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

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

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

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

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

• for every −1 ≤ i < j < 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).

 This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.[3] In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen. A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.

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

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

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 ${\displaystyle \mathbb {R} ^{1}}$, defined by its two end points or vertices in the line. Its Schläfli symbol is {} .

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

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

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

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

A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is 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

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 aperiogons also include 6-edge (hexagonal edges) elements.

Notations

Shephard's modified Schläfli notation

Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.

The number of vertices V is then g/p2 and the number of edges E is g/p1.

The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.

Coxeter's revised modified Schläfli notation

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

The symmetry group p1[q]p2 is represented by 2 generators R1, R2, where: 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

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.

Enumeration of regular complex polygons

12 irreducible Shephard groups with their subgroup index relations.[8] 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.
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 ${\displaystyle \mathbb {C} ^{2}}$. A regular complex polygon, p{q}r or , has p-edges, and q-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 real and unitary reflections.

For nonstarry groups, the order of the group p[q]r can be computed as ${\displaystyle g=8/q\cdot (1/p+2/q+1/r-1)^{-2}}$.[9]

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

The rank 2 solutions that generate complex polygons are:

 Group Order 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 2q 2p2 24 48 72 96 144 192 288 360 600 720 1200 1800

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.

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

Visualizations of regular complex polygons

Polygons of the form p{2r}q can be visualized by q color sets of p-edge. Each p-edge is seen as a regular polygon, while there are no faces.

2D orthogonal projections of complex polygons 2{r}q

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

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 overlappng vertices from the center.

3D perspective projections of complex polygons p{4}2
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

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}22{3}2 and diagram . Its symmetry group has diagram p[4]2[3]22[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}22{4}p and diagram .[21]

A 1-dimensional regular complex polytope in ${\displaystyle \mathbb {C} ^{1}}$ is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol 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.[22]

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-valance vertices.[23] 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

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

Coxeter enumerated this list of nonstarry regular complex polyhedra in ${\displaystyle \mathbb {C} ^{3}}$, including the 5 platonic solids in ${\displaystyle \mathbb {R} ^{3}}$.[24]

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

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

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

Space Group Order Coxeter number Polygon Vertices Edges Faces Vertex
figure
Van Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{3}}$ G(1,1,3)
2[3]2[3]2
= [3,3]
24 4 α3 = 2{3}2{3}2
= {3,3}
4 6 {} 4 {3} {3} none Real tetrahedron
Same as
${\displaystyle \mathbb {R} ^{3}}$ G23
2[3]2[5]2
= [3,5]
120 10 2{3}2{5}2 = {3,5} 12 30 {} 20 {3} {5} none Real icosahedron
2{5}2{3}2 = {5,3} 20 30 {} 12 {5} {3} none Real dodecahedron
${\displaystyle \mathbb {R} ^{3}}$ G(2,1,3)
2[3]2[4]2
= [3,4]
48 6 β2
3
= β3 = {3,4}
6 12 {} 8 {3} {4} {4} Real octahedron
Same as {}+{}+{}, order 8
Same as , order 24
${\displaystyle \mathbb {R} ^{3}}$ γ2
3
= γ3 = {4,3}
8 12 {} 6 {4} {3} none Real cube
Same as {}×{}×{} or
${\displaystyle \mathbb {C} ^{3}}$ G(p,1,3)
2[3]2[4]p
p=2,3,4,...
6p3 3p βp
3
= 2{3}2{4}p

