Complex polytope

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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[edit]

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

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

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

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

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

  • for every −1 ≤ i < j < kn, if F is a flat in P of dimension i and H is a flat in P of dimension k such that FH then there are at least two flats G in P of dimension j such that FGH;
  • for every i, j such that −1 ≤ i < j − 2, jn, if FG are flats of P of dimensions i, j, then the set of flats between F and G is connected, in the sense that one can get from any member of this set to any other by a sequence of containments; and
  • the subset of unitary transformations of V that fix P are transitive on the flags F0F1 ⊂ … ⊂Fn of flats of P (with Fi of dimension i for all i).

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

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

Three views of regular complex polygon 4{4}2, CDel 4node 1.pngCDel 3.pngCDel 4.pngCDel 3.pngCDel node.png
ComplexOctagon.svg
This complex polygon has 8 edges (complex lines), labeled as a..h, and 16 vertices. Four vertices lie in each edge and two edges intersect at each vertex. In the left image, the outlined squares are not elements of the polytope but are included merely to help identify vertices lying in the same complex line. The octagonal perimeter of the left image is not an element of the polytope, but it is a petrie polygon.[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.
Complex polygon 4-4-2-perspective-labeled.png
A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.

A complex polytope exists in the complex space of equivalent dimension. For example, the vertices of a complex polygon are points in the complex plane , and the edges are complex lines 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 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[edit]

Complex 1-polytopes represented in the Argand plane as regular polygons for p = 2, 3, 4, 5, and 6, with black vertices. The centroid of the p vertices is shown seen in red. The sides of the polygons represent one application of the symmetry generator, mapping each vertex to the next counterclockwise copy. These polygonal sides are not edge elements of the polytope, as a complex 1-polytope can have no edges (it often is a complex edge) and only contains vertex elements.

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

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

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

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

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

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

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

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[edit]

Shephard's modified Schläfli notation[edit]

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[edit]

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[edit]

Coxeter also generalised the use of Coxeter-Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by CDel pnode 1.pngCDel q.pngCDel rnode.png and the equivalent symmetry group, p[q]r, is a ringless diagram CDel pnode.pngCDel q.pngCDel rnode.png. The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or CDel node 1.pngCDel q.pngCDel node.png.

One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So CDel 3node 1.pngCDel 4.pngCDel node.png and CDel 3node 1.pngCDel 3.pngCDel 3node.png are ordinary, while CDel 4node 1.pngCDel 3.pngCDel node.png is starry.

Enumeration of regular complex polygons[edit]

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 . A regular complex polygon, p{q}r or CDel pnode 1.pngCDel q.pngCDel rnode.png, 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 .[9]

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

The rank 2 solutions that generate complex polygons are:

Group G3=G(q,1,1) G2=G(p,1,2) G4 G6 G5 G8 G14 G9 G10 G20 G16 G21 G17 G18
2[q]2, q=3,4... p[4]2, p=2,3... 3[3]3 3[6]2 3[4]3 4[3]4 3[8]2 4[6]2 4[4]3 3[5]3 5[3]5 3[10]2 5[6]2 5[4]3
CDel node.pngCDel q.pngCDel node.png CDel pnode.pngCDel 4.pngCDel node.png CDel 3node.pngCDel 3.pngCDel 3node.png CDel 3node.pngCDel 6.pngCDel node.png CDel 3node.pngCDel 4.pngCDel 3node.png CDel 4node.pngCDel 3.pngCDel 4node.png CDel 3node.pngCDel 8.pngCDel node.png CDel 4node.pngCDel 6.pngCDel node.png CDel 4node.pngCDel 4.pngCDel 3node.png CDel 3node.pngCDel 5.pngCDel 3node.png CDel 5node.pngCDel 3.pngCDel 5node.png CDel 3node.pngCDel 10.pngCDel node.png CDel 5node.pngCDel 6.pngCDel node.png CDel 5node.pngCDel 4.pngCDel 3node.png
Order 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: CDel 3node.pngCDel 3.pngCDel node.png, CDel 4node.pngCDel 3.pngCDel node.png, CDel 5node.pngCDel 3.pngCDel node.png, CDel 5node.pngCDel 3.pngCDel 3node.png, CDel 3node.pngCDel 5.pngCDel node.png, and CDel 5node.pngCDel 5.pngCDel node.png.

