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

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The complex line 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:[1][2]

  • for every −1 ≤ i < j < kn, if F is a flat in P of dimension i and H is a flat in P of dimension k such that FH then there are at least two flats G in P of dimension j such that FGH;
  • for every i, j such that −1 ≤ i < j − 2, jn, if FG 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 F0F1 ⊂ … ⊂Fn of flats of P (with Fi of dimension i for all i).

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

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

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

This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.[3] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen.


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

A complex polytope exists in the complex space of equivalent dimension. For example, the vertices of a complex polygon are points in the complex plane (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 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 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

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

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

A 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 . 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 has Coxeter-Dynkin diagram , for any positive integer p, 2 or greater, containing p vertices. p can be suppressed if it is 2. It can also be represented by an empty Schläfli symbol p{}, }p{, {}p, or p{2}1. The 1 is a notational placeholder, representing a nonexistent reflection, or a period 1 identity generator. (A 0-polytope, real or complex is a point, and is represented as } {, or 1{2}1.)

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

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

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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

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Shephard's modified Schläfli notation

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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

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A more modern notation p1{q}p2 is due to Coxeter,[7] and is based on group theory. As a symmetry group, its symbol is p1[q]p2.

The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: R1p1 = R2p2 = I. If q is even, (R2R1)q/2 = (R1R2)q/2. If q is odd, (R2R1)(q−1)/2R2 = (R1R2)(q−1)/2R1. When q is odd, p1=p2.

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

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

Coxeter-Dynkin diagrams

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

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

12 Irreducible Shephard groups

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12 irreducible Shephard groups with their subgroup index relations.[8] Subgroups index 2 relate by removing a real reflection:
p[2q]2p[q]p, index 2.
p[4]qp[q]p, index q.
p[4]2 subgroups: p=2,3,4...
p[4]2 → [p], index p
p[4]2p[]×p[], index 2

Coxeter enumerated this list of regular complex polygons in . A regular complex polygon, p{q}r or , has p-edges, and r-gonal vertex figures. p{q}r is a finite polytope if (p+r)q>pr(q-2).

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

For nonstarry groups, the order of the group p[q]r can be computed as .[9]

The Coxeter number for p[q]r is , so the group order can also be computed as . A regular complex polygon can be drawn in orthogonal projection with h-gonal symmetry.

The rank 2 solutions that generate complex polygons are:

Group G3=G(q,1,1) G2=G(p,1,2) G4 G6 G5 G8 G14 G9 G10 G20 G16 G21 G17 G18
2[q]2, q=3,4... p[4]2, p=2,3... 3[3]3 3[6]2 3[4]3 4[3]4 3[8]2 4[6]2 4[4]3 3[5]3 5[3]5 3[10]2 5[6]2 5[4]3
Order 2q 2p2 24 48 72 96 144 192 288 360 600 720 1200 1800
h q 2p 6 12 24 30 60

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

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

The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon is the same as quasiregular . As well, regular polygon with the same node orders, , have an alternated construction , allowing adjacent edges to be two different colors.[10]

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

Matrix generators

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The group p[q]r, , can be represented by two matrices:[11]

Name R1
R2
Order p r
Matrix

With

k=
Examples
Name R1
R2
Order p q
Matrix

Name R1
R2
Order p 2
Matrix

Name R1
R2
Order 3 3
Matrix

Name R1
R2
Order 4 4
Matrix

Name R1
R2
Order 4 2
Matrix

Name R1
R2
Order 3 2
Matrix

Enumeration of regular complex polygons

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

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

Visualizations of regular complex polygons

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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.

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.

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.
Other Complex polygons p{r}2
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.

Regular complex polytopes

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In general, a regular complex polytope is represented by Coxeter as p{z1}q{z2}r{z3}s... or Coxeter diagram ..., having symmetry p[z1]q[z2]r[z3]s... or ....[20]

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

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

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

Enumeration of regular complex polyhedra

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

Coxeter enumerated this list of nonstarry regular complex polyhedra in , including the 5 platonic solids in .[23]

A regular complex polyhedron, p{n1}q{n2}r or , has faces, edges, and vertex figures.

A complex regular polyhedron p{n1}q{n2}r requires both g1 = order(p[n1]q) and g2 = order(q[n2]r) be finite.

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

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

Visualizations of regular complex polyhedra

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2D orthogonal projections of complex polyhedra, p{s}t{r}r
Generalized octahedra

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

Generalized cubes

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

Enumeration of regular complex 4-polytopes

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Coxeter enumerated this list of nonstarry regular complex 4-polytopes in , including the 6 convex regular 4-polytopes in .[23]

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

Visualizations of regular complex 4-polytopes

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Generalized 4-orthoplexes

Generalized 4-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.

