel 3node 1.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 4.pngFile:CDel node.png, 72 vertices, 216 3-edges, and 54 faces.
Generalized octahedra File:CDel pnode.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png (regular)	File:2-generalized-3-orthoplex.svg Octahedron File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 6 vertices, 12 2-edges, and 8 {3} faces.	File:3-generalized-3-orthoplex.svg File:CDel 3node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 9 vertices, 27 2-edges, and 27 {3} faces.	File:4-generalized-3-orthoplex.svg File:CDel 4node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 12 vertices, 48 2-edges, and 64 {3} faces.	File:5-generalized-3-orthoplex.svg File:CDel 5node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 15 vertices, 75 2-edges, and 125 {3} faces.	File:6-generalized-3-orthoplex.svg File:CDel 6node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.png, 18 vertices, 108 2-edges, and 216 {3} faces.	File:Complex polyhedron 3-3-3-3-3b.png File:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node.pngFile:CDel 3.pngFile:CDel 3node 1.png, 27 vertices, 72 6-edges, and 27 faces.

Other complex polytopes with unitary reflections of period two

Other nonregular complex polytopes can be constructed within unitary reflection groups that don't make linear Coxeter graphs. In Coxeter diagrams with loops Coxeter marks a special period interior, like File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png or symbol (1₁ 1 1)³, and group [1 1 1]³. These complex polytopes have not been systematically explored beyond a few cases. The group File:CDel node.pngFile:CDel psplit1.pngFile:CDel branch.png is defined by 3 unitary reflections, R₁, R₂, R₃, all order 2: R₁² = R₁² = R₃² = (R₁R₂)³ = (R₂R₃)³ = (R₃R₁)³ = (R₁R₂R₃R₁)^p = 1. The period p can be seen as a double rotation in real ${\mathbb{ℝ}}^4$. As with all Wythoff constructions, polytopes generated by reflections, the number of vertices of a single-ringed Coxeter diagram polytope is equal to the order of the group divided by the order of the subgroup where the ringed node is removed. For example, a real cube has Coxeter diagram File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png, with octahedral symmetry File:CDel node.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png order 48, and subgroup dihedral symmetry File:CDel node.pngFile:CDel 3.pngFile:CDel node.png order 6, so the number of vertices of a cube is 48/6=8. Facets are constructed by removing one node furthest from the ringed node, for example File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.png for the cube. Vertex figures are generated by removing a ringed node and ringing one or more connected nodes, and File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.png for the cube. Coxeter represents these groups by the following symbols. Some groups have the same order, but a different structure, defining the same vertex arrangement in complex polytopes, but different edges and higher elements, like File:CDel node.pngFile:CDel psplit1.pngFile:CDel branch.png and File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png with p≠3.

Groups generated by unitary reflections

Coxeter diagram	Order	Symbol or Position in Table VII of Shephard and Todd (1954)
File:CDel branch.pngFile:CDel labelp.png, (File:CDel node.pngFile:CDel psplit1.pngFile:CDel branch.png and File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png), File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png, File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png ...	pn − 1 n!, p ≥ 3	G(p, p, n), [p], [1 1 1]p, [1 1 (n−2)p]3
File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.png, File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.png	72·6!, 108·9!	Nos. 33, 34, [1 2 2]3, [1 2 3]3
File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label4.png, (File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label5.png and File:CDel node.pngFile:CDel 5split1.pngFile:CDel branch.pngFile:CDel label4.png), (File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.png and File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch.png)	14·4!, 3·6!, 64·5!	Nos. 24, 27, 29

Coxeter calls some of these complex polyhedra almost regular because they have regular facets and vertex figures. The first is a lower symmetry form of the generalized cross-polytope in ${\mathbb{ℂ}}^3$. The second is a fractional generalized cube, reducing p-edges into single vertices leaving ordinary 2-edges. Three of them are related to the finite regular skew polyhedron in ${\mathbb{ℝ}}^4$.

