# 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.

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

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

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

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

## Regular complex one-dimensional polytopes

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

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

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

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

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

A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is e2πi/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

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.

GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(q,q,2)
2[q]2 = [q]
q=2,3,4,...
2qq2{q}2qq{}Real regular polygons
Same as
Same as if q even
GroupOrderCoxeter
number
PolygonVerticesEdgesNotes
G(p,1,2)
p[4]2
p=2,3,4,...
2p22pp(2p2)2p{4}2
p22pp{}same as p{}×p{} or
${\displaystyle \mathbb {R} ^{4}}$ representation as p-p duoprism
2(2p2)p2{4}p2pp2{}${\displaystyle \mathbb {R} ^{4}}$ representation as p-p duopyramid
G(2,1,2)
2[4]2 = [4]
842{4}2 = {4}44{}same as {}×{} or
Real square
G(3,1,2)
3[4]2
1866(18)23{4}2963{}same as 3{}×3{} or
${\displaystyle \mathbb {R} ^{4}}$ representation as 3-3 duoprism
2(18)32{4}369{}${\displaystyle \mathbb {R} ^{4}}$ representation as 3-3 duopyramid
G(4,1,2)
4[4]2
3288(32)24{4}21684{}same as 4{}×4{} or
${\displaystyle \mathbb {R} ^{4}}$ representation as 4-4 duoprism or {4,3,3}
2(32)42{4}4816{}${\displaystyle \mathbb {R} ^{4}}$ representation as 4-4 duopyramid or {3,3,4}
G(5,1,2)
5[4]2
50255(50)25{4}225105{}same as 5{}×5{} or
${\displaystyle \mathbb {R} ^{4}}$ representation as 5-5 duoprism
2(50)52{4}51025{}${\displaystyle \mathbb {R} ^{4}}$ representation as 5-5 duopyramid
G(6,1,2)
6[4]2
72366(72)26{4}236126{}same as 6{}×6{} or
${\displaystyle \mathbb {R} ^{4}}$ representation as 6-6 duoprism
2(72)62{4}61236{}${\displaystyle \mathbb {R} ^{4}}$ representation as 6-6 duopyramid
G4=G(1,1,2)
3[3]3
<2,3,3>
2463(24)33{3}3883{}Möbius–Kantor configuration
self-dual, same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,4}
G6
3[6]2
48123(48)23{6}224163{}same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,4,3}
3{3}2starry polygon
2(48)32{6}31624{}${\displaystyle \mathbb {R} ^{4}}$ representation as {4,3,3}
2{3}3starry polygon
G5
3[4]3
72123(72)33{4}324243{}self-dual, same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,4,3}
G8
4[3]4
96124(96)44{3}424244{}self-dual, same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,4,3}
G14
3[8]2
144243(144)23{8}272483{}same as
3{8/3}2starry polygon, same as
2(144)32{8}34872{}
2{8/3}3starry polygon
G9
4[6]2
192244(192)24{6}296484{}same as
2(192)42{6}44896{}
4{3}29648{}starry polygon
2{3}44896{}starry polygon
G10
4[4]3
288244(288)34{4}396724{}
124{8/3}3starry polygon
243(288)43{4}472963{}
123{8/3}4starry polygon
G20
3[5]3
360303(360)33{5}31201203{}self-dual, same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,5}
3{5/2}3self-dual, starry polygon
G16
5[3]5
600305(600)55{3}51201205{}self-dual, same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {3,3,5}
105{5/2}5self-dual, starry polygon
G21
3[10]2
720603(720)23{10}23602403{}same as
3{5}2starry polygon
3{10/3}2starry polygon, same as
3{5/2}2starry polygon
2(720)32{10}3240360{}
2{5}3starry polygon
2{10/3}3starry polygon
2{5/2}3starry polygon
G17
5[6]2
1200605(1200)25{6}26002405{}same as
${\displaystyle \mathbb {R} ^{4}}$ representation as {5,3,3}
205{5}2starry polygon
205{10/3}2starry polygon
605{3}2starry polygon
602(1200)52{6}5240600{}
202{5}5starry polygon
202{10/3}5starry polygon
602{3}5starry polygon
G18
5[4]3
1800605(1800)35{4}36003605{}${\displaystyle \mathbb {R} ^{4}}$ representation as {5,3,3}
155{10/3}3starry polygon
305{3}3starry polygon
305{5/2}3starry polygon
603(1800)53{4}53606003{}
153{10/3}5starry polygon
303{3}5starry polygon
303{5/2}5starry 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. 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

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-valence 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

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.