3p 3p2 {} p3 {3} 2{4}p 2{4}p Generalized octahedron
Same as p{}+p{}+p{}, order p3
Same as , order 6p2
${\displaystyle \mathbb {C} ^{3}}$ γp
3
= p{4}2{3}2
p3 3p2 p{} 3p p{4}2 {3} none Generalized cube
Same as p{}×p{}×p{} or
${\displaystyle \mathbb {C} ^{3}}$ G(3,1,3)
2[3]2[4]3
162 9 β3
3
= 2{3}2{4}3
9 27 {} 27 {3} 2{4}3 2{4}3 Same as 3{}+3{}+3{}, order 27
Same as , order 54
${\displaystyle \mathbb {C} ^{3}}$ γ3
3
= 3{4}2{3}2
27 27 3{} 9 3{4}2 {3} none Same as 3{}×3{}×3{} or
${\displaystyle \mathbb {C} ^{3}}$ G(4,1,3)
2[3]2[4]4
384 12 β4
3
= 2{3}2{4}4
12 48 {} 64 {3} 2{4}4 2{4}4 Same as 4{}+4{}+4{}, order 64
Same as , order 96
${\displaystyle \mathbb {C} ^{3}}$ γ4
3
= 4{4}2{3}2
64 48 4{} 12 4{4}2 {3} none Same as 4{}×4{}×4{} or
${\displaystyle \mathbb {C} ^{3}}$ G(5,1,3)
2[3]2[4]5
750 15 β5
3
= 2{3}2{4}5
15 75 {} 125 {3} 2{4}5 2{4}5 Same as 5{}+5{}+5{}, order 125
Same as , order 150
${\displaystyle \mathbb {C} ^{3}}$ γ5
3
= 5{4}2{3}2
125 75 5{} 15 5{4}2 {3} none Same as 5{}×5{}×5{} or
${\displaystyle \mathbb {C} ^{3}}$ G(6,1,3)
2[3]2[4]6
1296 18 β6
3
= 2{3}2{4}6
36 108 {} 216 {3} 2{4}6 2{4}6 Same as 6{}+6{}+6{}, order 216
Same as , order 216
${\displaystyle \mathbb {C} ^{3}}$ γ6
3
= 6{4}2{3}2
216 108 6{} 18 6{4}2 {3} none Same as 6{}×6{}×6{} or
${\displaystyle \mathbb {C} ^{3}}$ G25
3[3]3[3]3
648 9 3{3}3{3}3 27 72 3{} 27 3{3}3 3{3}3 3{4}2 Same as .
${\displaystyle \mathbb {R} ^{6}}$ representation as 221
Hessian polyhedron
G26
2[4]3[3]3
1296 18 2{4}3{3}3 54 216 {} 72 2{4}3 3{3}3 {6}
3{3}3{4}2 72 216 3{} 54 3{3}3 3{4}2 3{4}3 Same as [25]
${\displaystyle \mathbb {R} ^{6}}$ representation as 122

Visualizations of regular complex polyhedra

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

Coxeter enumerated this list of nonstarry regular complex 4-polytopes in ${\displaystyle \mathbb {C} ^{4}}$, including the 6 convex regular 4-polytopes in ${\displaystyle \mathbb {R} ^{4}}$.[30]

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

Visualizations of regular complex 4-polytopes

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

Regular complex 5-polytopes in ${\displaystyle \mathbb {C} ^{5}}$ 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
${\displaystyle \mathbb {R} ^{5}}$ 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
${\displaystyle \mathbb {R} ^{5}}$ 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
${\displaystyle \mathbb {R} ^{5}}$ 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
${\displaystyle \mathbb {C} ^{5}}$ 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
5p 10p2
{}
10p3
{3}
5p4
{3,3}
p5
{3,3,3}
2{4}p Generalized 5-orthoplex
Same as , order 120p4
${\displaystyle \mathbb {C} ^{5}}$ 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
${\displaystyle \mathbb {C} ^{5}}$ 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
15 90
{}
270
{3}
405
{3,3}
243
{3,3,3}
2{4}3 Same as , order 9720
${\displaystyle \mathbb {C} ^{5}}$ 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
${\displaystyle \mathbb {C} ^{5}}$ 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
20 160
{}
640
{3}
1280
{3,3}
1024
{3,3,3}
2{4}4 Same as , order 30720
${\displaystyle \mathbb {C} ^{5}}$ 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
${\displaystyle \mathbb {C} ^{5}}$ 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
25 250
{}
1250
{3}
3125
{3,3}
3125
{3,3,3}
2{5}5 Same as , order 75000
${\displaystyle \mathbb {C} ^{5}}$ 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
${\displaystyle \mathbb {C} ^{5}}$ 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
30 360
{}
2160
{3}
6480
{3,3}
7776
{3,3,3}
2{4}6 Same as , order 155520
${\displaystyle \mathbb {C} ^{5}}$ 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