The dual polygon of p{q}r is r{q}p. A polygon of the form p{q}p is self-dual. Groups of the form p[2q]2 have a half symmetry p[q]p, so a regular polygon CDel pnode 1.pngCDel 2x.pngCDel q.pngCDel node.png is the same as quasiregular CDel pnode 1.pngCDel q.pngCDel pnode 1.png. As well, regular polygon with the same node orders, CDel pnode 1.pngCDel q.pngCDel pnode.png, have an alternated construction CDel node h.pngCDel 2x.pngCDel q.pngCDel pnode.png, 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 CDel node 1.pngCDel q.pngCDel node.png q q {} Real regular polygons
Same as CDel node h.pngCDel 2x.pngCDel q.pngCDel node.png
Same as CDel node 1.pngCDel q.pngCDel rat.pngCDel 2x.pngCDel node 1.png if q even
Group Order Coxeter
number
Polygon Vertices Edges Notes
G(p,1,2)
p[4]2
p=2,3,4,...
2p2 2p p(2p2)2 p{4}2          
CDel pnode 1.pngCDel 4.pngCDel node.png
p2 2p p{} same as p{}×p{} or CDel pnode 1.pngCDel 2.pngCDel pnode 1.png
representation as p-p duoprism
2(2p2)p 2{4}p CDel node 1.pngCDel 4.pngCDel pnode.png 2p p2 {} representation as p-p duopyramid
G(2,1,2)
2[4]2 = [4]
8 4 2{4}2 = {4} CDel node 1.pngCDel 4.pngCDel node.png 4 4 {} same as {}×{} or CDel node 1.pngCDel 2.pngCDel node 1.png
Real square
G(3,1,2)
3[4]2
18 6 6(18)2 3{4}2 CDel 3node 1.pngCDel 4.pngCDel node.png 9 6 3{} same as 3{}×3{} or CDel 3node 1.pngCDel 2.pngCDel 3node 1.png
representation as 3-3 duoprism
2(18)3 2{4}3 CDel node 1.pngCDel 4.pngCDel 3node.png 6 9 {} representation as 3-3 duopyramid
G(4,1,2)
4[4]2
32 8 8(32)2 4{4}2 CDel 4node 1.pngCDel 4.pngCDel node.png 16 8 4{} same as 4{}×4{} or CDel 4node 1.pngCDel 2.pngCDel 4node 1.png
representation as 4-4 duoprism or {4,3,3}
2(32)4 2{4}4 CDel node 1.pngCDel 4.pngCDel 4node.png 8 16 {} representation as 4-4 duopyramid or {3,3,4}
G(5,1,2)
5[4]2
50 25 5(50)2 5{4}2 CDel 5node 1.pngCDel 4.pngCDel node.png 25 10 5{} same as 5{}×5{} or CDel 5node 1.pngCDel 2.pngCDel 5node 1.png
representation as 5-5 duoprism
2(50)5 2{4}5 CDel node 1.pngCDel 4.pngCDel 5node.png 10 25 {} representation as 5-5 duopyramid
G(6,1,2)
6[4]2
72 36 6(72)2 6{4}2 CDel 6node 1.pngCDel 4.pngCDel node.png 36 12 6{} same as 6{}×6{} or CDel 6node 1.pngCDel 2.pngCDel 6node 1.png
representation as 6-6 duoprism
2(72)6 2{4}6 CDel node 1.pngCDel 4.pngCDel 6node.png 12 36 {} representation as 6-6 duopyramid
G4=G(1,1,2)
3[3]3
<2,3,3>
24 6 3(24)3 3{3}3 CDel 3node 1.pngCDel 3.pngCDel 3node.png 8 8 3{} Möbius–Kantor configuration
self-dual, same as CDel node h.pngCDel 6.pngCDel 3node.png
representation as {3,3,4}
G6
3[6]2
48 12 3(48)2 3{6}2 CDel 3node 1.pngCDel 6.pngCDel node.png 24 16 3{} same as CDel 3node 1.pngCDel 3.pngCDel 3node 1.png
representation as {3,4,3}
3{3}2 CDel 3node 1.pngCDel 3.pngCDel node.png starry polygon
2(48)3 2{6}3 CDel node 1.pngCDel 6.pngCDel 3node.png 16 24 {} representation as {4,3,3}
2{3}3 CDel node 1.pngCDel 3.pngCDel 3node.png starry polygon
G5
3[4]3
72 12 3(72)3 3{4}3 CDel 3node 1.pngCDel 4.pngCDel 3node.png 24 24 3{} self-dual, same as CDel node h.pngCDel 8.pngCDel 3node.png
representation as {3,4,3}
G8
4[3]4
96 12 4(96)4 4{3}4 CDel 4node 1.pngCDel 3.pngCDel 4node.png 24 24 4{} self-dual, same as CDel node h.pngCDel 6.pngCDel 4node.png
representation as {3,4,3}
G14
3[8]2
144 24 3(144)2 3{8}2 CDel 3node 1.pngCDel 8.pngCDel node.png 72 48 3{} same as CDel 3node 1.pngCDel 4.pngCDel 3node 1.png
3{8/3}2 CDel 3node 1.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel node.png starry polygon, same as CDel 3node 1.pngCDel 4.pngCDel rat.pngCDel 3x.pngCDel 3node 1.png