SpaceGroupOrderCoxeter numberPolygonVerticesEdgesFacesVertex
figure
Van Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{3}}$G(1,1,3)
2[3]2[3]2
= [3,3]
244α3 = 2{3}2{3}2
= {3,3}
46{}4{3}{3}noneReal tetrahedron
Same as
${\displaystyle \mathbb {R} ^{3}}$G23
2[3]2[5]2
= [3,5]
120102{3}2{5}2 = {3,5}1230{}20{3}{5}noneReal icosahedron
2{5}2{3}2 = {5,3}2030{}12{5}{3}noneReal dodecahedron
${\displaystyle \mathbb {R} ^{3}}$G(2,1,3)
2[3]2[4]2
= [3,4]
486β2
3
= β3 = {3,4}
612{}8{3}{4}{4}Real octahedron
Same as {}+{}+{}, order 8
Same as , order 24
${\displaystyle \mathbb {R} ^{3}}$γ2
3
= γ3 = {4,3}
812{}6{4}{3}noneReal cube
Same as {}×{}×{} or
${\displaystyle \mathbb {C} ^{3}}$G(p,1,3)
2[3]2[4]p
p=2,3,4,...
6p33pβp
3
= 2{3}2{4}p

3p3p2{}p3{3}2{4}p2{4}pGeneralized octahedron
Same as p{}+p{}+p{}, order p3
Same as , order 6p2
${\displaystyle \mathbb {C} ^{3}}$γp
3
= p{4}2{3}2
p33p2p{}3pp{4}2{3}noneGeneralized cube
Same as p{}×p{}×p{} or
${\displaystyle \mathbb {C} ^{3}}$G(3,1,3)
2[3]2[4]3
1629β3
3
= 2{3}2{4}3
927{}27{3}2{4}32{4}3Same as 3{}+3{}+3{}, order 27
Same as , order 54
${\displaystyle \mathbb {C} ^{3}}$γ3
3
= 3{4}2{3}2
27273{}93{4}2{3}noneSame as 3{}×3{}×3{} or
${\displaystyle \mathbb {C} ^{3}}$G(4,1,3)
2[3]2[4]4
38412β4
3
= 2{3}2{4}4
1248{}64{3}2{4}42{4}4Same as 4{}+4{}+4{}, order 64
Same as , order 96
${\displaystyle \mathbb {C} ^{3}}$γ4
3
= 4{4}2{3}2
64484{}124{4}2{3}noneSame as 4{}×4{}×4{} or
${\displaystyle \mathbb {C} ^{3}}$G(5,1,3)
2[3]2[4]5
75015β5
3
= 2{3}2{4}5
1575{}125{3}2{4}52{4}5Same as 5{}+5{}+5{}, order 125
Same as , order 150
${\displaystyle \mathbb {C} ^{3}}$γ5
3
= 5{4}2{3}2
125755{}155{4}2{3}noneSame as 5{}×5{}×5{} or
${\displaystyle \mathbb {C} ^{3}}$G(6,1,3)
2[3]2[4]6
129618β6
3
= 2{3}2{4}6
36108{}216{3}2{4}62{4}6Same as 6{}+6{}+6{}, order 216
Same as , order 216
${\displaystyle \mathbb {C} ^{3}}$γ6
3
= 6{4}2{3}2
2161086{}186{4}2{3}noneSame as 6{}×6{}×6{} or
${\displaystyle \mathbb {C} ^{3}}$G25
3[3]3[3]3
64893{3}3{3}327723{}273{3}33{3}33{4}2Same as .
${\displaystyle \mathbb {R} ^{6}}$ representation as 221
Hessian polyhedron
G26
2[4]3[3]3
1296182{4}3{3}354216{}722{4}33{3}3{6}
3{3}3{4}2722163{}543{3}33{4}23{4}3Same 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]