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

Space Group Order Polytope Vertices Edges Faces Cells 4-faces 5-faces Van Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{6}}$ 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
${\displaystyle \mathbb {R} ^{6}}$ 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
${\displaystyle \mathbb {R} ^{6}}$ 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
${\displaystyle \mathbb {C} ^{6}}$ 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
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
${\displaystyle \mathbb {C} ^{6}}$ 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

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

Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs.[31]

For each dimension there are 12 apeirotopes symbolized as δp,r
n+1
exists in any dimensions ${\displaystyle \mathbb {C} ^{n}}$, or ${\displaystyle \mathbb {R} ^{n}}$ if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2.[32]

Each has proportional element counts given as:

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

Regular complex 1-polytopes

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

Regular complex apeirogons

Some subgroups of the apeirogonal shepherd 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).[33]

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 q-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 .[34]

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 ${\displaystyle \mathbb {C} ^{2}}$, 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 ${\displaystyle \mathbb {C} ^{1}}$.

Rank 2
Space Group Apeirogon Edge ${\displaystyle \mathbb {R} ^{2}}$ rep.[35] Picture Notes
${\displaystyle \mathbb {R} ^{1}}$ 2[∞]2 = [∞] δ2,2
2
= {∞}

{} Real apeirogon
Same as
${\displaystyle \mathbb {C} ^{2}}$ / ${\displaystyle \mathbb {C} ^{1}}$ [4]2 {4}2 {} {4,4} Same as
${\displaystyle \mathbb {C} ^{1}}$ [3]3 {3}3 {} {3,6} Same as
${\displaystyle \mathbb {C} ^{1}}$ p[q]r δp,r
2
= p{q}r
p{}
${\displaystyle \mathbb {C} ^{1}}$ 3[12]2 δ3,2
2
= 3{12}2
3{} r{3,6} Same as
δ2,3
2
= 2{12}3
{} {6,3}
${\displaystyle \mathbb {C} ^{1}}$ 3[6]3 δ3,3
2
= 3{6}3
3{} {3,6} Same as
${\displaystyle \mathbb {C} ^{1}}$ 4[8]2 δ4,2
2
= 4{8}2
4{} {4,4} Same as
δ2,4
2
= 2{8}4
{} {4,4}
${\displaystyle \mathbb {C} ^{1}}$ 4[4]4 δ4,4
2
= 4{4}4
4{} {4,4} Same as
${\displaystyle \mathbb {C} ^{1}}$ 6[6]2 δ6,2
2
= 6{6}2
6{} r{3,6} Same as
δ2,6
2
= 2{6}6
{} {3,6}
${\displaystyle \mathbb {C} ^{1}}$ 6[4]3 δ6,3
2
= 6{4}3
6{} {6,3}
δ3,6
2
= 3{4}6
3{} {3,6}
${\displaystyle \mathbb {C} ^{1}}$ 6[3]6 δ6,6
2
= 6{3}6
6{} {3,6} Same as

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 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 = .[36]

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

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 or or
Same as
Same as
Same as or or
Same as
Same as
Same as or or
Same as
Same as
Same as
Same as
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{2}}$ 2[4]r[4]2 2{4}r{4}2
2 {} 2 p{4}2' 2{4}r Same as and , r=2,3,4,6
${\displaystyle \mathbb {R} ^{2}}$ [4,4] {4,4} 2 4 {} 2 {4} {∞} Same as and
${\displaystyle \mathbb {C} ^{2}}$ 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