2(144)3 2{8}3 CDel node 1.pngCDel 8.pngCDel 3node.png 48 72 {}
2{8/3}3 CDel node 1.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel 3node.png starry polygon
G9
4[6]2
192 24 4(192)2 4{6}2 CDel 4node 1.pngCDel 6.pngCDel node.png 96 48 4{} same as CDel 4node 1.pngCDel 3.pngCDel 4node 1.png
2(192)4 2{6}4 CDel node 1.pngCDel 6.pngCDel 4node.png 48 96 {}
4{3}2 CDel 4node 1.pngCDel 3.pngCDel node.png 96 48 {} starry polygon
2{3}4 CDel node 1.pngCDel 3.pngCDel 4node.png 48 96 {} starry polygon
G10
4[4]3
288 24 4(288)3 4{4}3 CDel 4node 1.pngCDel 4.pngCDel 3node.png 96 72 4{}
12 4{8/3}3 CDel 4node 1.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel 3node.png starry polygon
24 3(288)4 3{4}4 CDel 3node 1.pngCDel 4.pngCDel 4node.png 72 96 3{}
12 3{8/3}4 CDel 3node 1.pngCDel 8.pngCDel rat.pngCDel 3x.pngCDel 4node.png starry polygon
G20
3[5]3
360 30 3(360)3 3{5}3 CDel 3node 1.pngCDel 5.pngCDel 3node.png 120 120 3{} self-dual, same as CDel node h.pngCDel 10.pngCDel 3node.png
representation as {3,3,5}
3{5/2}3 CDel 3node 1.pngCDel 5-2.pngCDel 3node.png self-dual, starry polygon
G16
5[3]5
600 30 5(600)5 5{3}5 CDel 5node 1.pngCDel 3.pngCDel 5node.png 120 120 5{} self-dual, same as CDel node h.pngCDel 6.pngCDel 5node.png
representation as {3,3,5}
10 5{5/2}5 CDel 5node 1.pngCDel 5-2.pngCDel 5node.png self-dual, starry polygon
G21
3[10]2
720 60 3(720)2 3{10}2 CDel 3node 1.pngCDel 10.pngCDel node.png 360 240 3{} same as CDel 3node 1.pngCDel 5.pngCDel 3node 1.png
3{5}2 CDel 3node 1.pngCDel 5.pngCDel node.png starry polygon
3{10/3}2 CDel 3node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node.png starry polygon, same as CDel 3node 1.pngCDel 5.pngCDel rat.pngCDel 3x.pngCDel 3node 1.png
3{5/2}2 CDel 3node 1.pngCDel 5-2.pngCDel node.png starry polygon
2(720)3 2{10}3 CDel node 1.pngCDel 10.pngCDel 3node.png 240 360 {}
2{5}3 CDel node 1.pngCDel 5.pngCDel 3node.png starry polygon
2{10/3}3 CDel node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel 3node.png starry polygon
2{5/2}3 CDel node 1.pngCDel 5-2.pngCDel 3node.png starry polygon
G17
5[6]2
1200 60 5(1200)2 5{6}2 CDel 5node 1.pngCDel 6.pngCDel node.png 600 240 5{} same as CDel 5node 1.pngCDel 3.pngCDel 5node 1.png
representation as {5,3,3}
20 5{5}2 CDel 5node 1.pngCDel 5.pngCDel node.png starry polygon
20 5{10/3}2 CDel 5node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel node.png starry polygon
60 5{3}2 CDel 5node 1.pngCDel 3.pngCDel node.png starry polygon
60 2(1200)5 2{6}5 CDel node 1.pngCDel 6.pngCDel 5node.png 240 600 {}
20 2{5}5 CDel node 1.pngCDel 5.pngCDel 5node.png starry polygon
20 2{10/3}5 CDel node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel 5node.png starry polygon
60 2{3}5 CDel node 1.pngCDel 3.pngCDel 5node.png starry polygon
G18
5[4]3
1800 60 5(1800)3 5{4}3 CDel 5node 1.pngCDel 4.pngCDel 3node.png 600 360 5{} representation as {5,3,3}
15 5{10/3}3 CDel 5node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel 3node.png starry polygon
30 5{3}3 CDel 5node 1.pngCDel 3.pngCDel 3node.png starry polygon
30 5{5/2}3 CDel 5node 1.pngCDel 5-2.pngCDel 3node.png starry polygon
60 3(1800)5 3{4}5 CDel 3node 1.pngCDel 4.pngCDel 5node.png 360 600 3{}
15 3{10/3}5 CDel 3node 1.pngCDel 10.pngCDel rat.pngCDel 3x.pngCDel 5node.png starry polygon
30 3{3}5 CDel 3node 1.pngCDel 3.pngCDel 5node.png starry polygon
30 3{5/2}5 CDel 3node 1.pngCDel 5-2.pngCDel 5node.png starry polygon