SpaceGroupOrderCoxeter
number
PolytopeVerticesEdgesFacesCellsVan Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{4}}$G(1,1,4)
2[3]2[3]2[3]2
= [3,3,3]
1205α4 = 2{3}2{3}2{3}2
= {3,3,3}
510
{}
10
{3}
5
{3,3}
noneReal 5-cell (simplex)
${\displaystyle \mathbb {R} ^{4}}$G28
2[3]2[4]2[3]2
= [3,4,3]
1152122{3}2{4}2{3}2 = {3,4,3}
2496
{}
96
{3}
24
{3,4}
{6}Real 24-cell
G30
2[3]2[3]2[5]2
= [3,3,5]
14400302{3}2{3}2{5}2 = {3,3,5}
120720
{}
1200
{3}
600
{3,3}
{10}Real 600-cell
2{5}2{3}2{3}2 = {5,3,3}
6001200
{}
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]
3848β2
4
= β4 = {3,3,4}
824
{}
32
{3}
16
{3,3}
{4}Real 16-cell
Same as , order 192
${\displaystyle \mathbb {R} ^{4}}$γ2
4
= γ4 = {4,3,3}
1632
{}
24
{4}
8
{4,3}
noneReal 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,...
24p44pβp
4
= 2{3}2{3}2{4}p
4p6p2
{}
4p3
{3}
p4
{3,3}
2{4}pGeneralized 4-orthoplex
Same as , order 24p3
${\displaystyle \mathbb {C} ^{4}}$γp
4
= p{4}2{3}2{3}2
p44p3
p{}
6p2
p{4}2
4p
p{4}2{3}2
noneGeneralized tesseract
Same as p{}4 or , order p4
${\displaystyle \mathbb {C} ^{4}}$G(3,1,4)
2[3]2[3]2[4]3
194412β3
4
= 2{3}2{3}2{4}3
1254
{}
108
{3}
81
{3,3}
2{4}3Generalized 4-orthoplex
Same as , order 648
${\displaystyle \mathbb {C} ^{4}}$γ3
4
= 3{4}2{3}2{3}2
81108
3{}
54
3{4}2
12
3{4}2{3}2
noneSame as 3{}4 or , order 81
${\displaystyle \mathbb {C} ^{4}}$G(4,1,4)
2[3]2[3]2[4]4
614416β4
4
= 2{3}2{3}2{4}4
1696
{}
256
{3}
64
{3,3}
2{4}4Same as , order 1536
${\displaystyle \mathbb {C} ^{4}}$γ4
4
= 4{4}2{3}2{3}2
256256
4{}
96
4{4}2
16
4{4}2{3}2
noneSame as 4{}4 or , order 256
${\displaystyle \mathbb {C} ^{4}}$G(5,1,4)
2[3]2[3]2[4]5
1500020β5
4
= 2{3}2{3}2{4}5
20150
{}
500
{3}
625
{3,3}
2{4}5Same as , order 3000
${\displaystyle \mathbb {C} ^{4}}$γ5
4
= 5{4}2{3}2{3}2
625500
5{}
150
5{4}2
20
5{4}2{3}2
noneSame as 5{}4 or , order 625
${\displaystyle \mathbb {C} ^{4}}$G(6,1,4)
2[3]2[3]2[4]6
3110424β6
4
= 2{3}2{3}2{4}6
24216
{}
864
{3}
1296
{3,3}
2{4}6Same as , order 5184
${\displaystyle \mathbb {C} ^{4}}$γ6
4
= 6{4}2{3}2{3}2
1296864
6{}
216
6{4}2
24
6{4}2{3}2
noneSame as 6{}4 or , order 1296
${\displaystyle \mathbb {C} ^{4}}$G32
3[3]3[3]3[3]3
155520303{3}3{3}3{3}3
2402160
3{}
2160
3{3}3
240
3{3}3{3}3
3{4}3Witting 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.