Some almost regular complex polyhedra

Space	Group	Order	Coxeter symbols	Vertices	Edges	Faces	Vertex figure	Notes
${\mathbb{ℂ}}^3$	[1 1 1p]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png p=2,3,4...	6p2	(1 1 11p)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	3p	3p2	{3}	{2p}	Shephard symbol (1 1; 11)p same as βp 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png
			(11 1 1p)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10l.pngFile:CDel labelp.png	p2		{3}	{6}	Shephard symbol (11 1; 1)p 1/p γp 3
${\mathbb{ℝ}}^3$	[1 1 12]3 File:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	24	(1 1 112)3 File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png	6	12	8 {3}	{4}	Same as β2 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png = real octahedron
			(11 1 12)3 File:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png	4	6	4 {3}	{3}	1/2 γ2 3 = File:CDel node h.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png = α3 = real tetrahedron
${\mathbb{ℂ}}^3$	[1 1 1]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png	54	(1 1 11)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png	9	27	{3}	{6}	Shephard symbol (1 1; 11)3 same as β3 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png
			(11 1 1)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10l.png	9	27	{3}	{6}	Shephard symbol (11 1; 1)3 1/3 γ3 3 = β3 3
${\mathbb{ℂ}}^3$	[1 1 14]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png	96	(1 1 114)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png	12	48	{3}	{8}	Shephard symbol (1 1; 11)4 same as β4 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png
			(11 1 14)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10l.pngFile:CDel label4.png	16		{3}	{6}	Shephard symbol (11 1; 1)4 1/4 γ4 3
${\mathbb{ℂ}}^3$	[1 1 15]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png	150	(1 1 115)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label5.png	15	75	{3}	{10}	Shephard symbol (1 1; 11)5 same as β5 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 5node.png
			(11 1 15)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10l.pngFile:CDel label5.png	25		{3}	{6}	Shephard symbol (11 1; 1)5 1/5 γ5 3
${\mathbb{ℂ}}^3$	[1 1 16]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png	216	(1 1 116)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label6.png	18	216	{3}	{12}	Shephard symbol (1 1; 11)6 same as β6 3 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 6node.png
			(11 1 16)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10l.pngFile:CDel label6.png	36		{3}	{6}	Shephard symbol (11 1; 1)6 1/6 γ6 3
${\mathbb{ℂ}}^3$	[1 1 14]4 File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label4.png	336	(1 1 114)4 File:CDel node 1.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label4.png	42	168	112 {3}	{8}	${\mathbb{ℝ}}^4$ representation {3,8|,4} = {3,8}8
			(11 1 14)4 File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch 10l.pngFile:CDel label4.png	56		{3}	{6}
${\mathbb{ℂ}}^3$	[1 1 15]4 File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label5.png	2160	(1 1 115)4 File:CDel node 1.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label5.png	216	1080	720 {3}	{10}	${\mathbb{ℝ}}^4$ representation {3,10|,4} = {3,10}8
			(11 1 15)4 File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch 10l.pngFile:CDel label5.png	360		{3}	{6}
${\mathbb{ℂ}}^3$	[1 1 14]5 File:CDel node.pngFile:CDel 5split1.pngFile:CDel branch.pngFile:CDel label4.png		(1 1 114)5 File:CDel node 1.pngFile:CDel 5split1.pngFile:CDel branch.pngFile:CDel label4.png	270	1080	720 {3}	{8}	${\mathbb{ℝ}}^4$ representation {3,8|,5} = {3,8}10
			(11 1 14)5 File:CDel node.pngFile:CDel 5split1.pngFile:CDel branch 10l.pngFile:CDel label4.png	360		{3}	{6}

Coxeter defines other groups with anti-unitary constructions, for example these three. The first was discovered and drawn by Peter McMullen in 1966.