Visualizations of regular complex polygons[edit]

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[edit]

In general, a regular complex polytope is represented by Coxeter as p{z1}q{z2}r{z3}s… or Coxeter diagram CDel pnode 1.pngCDel 3.pngCDel z.pngCDel 1x.pngCDel 3.pngCDel qnode.pngCDel 3.pngCDel z.pngCDel 2x.pngCDel 3.pngCDel rnode.pngCDel 3.pngCDel z.pngCDel 3x.pngCDel 3.pngCDel snode.png…, having symmetry p[z1]q[z2]r[z3]s… or CDel pnode.pngCDel 3.pngCDel z.pngCDel 1x.pngCDel 3.pngCDel qnode.pngCDel 3.pngCDel z.pngCDel 2x.pngCDel 3.pngCDel rnode.pngCDel 3.pngCDel z.pngCDel 3x.pngCDel 3.pngCDel snode.png….[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 CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png. 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 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel 3.pngCDel node.pngCDel 4.pngCDel pnode.png.[21]

A 1-dimensional regular complex polytope in is represented as CDel pnode 1.png, 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 CDel pnode.png, a cyclic group of order p. In a higher polytope, p{} or CDel pnode 1.png represents a p-edge element, with a 2-edge, {} or CDel node 1.png, 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[edit]

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

Coxeter enumerated this list of nonstarry regular complex polyhedra in , including the 5 platonic solids in .[24]

A regular complex polyhedron, p{n1}q{n2}r or CDel pnode 1.pngCDel 3.pngCDel n.pngCDel 1x.pngCDel 3.pngCDel qnode.pngCDel 3.pngCDel n.pngCDel 2x.pngCDel 3.pngCDel rnode.png, has CDel pnode 1.pngCDel 3.pngCDel n.pngCDel 1x.pngCDel 3.pngCDel qnode.png faces, CDel pnode 1.png edges, and CDel qnode 1.pngCDel 3.pngCDel n.pngCDel 2x.pngCDel 3.pngCDel rnode.png vertex figures.