2 9
16
36
{} 2 2{4}3
2{4}4
2{4}6
2{q}r Same as and
Same as and
Same as and [37]
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
${\displaystyle \mathbb {R} ^{2}}$ 2[6]2[3]2
= [6,3]
{3,6}
1 3 {} 2 {3} {∞} Real triangular tiling
{6,3} 2 3 {} 1 {6} none Real hexagonal tiling
${\displaystyle \mathbb {C} ^{2}}$ 3[4]3[3]3 3{3}3{4}3 1 8 3{} 3 3{3}3 3{4}6 Same as
3{4}3{3}3 3 8 3{} 2 3{4}3 3{12}2
${\displaystyle \mathbb {C} ^{2}}$ 4[3]4[3]4 4{3}4{3}4 1 6 4{} 1 4{3}4 4{4}4 Self-dual, same as
${\displaystyle \mathbb {C} ^{2}}$ 4[3]4[4]2 4{3}4{4}2 1 12 4{} 3 4{3}4 2{8}4 Same as
2{4}4{3}4 3 12 {} 1 2{4}4 4{4}4

Regular complex 3-apeirotopes

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

Rank 4
Space Group 3-apeirotope Vertex Edge Face Cell van Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{3}}$ p[4]2[3]2[4]r δp,r
3
= p{4}2{3}2{4}r
p{} p{4}2 p{4}2{3}2 p{q}r Same as
${\displaystyle \mathbb {R} ^{3}}$ 2[4]2[3]2[4]2
=[4,3,4]
δ2,2
3
= 2{4}2{3}2{4}2
{} {4} {4,3} Cubic honeycomb
Same as or or
${\displaystyle \mathbb {C} ^{3}}$ 3[4]2[3]2[4]2 δ3,2
3
= 3{4}2{3}2{4}2
3{} 3{4}2 3{4}2{3}2 Same as or or
δ2,3
3
= 2{4}2{3}2{4}3
{} {4} {4,3} Same as
${\displaystyle \mathbb {C} ^{3}}$ 3[4]2[3]2[4]3 δ3,3
3
= 3{4}2{3}2{4}3
3{} 3{4}2 3{4}2{3}2 Same as
${\displaystyle \mathbb {C} ^{3}}$ 4[4]2[3]2[4]2 δ4,2
3
= 4{4}2{3}2{4}2
4{} 4{4}2 4{4}2{3}2 Same as or or
δ2,4
3
= 2{4}2{3}2{4}4
{} {4} {4,3} Same as
${\displaystyle \mathbb {C} ^{3}}$ 4[4]2[3]2[4]4 δ4,4
3
= 4{4}2{3}2{4}4
4{} 4{4}2 4{4}2{3}2 Same as
${\displaystyle \mathbb {C} ^{3}}$ 6[4]2[3]2[4]2 δ6,2
3
= 6{4}2{3}2{4}2
6{} 6{4}2 6{4}2{3}2 Same as or or
δ2,6
3
= 2{4}2{3}2{4}6
{} {4} {4,3} Same as
${\displaystyle \mathbb {C} ^{3}}$ 6[4]2[3]2[4]3 δ6,3
3
= 6{4}2{3}2{4}3
6{} 6{4}2 6{4}2{3}2 Same as
δ3,6
3
= 3{4}2{3}2{4}6
3{} 3{4}2 3{4}2{3}2 Same as
${\displaystyle \mathbb {C} ^{3}}$ 6[4]2[3]2[4]6 δ6,6
3
= 6{4}2{3}2{4}6
6{} 6{4}2 6{4}2{3}2 Same as
Rank 4, exceptional cases
Space Group 3-apeirotope Vertex Edge Face Cell van Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{3}}$ 2[4]3[3]3[3]3 3{3}3{3}3{4}2
1 24 3{} 27 3{3}3 2 3{3}3{3}3 3{4}6 Same as
2{4}3{3}3{3}3
2 27 {} 24 2{4}3 1 2{4}3{3}3 2{12}3
${\displaystyle \mathbb {C} ^{3}}$ 2[3]2[4]3[3]3 2{3}2{4}3{3}3
1 27 {} 72 2{3}2 8 2{3}2{4}3 2{6}6
3{3}3{4}2{3}2
8 72 3{} 27 3{3}3 1 3{3}3{4}2 3{6}3 Same as or