More almost regular complex polyhedra

Space	Group	Order	Coxeter symbols	Vertices	Edges	Faces	Vertex figure	Notes
${\mathbb{ℂ}}^3$	[1 14 14](3) File:CDel node.pngFile:CDel anti3split1-44.pngFile:CDel branch.png	336	(11 14 14)(3) File:CDel node 1.pngFile:CDel anti3split1-44.pngFile:CDel branch.png	56	168	84 {4}	{6}	${\mathbb{ℝ}}^4$ representation {4,6|,3} = {4,6}6
${\mathbb{ℂ}}^3$	[15 14 14](3) File:CDel node.pngFile:CDel anti3split1-44.pngFile:CDel branch.pngFile:CDel label5.png	2160	(115 14 14)(3) File:CDel node 1.pngFile:CDel anti3split1-44.pngFile:CDel branch.pngFile:CDel label5.png	216	1080	540 {4}	{10}	${\mathbb{ℝ}}^4$ representation {4,10|,3} = {4,10}6
${\mathbb{ℂ}}^3$	[14 15 15](3) File:CDel node.pngFile:CDel anti3split1-55.pngFile:CDel branch.pngFile:CDel label4.png		(114 15 15)(3) File:CDel node 1.pngFile:CDel anti3split1-55.pngFile:CDel branch.pngFile:CDel label4.png	270	1080	432 {5}	{8}	${\mathbb{ℝ}}^4$ representation {5,8|,3} = {5,8}6

Some complex 4-polytopes

Space	Group	Order	Coxeter symbols	Vertices	Other elements	Cells	Vertex figure	Notes
${\mathbb{ℂ}}^4$	[1 1 2p]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png p=2,3,4...	24p3	(1 1 22p)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	4p		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	Shephard (22 1; 1)p same as βp 4 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png
			(11 1 2p )3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	p3		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (2 1; 11)p 1/p γp 4
${\mathbb{ℝ}}^4$	[1 1 22]3 =[31,1,1] File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	192	(1 1 222)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes.png	8	24 edges 32 faces	16 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel split1.pngFile:CDel nodes.png	β2 4 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node.png, real 16-cell
			(11 1 22 )3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel split1.pngFile:CDel nodes 10lu.png					1/2 γ2 4 = File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel node h.png = β2 4, real 16-cell
${\mathbb{ℂ}}^4$	[1 1 2]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png	648	(1 1 22)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.png	12		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.png	Shephard (22 1; 1)3 same as β3 4 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 3node.png
			(11 1 23)3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.png	27		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (2 1; 11)3 1/3 γ3 4
${\mathbb{ℂ}}^4$	[1 1 24]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png	1536	(1 1 224)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png	16		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel label4.png	Shephard (22 1; 1)4 same as β4 4 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel 4node.png
			(11 1 24 )3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel label4.png	64		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel label4.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (2 1; 11)4 1/4 γ4 4
${\mathbb{ℂ}}^4$	[14 1 2]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch.png	7680	(22 14 1)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch.png	80		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3split1-43.pngFile:CDel branch.png	Shephard (22 1; 1)4
			(114 1 2)3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch 01l.png	160		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch 01l.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 4.pngFile:CDel node 1.png	Shephard (2 1; 11)4
			(11 14 2)3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch 10l.png	320		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3split1-43.pngFile:CDel branch 10l.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (2 11; 1)4
${\mathbb{ℂ}}^4$	[1 1 2]4 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.png		(1 1 22)4 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.png	80	640 edges 1280 triangles	640 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 4split1.pngFile:CDel branch.png
			(11 1 2)4 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch 10lu.png	320		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 4split1.pngFile:CDel branch 10lu.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png

Some complex 5-polytopes

Space	Group	Order	Coxeter symbols	Vertices	Edges	Facets	Vertex figure	Notes
${\mathbb{ℂ}}^5$	[1 1 3p]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png p=2,3,4...	120p4	(1 1 33p)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	5p		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	Shephard (33 1; 1)p same as βp 5 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png
			(11 1 3p)3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	p4		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (3 1; 11)p 1/p γp 5
${\mathbb{ℂ}}^5$	[2 2 1]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.png	51840	(2 1 22)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png	80		File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea 1.png File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png	File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lr.pngFile:CDel 3b.pngFile:CDel nodeb.png	Shephard (2 1; 22)3
			(2 11 2)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.png	432		File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.png	Shephard (2 11; 2)3