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

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

Space Group Order Coxeter number Polygon Vertices Edges Faces Vertex
figure
Van Oss
polygon
Notes
G(1,1,3)
2[3]2[3]2
= [3,3]
24 4 α3 = 2{3}2{3}2
= {3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 4 6 {} 4 {3} {3} none Real tetrahedron
Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
G23
2[3]2[5]2
= [3,5]
120 10 2{3}2{5}2 = {3,5} CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 12 30 {} 20 {3} {5} none Real icosahedron
2{5}2{3}2 = {5,3} CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png 20 30 {} 12 {5} {3} none Real dodecahedron
G(2,1,3)
2[3]2[4]2
= [3,4]
48 6 β2
3
= β3 = {3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png 6 12 {} 8 {3} {4} {4} Real octahedron
Same as {}+{}+{}, order 8
Same as CDel node 1.pngCDel split1.pngCDel nodes.png, order 24
γ2
3
= γ3 = {4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 8 12 {} 6 {4} {3} none Real cube
Same as {}×{}×{} or CDel node 1.pngCDel 2c.pngCDel node 1.pngCDel 2c.pngCDel node 1.png
G(p,1,3)
2[3]2[4]p
p=2,3,4,...
6p3 3p βp
3
= 2{3}2{4}p
          
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel pnode.png
3p 3p2 {} p3 {3} 2{4}p 2{4}p Generalized octahedron
Same as p{}+p{}+p{}, order p3
Same as CDel node 1.pngCDel 3split1.pngCDel branch.pngCDel labelp.png, order 6p2
γp
3
= p{4}2{3}2
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png p3 3p2 p{} 3p p{4}2 {3} none Generalized cube
Same as p{}×p{}×p{} or CDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.png
G(3,1,3)
2[3]2[4]3
162 9 β3
3
= 2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png 9 27 {} 27 {3} 2{4}3 2{4}3 Same as 3{}+3{}+3{}, order 27
Same as CDel node 1.pngCDel 3split1.pngCDel branch.png, order 54
γ3
3
= 3{4}2{3}2
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 27 27 3{} 9 3{4}2 {3} none Same as 3{}×3{}×3{} or CDel 3node 1.pngCDel 2c.pngCDel 3node 1.pngCDel 2c.pngCDel 3node 1.png
G(4,1,3)
2[3]2[4]4
384 12 β4
3
= 2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png 12 48 {} 64 {3} 2{4}4 2{4}4 Same as 4{}+4{}+4{}, order 64
Same as CDel node 1.pngCDel 3split1.pngCDel branch.pngCDel label4.png, order 96
γ4
3
= 4{4}2{3}2
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 64 48 4{} 12 4{4}2 {3} none Same as 4{}×4{}×4{} or CDel 4node 1.pngCDel 2c.pngCDel 4node 1.pngCDel 2c.pngCDel 4node 1.png
G(5,1,3)
2[3]2[4]5
750 15 β5
3
= 2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png 15 75 {} 125 {3} 2{4}5 2{4}5 Same as 5{}+5{}+5{}, order 125
Same as CDel node 1.pngCDel 3split1.pngCDel branch.pngCDel label5.png, order 150
γ5
3
= 5{4}2{3}2
CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 125 75 5{} 15 5{4}2 {3} none Same as 5{}×5{}×5{} or CDel 5node 1.pngCDel 2c.pngCDel 5node 1.pngCDel 2c.pngCDel 5node 1.png
G(6,1,3)
2[3]2[4]6
1296 18 β6
3
= 2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png 36 108 {} 216 {3} 2{4}6 2{4}6 Same as 6{}+6{}+6{}, order 216
Same as CDel node 1.pngCDel 3split1.pngCDel branch.pngCDel label6.png, order 216
γ6
3
= 6{4}2{3}2
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 216 108 6{} 18 6{4}2 {3} none Same as 6{}×6{}×6{} or CDel 6node 1.pngCDel 2c.pngCDel 6node 1.pngCDel 2c.pngCDel 6node 1.png
G25
3[3]3[3]3
648 9 3{3}3{3}3 CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png 27 72 3{} 27 3{3}3 3{3}3 3{4}2 Same as CDel node h.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.png.
representation as 221
Hessian polyhedron
G26
2[4]3[3]3
1296 18 2{4}3{3}3 CDel node 1.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.png 54 216 {} 72 2{4}3 3{3}3 {6}
3{3}3{4}2 CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png 72 216 3{} 54 3{3}3 3{4}2 3{4}3 Same as CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png[25]
representation as 122