SpaceGroupOrderPolytopeVerticesEdgesFacesCells4-facesVan Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{5}}$G(1,1,5)
= [3,3,3,3]
720α5 = {3,3,3,3}
615
{}
20
{3}
15
{3,3}
6
{3,3,3}
noneReal 5-simplex
${\displaystyle \mathbb {R} ^{5}}$G(2,1,5)
=[3,3,3,4]
3840β2
5
= β5 = {3,3,3,4}
1040
{}
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}
3280
{}
80
{4}
40
{4,3}
10
{4,3,3}
noneReal 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
120p5βp
5
= 2{3}2{3}2{3}2{4}p
5p10p2
{}
10p3
{3}
5p4
{3,3}
p5
{3,3,3}
2{4}pGeneralized 5-orthoplex
Same as , order 120p4
${\displaystyle \mathbb {C} ^{5}}$γp
5
= p{4}2{3}2{3}2{3}2
p55p4
p{}
10p3
p{4}2
10p2
p{4}2{3}2
5p
p{4}2{3}2{3}2
noneGeneralized 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
29160β3
5
= 2{3}2{3}2{3}2{4}3
1590
{}
270
{3}
405
{3,3}
243
{3,3,3}
2{4}3Same as , order 9720
${\displaystyle \mathbb {C} ^{5}}$γ3
5
= 3{4}2{3}2{3}2{3}2
243405
3{}
270
3{4}2
90
3{4}2{3}2
15
3{4}2{3}2{3}2
noneSame as 3{}5 or , order 243
${\displaystyle \mathbb {C} ^{5}}$G(4,1,5)
2[3]2[3]2[3]2[4]4
122880β4
5
= 2{3}2{3}2{3}2{4}4
20160
{}
640
{3}
1280
{3,3}
1024
{3,3,3}
2{4}4Same as , order 30720
${\displaystyle \mathbb {C} ^{5}}$γ4
5
= 4{4}2{3}2{3}2{3}2
10241280
4{}
640
4{4}2
160
4{4}2{3}2
20
4{4}2{3}2{3}2
noneSame as 4{}5 or , order 1024
${\displaystyle \mathbb {C} ^{5}}$G(5,1,5)
2[3]2[3]2[3]2[4]5
375000β5
5
= 2{3}2{3}2{3}2{5}5
25250
{}
1250
{3}
3125
{3,3}
3125
{3,3,3}
2{5}5Same as , order 75000
${\displaystyle \mathbb {C} ^{5}}$γ5
5
= 5{4}2{3}2{3}2{3}2
31253125
5{}
1250
5{5}2
250
5{5}2{3}2
25
5{4}2{3}2{3}2
noneSame as 5{}5 or , order 3125
${\displaystyle \mathbb {C} ^{5}}$G(6,1,5)
2[3]2[3]2[3]2[4]6
933210β6
5
= 2{3}2{3}2{3}2{4}6
30360
{}
2160
{3}
6480
{3,3}
7776
{3,3,3}
2{4}6Same as , order 155520
${\displaystyle \mathbb {C} ^{5}}$γ6
5
= 6{4}2{3}2{3}2{3}2
77766480
6{}
2160
6{4}2
360
6{4}2{3}2
30
6{4}2{3}2{3}2
noneSame 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

SpaceGroupOrderPolytopeVerticesEdgesFacesCells4-faces5-facesVan Oss
polygon
Notes
${\displaystyle \mathbb {R} ^{6}}$G(1,1,6)
= [3,3,3,3,3]
720α6 = {3,3,3,3,3}
721
{}
35
{3}
35
{3,3}
21
{3,3,3}
7
{3,3,3,3}
noneReal 6-simplex
${\displaystyle \mathbb {R} ^{6}}$G(2,1,6)
[3,3,3,4]
46080β2
6
= β6 = {3,3,3,4}
1260
{}
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}
64192
{}
240
{4}
160
{4,3}
60
{4,3,3}
12
{4,3,3,3}
noneReal 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
720p6βp
6
= 2{3}2{3}2{3}2{4}p
6p15p2
{}
20p3
{3}
15p4
{3,3}
6p5
{3,3,3}
p6
{3,3,3,3}
2{4}pGeneralized 6-orthoplex
Same as , order 720p5
${\displaystyle \mathbb {C} ^{6}}$γp
6
= p{4}2{3}2{3}2{3}2
p66p5
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
noneGeneralized 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