Generalized 4-cubes

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

Enumeration of regular complex 5-polytopes

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Regular complex 5-polytopes in 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
G(1,1,5)
= [3,3,3,3]
720 α5 = {3,3,3,3}
6 15
{}
20
{3}
15
{3,3}
6
{3,3,3}
none Real 5-simplex
G(2,1,5)
=[3,3,3,4]
3840 β2
5
= β5 = {3,3,3,4}
10 40
{}
80
{3}
80
{3,3}
32
{3,3,3}
{4} Real 5-orthoplex
Same as , order 1920
γ2
5
= γ5 = {4,3,3,3}
32 80
{}
80
{4}
40
{4,3}
10
{4,3,3}
none Real 5-cube
Same as {}5 or , order 32
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
5p 10p2
{}
10p3
{3}
5p4
{3,3}
p5
{3,3,3}
2{4}p Generalized 5-orthoplex
Same as , order 120p4
γp
5
= p{4}2{3}2{3}2{3}2
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 , order p5
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
15 90
{}
270
{3}
405
{3,3}
243
{3,3,3}
2{4}3 Same as , order 9720
γ3
5
= 3{4}2{3}2{3}2{3}2
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 , order 243
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
20 160
{}
640
{3}
1280
{3,3}
1024
{3,3,3}
2{4}4 Same as , order 30720
γ4
5
= 4{4}2{3}2{3}2{3}2
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 , order 1024
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
25 250
{}
1250
{3}
3125
{3,3}
3125
{3,3,3}
2{5}5 Same as , order 75000
γ5
5
= 5{4}2{3}2{3}2{3}2
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 , order 3125
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
30 360
{}
2160
{3}
6480
{3,3}
7776
{3,3,3}
2{4}6 Same as , order 155520
γ6
5
= 6{4}2{3}2{3}2{3}2
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 , order 7776

Visualizations of regular complex 5-polytopes

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Generalized 5-orthoplexes

Generalized 5-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.

Generalized 5-cubes

Generalized 5-cubes have a regular construction as and prismatic construction as , a product of five p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

Enumeration of regular complex 6-polytopes

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Space Group Order Polytope Vertices Edges Faces Cells 4-faces 5-faces Van Oss
polygon
Notes
G(1,1,6)
= [3,3,3,3,3]
720 α6 = {3,3,3,3,3}
7 21
{}
35
{3}
35
{3,3}
21
{3,3,3}
7
{3,3,3,3}
none Real 6-simplex
G(2,1,6)
[3,3,3,4]
46080 β2
6
= β6 = {3,3,3,4}
12 60
{}
160
{3}
240
{3,3}
192
{3,3,3}
64
{3,3,3,3}
{4} Real 6-orthoplex
Same as , order 23040
γ2
6
= γ6 = {4,3,3,3}
64 192
{}
240
{4}
160
{4,3}
60
{4,3,3}
12
{4,3,3,3}
none Real 6-cube
Same as {}6 or , order 64
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
6p 15p2
{}
20p3
{3}
15p4
{3,3}
6p5
{3,3,3}
p6
{3,3,3,3}
2{4}p Generalized 6-orthoplex
Same as , order 720p5
γp
6
= p{4}2{3}2{3}2{3}2
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 , order p6

Visualizations of regular complex 6-polytopes

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Generalized 6-orthoplexes

Generalized 6-orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.

Generalized 6-cubes

Generalized 6-cubes have a regular construction as and prismatic construction as , a product of six p-gonal 1-polytopes. Elements are lower dimensional generalized cubes.

Enumeration of regular complex apeirotopes

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Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs.[28]

For each dimension there are 12 apeirotopes symbolized as δp,r
n+1
exists in any dimensions , or if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2.[29]

Each has proportional element counts given as:

k-faces = , where and n! denotes the factorial of n.

Regular complex 1-polytopes

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The only regular complex 1-polytope is {}, or . Its real representation is an apeirogon, {∞}, or .

Regular complex apeirogons

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Some subgroups of the apeirogonal Shephard groups
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.
A quasiregular apeirogon is a mixture of two regular apeirogons and , seen here with blue and pink edges. 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 δp,r
2
where q is constrained to satisfy q = 2/(1 – (p + r)/pr).[30]

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

There are two excluded solutions odd q and unequal p and r: 10[5]2 and 12[3]4, or and .

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 is the same as quasiregular .[31]

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 , there are 3 more infinite solutions: [2], [4]2, [3]3, and , , and . The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in .

Rank 2
Space Group Apeirogon Edge rep.[32] Picture Notes
2[∞]2 = [∞] δ2,2
2
= {∞}
       
{} Real apeirogon
Same as
/ [4]2 {4}2 {} {4,4} Same as
[3]3 {3}3 {} {3,6} Same as
p[q]r δp,r
2
= p{q}r
p{}
3[12]2 δ3,2
2
= 3{12}2
3{} r{3,6} Same as
δ2,3
2
= 2{12}3
{} {6,3}
3[6]3 δ3,3
2
= 3{6}3
3{} {3,6} Same as
4[8]2 δ4,2
2
= 4{8}2
4{} {4,4} Same as
δ2,4
2
= 2{8}4
{} {4,4}
4[4]4 δ4,4
2
= 4{4}4
4{} {4,4} Same as
6[6]2 δ6,2
2
= 6{6}2
6{} r{3,6} Same as
δ2,6
2
= 2{6}6
{} {3,6}
6[4]3 δ6,3
2
= 6{4}3
6{} {6,3}
δ3,6
2
= 3{4}6
3{} {3,6}
6[3]6 δ6,6
2
= 6{3}6
6{} {3,6} Same as

Regular complex apeirohedra

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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 δp,r
3
or p{4}2{4}r is the regular form of the product apeirotope δp,r
2
× δp,r
2
or p{q}r × p{q}r, where q is determined from p and r.

is the same as , as well as , for p,r=2,3,4,6. Also = .[33]

Rank 3
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
2[3]2[4] {4}2{3}2 {} {4}2 Same as {}×{}×{} or
Real representation {4,3,4}
p[4]2[4]r p{4}2{4}r            
p2 2pr p{} r2 p{4}2 2{q}r Same as , p,r=2,3,4,6
[4,4] δ2,2
3
= {4,4}
4 8 {} 4 {4} {∞} Real square tiling
Same as or or
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