Visualizations of regular complex polyhedra[edit]

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

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

Generalized cubes

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

Enumeration of regular complex 4-polytopes[edit]

Coxeter enumerated this list of nonstarry regular complex 4-polytopes in , including the 6 convex regular 4-polytopes in .[30]

Space Group Order Coxeter
number
Polytope Vertices Edges Faces Cells Van Oss
polygon
Notes
G(1,1,4)
2[3]2[3]2[3]2
= [3,3,3]
120 5 α4 = 2{3}2{3}2{3}2
= {3,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
5 10
{}
10
{3}
5
{3,3}
none Real 5-cell (simplex)
G28
2[3]2[4]2[3]2
= [3,4,3]
1152 12 2{3}2{4}2{3}2 = {3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
24 96
{}
96
{3}
24
{3,4}
{6} Real 24-cell
G30
2[3]2[3]2[5]2
= [3,3,5]
14400 30 2{3}2{3}2{5}2 = {3,3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png
120 720
{}
1200
{3}
600
{3,3}
{10} Real 600-cell
2{5}2{3}2{3}2 = {5,3,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
600 1200
{}
720
{5}
120
{5,3}
Real 120-cell
G(2,1,4)
2[3]2[3]2[4]p
=[3,3,4]
384 8 β2
4
= β4 = {3,3,4}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8 24
{}
32
{3}
16
{3,3}
{4} Real 16-cell
Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png, order 192
γ2
4
= γ4 = {4,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
16 32
{}
24
{4}
8
{4,3}
none Real tesseract
Same as {}4 or CDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.png, order 16
G(p,1,4)
2[3]2[3]2[4]p
p=2,3,4,...
24p4 4p βp
4
= 2{3}2{3}2{4}p
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel pnode.png
4p 6p2
{}
4p3
{3}
p4
{3,3}
2{4}p Generalized 4-orthoplex
Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel labelp.png, order 24p3
γp
4
= p{4}2{3}2{3}2
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
p4 4p3
p{}
6p2
p{4}2
4p
p{4}2{3}2
none Generalized tesseract
Same as p{}4 or CDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.png, order p4
G(3,1,4)
2[3]2[3]2[4]3
1944 12 β3
4
= 2{3}2{3}2{4}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
12 54
{}
108
{3}
81
{3,3}
2{4}3 Generalized 4-orthoplex
Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.png, order 648
γ3
4
= 3{4}2{3}2{3}2
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
81 108
3{}
54
3{4}2
12
3{4}2{3}2
none Same as 3{}4 or CDel 3node 1.pngCDel 2c.pngCDel 3node 1.pngCDel 2c.pngCDel 3node 1.pngCDel 2c.pngCDel 3node 1.png, order 81
G(4,1,4)
2[3]2[3]2[4]4
6144 16 β4
4
= 2{3}2{3}2{4}4
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
16 96
{}
256
{3}
64
{3,3}
2{4}4 Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel label4.png, order 1536
γ4
4
= 4{4}2{3}2{3}2
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
256 256
4{}
96
4{4}2
16
4{4}2{3}2
none Same as 4{}4 or CDel 4node 1.pngCDel 2c.pngCDel 4node 1.pngCDel 2c.pngCDel 4node 1.pngCDel 2c.pngCDel 4node 1.png, order 256
G(5,1,4)
2[3]2[3]2[4]5
15000 20 β5
4
= 2{3}2{3}2{4}5
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 5node.png
20 150
{}
500
{3}
625
{3,3}
2{4}5 Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel label5.png, order 3000
γ5
4
= 5{4}2{3}2{3}2
CDel 5node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
625 500
5{}
150
5{4}2
20
5{4}2{3}2
none Same as 5{}4 or CDel 5node 1.pngCDel 2c.pngCDel 5node 1.pngCDel 2c.pngCDel 5node 1.pngCDel 2c.pngCDel 5node 1.png, order 625
G(6,1,4)
2[3]2[3]2[4]6
31104 24 β6
4
= 2{3}2{3}2{4}6
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
24 216
{}
864
{3}
1296
{3,3}
2{4}6 Same as CDel node 1.pngCDel 3.pngCDel node.pngCDel 3split1.pngCDel branch.pngCDel label6.png, order 5184
γ6
4
= 6{4}2{3}2{3}2
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
1296 864
6{}
216
6{4}2
24
6{4}2{3}2
none Same as 6{}4 or CDel 6node 1.pngCDel 2c.pngCDel 6node 1.pngCDel 2c.pngCDel 6node 1.pngCDel 2c.pngCDel 6node 1.png, order 1296
G32
3[3]3[3]3[3]3
155520 30 3{3}3{3}3{3}3
CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
240 2160
3{}
2160
3{3}3
240
3{3}3{3}3
3{4}3 Witting polytope
representation as 421