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
SpaceGroupApeirogonEdge${\displaystyle \mathbb {R} ^{2}}$ rep.[35]PictureNotes
${\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]rδp,r
2
= p{q}r
p{}
${\displaystyle \mathbb {C} ^{1}}$3[12]2δ3,2
2
= 3{12}2
3{}r{3,6}Same as
δ2,3
2
= 2{12}3
{}{6,3}
${\displaystyle \mathbb {C} ^{1}}$3[6]3δ3,3
2
= 3{6}3
3{}{3,6}Same as
${\displaystyle \mathbb {C} ^{1}}$4[8]2δ4,2
2
= 4{8}2
4{}{4,4}Same as
δ2,4
2
= 2{8}4
{}{4,4}
${\displaystyle \mathbb {C} ^{1}}$4[4]4δ4,4
2
= 4{4}4
4{}{4,4}Same as
${\displaystyle \mathbb {C} ^{1}}$6[6]2δ6,2
2
= 6{6}2
6{}r{3,6}Same as
δ2,6
2
= 2{6}6
{}{3,6}
${\displaystyle \mathbb {C} ^{1}}$6[4]3δ6,3
2
= 6{4}3
6{}{6,3}
δ3,6
2
= 3{4}6
3{}{3,6}
${\displaystyle \mathbb {C} ^{1}}$6[3]6δ6,6
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
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{3}}$2[3]2[4]{4}2{3}2{}{4}2Same as {}×{}×{} or
Real representation {4,3,4}
${\displaystyle \mathbb {C} ^{2}}$p[4]2[4]rp{4}2{4}r
p22pqp{}r2p{4}22{q}rSame as , p,r=2,3,4,6
${\displaystyle \mathbb {R} ^{2}}$[4,4]δ2,2
3
= {4,4}
48{}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
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{2}}$2[4]r[4]22{4}r{4}2
2{}2p{4}2'2{4}rSame as and , r=2,3,4,6
${\displaystyle \mathbb {R} ^{2}}$[4,4]{4,4}24{}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

29
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]
SpaceGroupApeirohedronVertexEdgeFacevan Oss
apeirogon
Notes
${\displaystyle \mathbb {R} ^{2}}$ 2[6]2[3]2
= [6,3]
{3,6}
13{}2{3}{}Real triangular tiling
{6,3}23{}1{6}noneReal hexagonal tiling
${\displaystyle \mathbb {C} ^{2}}$ 3[4]3[3]33{3}3{4}3183{}33{3}33{4}6Same as
3{4}3{3}3383{}23{4}33{12}2
${\displaystyle \mathbb {C} ^{2}}$ 4[3]4[3]44{3}4{3}4164{}14{3}44{4}4Self-dual, same as
${\displaystyle \mathbb {C} ^{2}}$ 4[3]4[4]24{3}4{4}21124{}34{3}42{8}4Same as
2{4}4{3}4312{}12{4}44{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
SpaceGroup3-apeirotopeVertexEdgeFaceCellvan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{3}}$p[4]2[3]2[4]rδp,r
3
= p{4}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{q}rSame 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]2δ3,2
3
= 3{4}2{3}2{4}2
3{}3{4}23{4}2{3}2Same 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]3δ3,3
3
= 3{4}2{3}2{4}3
3{}3{4}23{4}2{3}2Same as
${\displaystyle \mathbb {C} ^{3}}$4[4]2[3]2[4]2δ4,2
3
= 4{4}2{3}2{4}2
4{}4{4}24{4}2{3}2Same 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]4δ4,4
3
= 4{4}2{3}2{4}4
4{}4{4}24{4}2{3}2Same as
${\displaystyle \mathbb {C} ^{3}}$6[4]2[3]2[4]2δ6,2
3
= 6{4}2{3}2{4}2
6{}6{4}26{4}2{3}2Same 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]3δ6,3
3
= 6{4}2{3}2{4}3
6{}6{4}26{4}2{3}2Same as
δ3,6
3
= 3{4}2{3}2{4}6
3{}3{4}23{4}2{3}2Same as
${\displaystyle \mathbb {C} ^{3}}$6[4]2[3]2[4]6δ6,6
3
= 6{4}2{3}2{4}6
6{}6{4}26{4}2{3}2Same as
Rank 4, exceptional cases
SpaceGroup3-apeirotopeVertexEdgeFaceCellvan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{3}}$2[4]3[3]3[3]33{3}3{3}3{4}2
124 3{}27 3{3}32 3{3}3{3}33{4}6Same as
2{4}3{3}3{3}3
227 {}24 2{4}31 2{4}3{3}32{12}3
${\displaystyle \mathbb {C} ^{3}}$2[3]2[4]3[3]32{3}2{4}3{3}3
127 {}72 2{3}28 2{3}2{4}32{6}6
3{3}3{4}2{3}2
872 3{}27 3{3}31 3{3}3{4}23{6}3Same 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
SpaceGroup4-apeirotopeVertexEdgeFaceCell4-facevan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{4}}$p[4]2[3]2[3]2[4]rδp,r
4
= p{4}2{3}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{4}2{3}2{3}2p{q}rSame as
${\displaystyle \mathbb {R} ^{4}}$2[4]2[3]2[3]2[4]2δ2,2
4
= {4,3,3,3}
{}{4}{4,3}{4,3,3}{}Tesseractic honeycomb
Same as
${\displaystyle \mathbb {R} ^{4}}$2[3]2[4]2[3]2[3]2
=[3,4,3,3]
{3,3,4,3}
112 {}32 {3}24 {3,3}3 {3,3,4}Real 16-cell honeycomb
Same as
{3,4,3,3}
324 {}32 {3}12 {3,4}1 {3,4,3}Real 24-cell honeycomb
Same as or
${\displaystyle \mathbb {C} ^{4}}$3[3]3[3]3[3]3[3]33{3}3{3}3{3}3{3}3
180 3{}270 3{3}380 3{3}3{3}31 3{3}3{3}3{3}33{4}6${\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
SpaceGroup5-apeirotopesVerticesEdgeFaceCell4-face5-facevan Oss
apeirogon
Notes
${\displaystyle \mathbb {C} ^{5}}$p[4]2[3]2[3]2[3]2[4]rδp,r
5
= p{4}2{3}2{3}2{3}2{4}r
p{}p{4}2p{4}2{3}2p{4}2{3}2{3}2p{4}2{3}2{3}2{3}2p{q}rSame 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 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]r2[4]23[4]24[4]25[4]26[4]27[4]28[4]23[3]33[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]r4[8]24[4]46[6]26[4]33[12]23[6]36[3]6
Regular
or p{q}r