Some complex 6-polytopes

Space	Group	Order	Coxeter symbols	Vertices	Edges	Facets	Vertex figure	Notes
${\mathbb{ℂ}}^6$	[1 1 4p]3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png p=2,3,4...	720p5	(1 1 44p)3 File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	6p		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png	File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel labelp.png	Shephard (44 1; 1)p same as βp 6 = File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4.pngFile:CDel pnode.png
			(11 1 4p)3 File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	p5		File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 10lu.pngFile:CDel labelp.png	File:CDel node.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node 1.pngFile:CDel 3.pngFile:CDel node 1.png	Shephard (4 1; 11)p 1/p γp 6
${\mathbb{ℂ}}^6$	[1 2 3]3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.png	39191040	(2 1 33)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea 1.png	756		File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea 1.png File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea 1.png	File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 10l.png	Shephard (2 1; 33)3
			(22 1 3)3 File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 01lr.pngFile:CDel 3a.pngFile:CDel nodea.png	4032		File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 01l.png File:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes 01lr.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel node.pngFile:CDel 3split1.pngFile:CDel branch 01lr.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	Shephard (22 1; 3)3
			(2 11 3)3 File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.png	54432		File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3ab.pngFile:CDel nodes.png File:CDel node 1.pngFile:CDel 3split1.pngFile:CDel branch.pngFile:CDel 3a.pngFile:CDel nodea.pngFile:CDel 3a.pngFile:CDel nodea.png	File:CDel branch 11.pngFile:CDel 3ab.pngFile:CDel nodes.pngFile:CDel 3a.pngFile:CDel nodea.png	Shephard (2 11; 3)3

Visualizations

• (1 1 114)4, File:CDel node 1.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label4.png has 42 vertices, 168 edges and 112 triangular faces, seen in this 14-gonal projection.
  (1 1 1₁⁴)⁴, File:CDel node 1.pngFile:CDel 4split1.pngFile:CDel branch.pngFile:CDel label4.png has 42 vertices, 168 edges and 112 triangular faces, seen in this 14-gonal projection.
• (14 14 11)(3), File:CDel node 1.pngFile:CDel anti3split1-44.pngFile:CDel branch.png has 56 vertices, 168 edges and 84 square faces, seen in this 14-gonal projection.
  (1⁴ 1⁴ 1₁)⁽³⁾, File:CDel node 1.pngFile:CDel anti3split1-44.pngFile:CDel branch.png has 56 vertices, 168 edges and 84 square faces, seen in this 14-gonal projection.
• (1 1 22)4, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.png has 80 vertices, 640 edges, 1280 triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection.[43]
  (1 1 2₂)⁴, File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 4split1.pngFile:CDel branch.png has 80 vertices, 640 edges, 1280 triangular faces and 640 tetrahedral cells, seen in this 20-gonal projection.

See also

• Quaternionic polytope

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. (1991), Regular Complex Polytopes, Cambridge University Press, ISBN 0-521-39490-2
• 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.
• G. C. Shephard, J. A. Todd, Finite unitary reflection groups, Canadian Journal of Mathematics. 6(1954), 274-304, doi:10.4153/CJM-1954-028-3
• Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009

Further reading

• F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H.S.M. Coxeter., Paper 25, Finite groups generated by unitary reflections, p 415-425, John Wiley, 1995, ISBN 0-471-01003-0
• McMullen, Peter; Schulte, Egon (December 2002), Abstract Regular Polytopes (1st ed.), Cambridge University Press, ISBN 0-521-81496-0 Chapter 9 Unitary Groups and Hermitian Forms, pp. 289–298