Regular complex 4-apeirotopes

There are 15 regular complex apeirotopes in ${\displaystyle \mathbb {C} ^{4}}$. Coxeter expresses 12 of them by δp,r
4
where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: = . The first case is the ${\displaystyle \mathbb {R} ^{4}}$ 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
${\displaystyle \mathbb {C} ^{4}}$ p[4]2[3]2[3]2[4]r δp,r
4
= p{4}2{3}2{3}2{4}r
p{} p{4}2 p{4}2{3}2 p{4}2{3}2{3}2 p{q}r Same as
${\displaystyle \mathbb {R} ^{4}}$ 2[4]2[3]2[3]2[4]2 δ2,2
4
= {4,3,3,3}
{} {4} {4,3} {4,3,3} {∞} Tesseractic honeycomb
Same as
${\displaystyle \mathbb {R} ^{4}}$ 2[4]2[3]2[3]2[3]2
=[3,4,3,3]
{3,3,4,3}
1 12 {} 32 {3} 24 {3,3} 3 {3,3,4} Real 16-cell honeycomb
Same as
{3,4,3,3}
3 24 {} 32 {3} 12 {3,4} 1 {3,4,3} Real 24-cell honeycomb
Same as or
${\displaystyle \mathbb {C} ^{4}}$ 3[3]3[3]3[3]3[3]3 3{3}3{3}3{3}3{3}3
1 80 3{} 270 3{3}3 80 3{3}3{3}3 1 3{3}3{3}3{3}3 3{4}6 ${\displaystyle \mathbb {R} ^{8}}$ representation 521

Regular complex 5-apeirotopes and higher

There are only 12 regular complex apeirotopes in ${\displaystyle \mathbb {C} ^{5}}$ or higher,[38] expressed δp,r
n
where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed a product of n apeirogons: ... = ... . The first case is the real ${\displaystyle \mathbb {R} ^{n}}$ hypercube honeycomb.

Rank 6
Space Group 5-apeirotopes Vertices Edge Face Cell 4-face 5-face van Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{5}}$ 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
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
${\displaystyle \mathbb {R} ^{5}}$ 2[4]2[3]2[3]2[3]2[4]2
=[4,3,3,3,4]
δ2,2
5
= {4,3,3,3,4}
{} {4} {4,3} {4,3,3} {4,3,3,3} {∞} 5-cubic honeycomb
Same as

van Oss polygon

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 ${\displaystyle \mathbb {R} ^{2}}$, or unitary plane ${\displaystyle \mathbb {C} ^{2}}$) 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.[39]

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

 Complex product polygon or {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges, with its real representation as a 3-dimensional pentagonal prism. 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 of two 1-dimensional polytopes is the same as the regular p{4}2 or . 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 .

Similarly, a ${\displaystyle \mathbb {C} ^{3}}$ complex polyhedron can be constructed as a triple product: p{}×p{}×p{} or is the same as the regular generalized cube, p{4}2{3}2 or , as well as product p{4}2×p{} or .[40]

Quasiregular polygons

A quasiregular polygon is a truncation of a regular polygon. A quasiregular polygon contains alternate edges of the regular polygons and . The quasiregular polygon has p vertices on the p-edges of the regular form.