Quasiregular

=

=

=
Regular dual
or r{q}p

### Quasiregular polyhedra

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/octahedraHessian polyhedron
p=2 (real)p=3p=4p=5p=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 diagramOrderSymbol or Position in Table VII of Shephard and Todd (1954)
, ( and ), , ...
pn − 1 n!, p ≥ 3G(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]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacesVertex
figure
Notes
${\displaystyle \mathbb {C} ^{3}}$[1 1 1p]3

p=2,3,4...
6p2(1 1 11p)3
3p3p2{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
24(1 1 112)3
6128 {3}{4}Same as β2
3
= = real octahedron
(11 1 12)3
464 {3}{3}1/2 γ2
3
= = α3 = real tetrahedron
${\displaystyle \mathbb {C} ^{3}}$[1 1 1]3
54(1 1 11)3
927{3}{6}Shephard symbol (1 1; 11)3
same as β3
3
=
(11 1 1)3
927{3}{6}Shephard symbol (11 1; 1)3
1/3 γ3
3
= β3
3
${\displaystyle \mathbb {C} ^{3}}$[1 1 14]3
96(1 1 114)3
1248{3}{8}Shephard symbol (1 1; 11)4
same as β4
3
=
(11 1 14)3
16{3}{6}Shephard symbol (11 1; 1)4
1/4 γ4
3
${\displaystyle \mathbb {C} ^{3}}$[1 1 15]3
150(1 1 115)3
1575{3}{10}Shephard symbol (1 1; 11)5
same as β5
3
=
(11 1 15)3
25{3}{6}Shephard symbol (11 1; 1)5
1/5 γ5
3
${\displaystyle \mathbb {C} ^{3}}$[1 1 16]3
216(1 1 116)3
18216{3}{12}Shephard symbol (1 1; 11)6
same as β6
3
=
(11 1 16)3
36{3}{6}Shephard symbol (11 1; 1)6
1/6 γ6
3
${\displaystyle \mathbb {C} ^{3}}$[1 1 14]4
336(1 1 114)4
42168112 {3}{8}${\displaystyle \mathbb {R} ^{4}}$ representation {3,8|,4} = {3,8}8
(11 1 14)4
56{3}{6}
${\displaystyle \mathbb {C} ^{3}}$[1 1 15]4
2160(1 1 115)4
2161080720 {3}{10}${\displaystyle \mathbb {R} ^{4}}$ representation {3,10|,4} = {3,10}8
(11 1 15)4
360{3}{6}
${\displaystyle \mathbb {C} ^{3}}$[1 1 14]5
(1 1 114)5
2701080720 {3}{8}${\displaystyle \mathbb {R} ^{4}}$ representation {3,8|,5} = {3,8}10
(11 1 14)5
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.[45]

More almost regular complex polyhedra[46]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacesVertex
figure
Notes
${\displaystyle \mathbb {C} ^{3}}$[1 14 14](3)
336(11 14 14)(3)
5616884 {4}{6}${\displaystyle \mathbb {R} ^{4}}$ representation {4,6|,3} = {4,6}6
${\displaystyle \mathbb {C} ^{3}}$[15 14 14](3)
2160(115 14 14)(3)
2161080540 {4}{10}${\displaystyle \mathbb {R} ^{4}}$ representation {4,10|,3} = {4,10}6
${\displaystyle \mathbb {C} ^{3}}$[14 15 15](3)
(114 15 15)(3)
2701080432 {5}{8}${\displaystyle \mathbb {R} ^{4}}$ representation {5,8|,3} = {5,8}6
Some complex 4-polytopes[47]
SpaceGroupOrderCoxeter
symbols
VerticesOther
elements
CellsVertex
figure
Notes
${\displaystyle \mathbb {C} ^{4}}$[1 1 2p]3

p=2,3,4...
24p3(1 1 22p)3
4pShephard (22 1; 1)p
same as βp
4
=
(11 1 2p )3
p3
Shephard (2 1; 11)p
1/p γp
4
${\displaystyle \mathbb {R} ^{4}}$[1 1 22]3
=[31,1,1]
192(1 1 222)3
824 edges
32 faces
16 β2
4
= , real 16-cell
(11 1 22 )3
1/2 γ2
4
= = β2
4
, real 16-cell
${\displaystyle \mathbb {C} ^{4}}$[1 1 2]3
648(1 1 22)3
12Shephard (22 1; 1)3
same as β3
4
=
(11 1 23)3
27
Shephard (2 1; 11)3
1/3 γ3
4
${\displaystyle \mathbb {C} ^{4}}$[1 1 24]3
1536(1 1 224)3
16Shephard (22 1; 1)4
same as β4
4
=
(11 1 24 )3
64
Shephard (2 1; 11)4
1/4 γ4
4
${\displaystyle \mathbb {C} ^{4}}$[14 1 2]3
7680(22 14 1)3
80Shephard (22 1; 1)4
(114 1 2)3
160
Shephard (2 1; 11)4
(11 14 2)3
320
Shephard (2 11; 1)4
${\displaystyle \mathbb {C} ^{4}}$[1 1 2]4
(1 1 22)4
80640 edges
1280 triangles
640
(11 1 2)4
320
Some complex 5-polytopes[48]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacetsVertex
figure
Notes
${\displaystyle \mathbb {C} ^{5}}$[1 1 3p]3

p=2,3,4...
120p4(1 1 33p)3
5pShephard (33 1; 1)p
same as βp
5
=
(11 1 3p)3
p4
Shephard (3 1; 11)p
1/p γp
5
${\displaystyle \mathbb {C} ^{5}}$[2 2 1]3
51840(2 1 22)3
80
Shephard (2 1; 22)3
(2 11 2)3
432Shephard (2 11; 2)3
Some complex 6-polytopes[49]
SpaceGroupOrderCoxeter
symbols
VerticesEdgesFacetsVertex
figure
Notes
${\displaystyle \mathbb {C} ^{6}}$[1 1 4p]3

p=2,3,4...
720p5(1 1 44p)3
6pShephard (44 1; 1)p
same as βp
6
=
(11 1 4p)3
p5
Shephard (4 1; 11)p
1/p γp
6
${\displaystyle \mathbb {C} ^{6}}$[1 2 3]3
39191040(2 1 33)3
756
Shephard (2 1; 33)3
(22 1 3)3
4032
Shephard (22 1; 3)3
(2 11 3)3
54432
Shephard (2 11; 3)3