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

4 2-edges

9 3-edges

16 4-edges

25 5-edges

36 6-edges

49 8-edges

64 8-edges

Quasiregular

=
4+4 2-edges

6 2-edges
9 3-edges

8 2-edges
16 4-edges

10 2-edges
25 5-edges

12 2-edges
36 6-edges

14 2-edges
49 7-edges

16 2-edges
64 8-edges

=

=
Regular

4 2-edges

6 2-edges

8 2-edges

10 2-edges

12 2-edges

14 2-edges

16 2-edges

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: =

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
or p{q}r

Quasiregular

=

=

=
Regular dual
or r{q}p

Quasiregular polyhedra

Example truncation of 3-generalized octahedron, 2{3}2{4}3, , to its rectified limit, showing outlined-green triangles faces at the start, and blue 2{4}3, , 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, , has p3 vertices, 3p2 edges, and 3p p-generalized square faces, while the p-generalized octahedron, , has 3p vertices, 3p2 edges and p3 triangular faces. The middle quasiregular form p-generalized cuboctahedron, , has 3p2 vertices, 3p3 edges, and 3p+p3 faces.

Also the rectification of the Hessian polyhedron , is , a quasiregular form sharing the geometry of the regular complex polyhedron .

Quasiregular examples
Generalized cube/octahedra Hessian polyhedron
p=2 (real) p=3 p=4 p=5 p=6
Generalized
cubes

(regular)

Cube
, 8 vertices, 12 2-edges, and 6 faces.

, 27 vertices, 27 3-edges, and 9 faces, with one face blue and red

, 64 vertices, 48 4-edges, and 12 faces.

, 125 vertices, 75 5-edges, and 15 faces.

, 216 vertices, 108 6-edges, and 18 faces.

, 27 vertices, 72 6-edges, and 27 faces.
Generalized
cuboctahedra

(quasiregular)

Cuboctahedron
, 12 vertices, 24 2-edges, and 6+8 faces.

, 27 vertices, 81 2-edges, and 9+27 faces, with one face blue

, 48 vertices, 192 2-edges, and 12+64 faces, with one face blue

, 75 vertices, 375 2-edges, and 15+125 faces.

, 108 vertices, 648 2-edges, and 18+216 faces.

= , 72 vertices, 216 3-edges, and 54 faces.
Generalized
octahedra

(regular)

Octahedron
, 6 vertices, 12 2-edges, and 8 {3} faces.

, 9 vertices, 27 2-edges, and 27 {3} faces.

, 12 vertices, 48 2-edges, and 64 {3} faces.

, 15 vertices, 75 2-edges, and 125 {3} faces.

, 18 vertices, 108 2-edges, and 216 {3} faces.

, 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 or symbol (11 1 1)3, and group [1 1 1]3.[41][42] These complex polytopes have not been systematically explored beyond a few cases.

The group 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 ${\displaystyle \mathbb {R} ^{4}}$.

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 , with octahedral symmetry order 48, and subgroup dihedral symmetry 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 for the cube. Vertex figures are generated by removing a ringed node and ringing one or more connected nodes, and 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 and with p≠3.[43]

Groups generated by unitary reflections
Coxeter diagram Order Symbol or Position in Table VII of Shephard and Todd (1954)
, ( and ), , ...
pn − 1 n!, p ≥ 3 G(p, p, n), [p], [1 1 1]p, [1 1 (n−2)p]3
, 72·6!, 108·9! Nos. 33, 34, [1 2 2]3, [1 2 3]3
, ( and ), ( and ) 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 ${\displaystyle \mathbb {C} ^{3}}$. 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 ${\displaystyle \mathbb {R} ^{4}}$.

Some almost regular complex polyhedra[44]
Space Group Order Coxeter
symbols
Vertices Edges Faces Vertex
figure
Notes
${\displaystyle \mathbb {C} ^{3}}$ [1 1 1p]3

p=2,3,4...
6p2 (1 1 11p)3
3p 3p2 {3} {2p} Shephard symbol (1 1; 11)p
same as βp
3
=
(11 1 1p)3
p2 {3} {6} Shephard symbol (11 1; 1)p
1/p γp
3
${\displaystyle \mathbb {R} ^{3}}$ [1 1 12]3