## Notes

1. Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp 477–492. DOI:10.1007/s002080200001
2. Coxeter, Regular Complex Polytopes, p. 115
3. Coxeter, Regular Complex Polytopes, 11.3 Petrie Polygon, a simple h-gon formed by the orbit of the flag (O0,O0O1) for the product of the two generating reflections of any nonstarry regular complex polygon, p1{q}p2.
4. Complex Regular Polytopes,11.1 Regular complex polygons p.103
5. Shephard, 1952; "It is from considerations such as these that we derive the notion of the interior of a polytope, and it will be seen that in unitary space where the numbers cannot be so ordered such a concept of interior is impossible. [Para break] Hence ... we have to consider unitary polytopes as configurations."
6. Coxeter, Regular Complex polytopes, p. 96
7. Coxeter, Regular Complex Polytopes, p. xiv
8. Coxeter, Complex Regular Polytopes, p. 177, Table III
9. Lehrer & Taylor 2009, p.87
10. Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
11. Coxeter, Regular Complex Polytopes, p. 108
12. Coxeter, Regular Complex Polytopes, p. 108
13. Coxeter, Regular Complex Polytopes, p. 109
14. Coxeter, Regular Complex Polytopes, p. 111
15. Coxeter, Regular Complex Polytopes, p. 30 diagram and p. 47 indices for 8 3-edges
16. Coxeter, Regular Complex Polytopes, p. 110
17. Coxeter, Regular Complex Polytopes, p. 110
18. Coxeter, Regular Complex Polytopes, p. 48
19. Coxeter, Regular Complex Polytopes, p. 49
20. Coxeter, Regular Complex Polytopes, pp. 116–140.
21. Coxeter, Regular Complex Polytopes, pp. 118–119.
22. Coxeter, Regular Complex Polytopes, pp. 118-119
23. Complex Regular Polytopes, p.29
24. Coxeter, Regular Complex Polytopes, Table V. The nonstarry regular polyhedra and 4-polytopes. p. 180.
25. Coxeter, Kaleidoscopes — Selected Writings of H.S.M. Coxeter, Paper 25 Surprising relationships among unitary reflection groups, p. 431.
26. Coxeter, Regular Complex Polytopes, p. 131
27. Coxeter, Regular Complex Polytopes, p. 126
28. Coxeter, Regular Complex Polytopes, p. 125
29. Coxeter, Regular Complex Polytopes, p. 131
30. Coxeter, Regular Complex Polytopes, Table V. The nonstarry regular polyhedra and 4-polytopes. p. 180.
31. Coxeter, Regular Complex Polytopes, Table VI. The regular honeycombs. p. 180.
32. Complex regular polytope, p.174
33. Coxeter, Regular Complex Polytopes, Table VI. The regular honeycombs. p. 111, 136.
34. Coxeter, Regular Complex Polytopes, Table IV. The regular polygons. pp. 178–179
35. Coxeter, Regular Complex Polytopes, 11.6 Apeirogons, pp. 111-112
36. Coxeter, Complex Regular Polytopes, p.140
37. Coxeter, Regular Complex Polytopes, pp. 139-140
38. Complex Regular Polytopes, p.146
39. Complex Regular Polytopes, p.141
40. Coxeter, Regular Complex Polytopes, pp. 118–119, 138.
41. Coxeter, Regular Complex Polytopes, Chapter 14, Almost regular polytopes, pp. 156–174.
42. Coxeter, Groups Generated by Unitary Reflections of Period Two, 1956
43. Coxeter, Finite Groups Generated by Unitary Reflections, 1966, 4. The Graphical Notation, Table of n-dimensional groups generated by n Unitary Reflections. pp. 422-423
44. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
45. Coxeter, Complex Regular Polytopes, (1991), 14.6 McMullen's two polyhedral with 84 square faces, pp.166-171
46. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
47. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
48. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
49. Coxeter, Groups generated by Unitary Reflections of Period Two (1956), Table III: Some Complex Polytopes, p.413
50. Coxeter, Complex Regular Polytopes, pp.172-173

## 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
• Gustav I. Lehrer and Donald E. Taylor, Unitary Reflection Groups, Cambridge University Press 2009
• 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