Visualizations of regular complex 4-polytopes[edit]

Generalized 4-orthoplexes

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

Generalized 4-cubes

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

Enumeration of regular complex 5-polytopes[edit]

Regular complex 5-polytopes in or higher exist in three families, the real simplexes and the generalized hypercube, and orthoplex.

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

Visualizations of regular complex 5-polytopes[edit]

Generalized 5-orthoplexes

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

Generalized 5-cubes

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

Enumeration of regular complex 6-polytopes[edit]

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

Visualizations of regular complex 6-polytopes[edit]

Generalized 6-orthoplexes

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

Generalized 6-cubes

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

Enumeration of regular complex apeirotopes[edit]

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 , or if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2.[32]

Each has proportional element counts given as:

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

Regular complex 1-polytopes[edit]

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

Regular complex apeirogons[edit]

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

Rank 2 complex apeirogons have symmetry p[q]r, where 1/p + 2/q + 1/r = 1. Coxeter expresses them as 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
CDel node.pngCDel infin.pngCDel node.png CDel 3node.pngCDel 12.pngCDel node.png CDel 4node.pngCDel 8.pngCDel node.png CDel 6node.pngCDel 6.pngCDel node.png CDel 3node.pngCDel 6.pngCDel 3node.png CDel 6node.pngCDel 4.pngCDel 3node.png CDel 4node.pngCDel 4.pngCDel 4node.png CDel 6node.pngCDel 3.pngCDel 6node.png

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

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 CDel pnode 1.pngCDel 2x.pngCDel q.pngCDel node.png is the same as quasiregular CDel pnode 1.pngCDel q.pngCDel pnode 1.png.[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 , there are 3 more infinite solutions: [2], [4]2, [3]3, and CDel infinnode 1.pngCDel 2.pngCDel infinnode 1.png, CDel infinnode 1.pngCDel 4.pngCDel node.png, and CDel infinnode 1.pngCDel 3.pngCDel 3node.png. The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in .

Rank 2
Space Group Apeirogon Edge rep.[35] Picture Notes
2[∞]2 = [∞] δ2,2
2
= {∞}
       
CDel node 1.pngCDel infin.pngCDel node.png
{} Regular apeirogon.png Real apeirogon
Same as CDel node 1.pngCDel infin.pngCDel node 1.png
/ [4]2 {4}2 CDel infinnode 1.pngCDel 4.pngCDel node.png {} {4,4} Complex polygon i-4-2.png Same as CDel infinnode 1.pngCDel 2.pngCDel infinnode 1.png Truncated complex polygon i-2-i.png
[3]3 {3}3 CDel infinnode 1.pngCDel 3.pngCDel 3node.png {} {3,6} Complex apeirogon 2-6-6.png Same as CDel infinnode 1.pngCDel split1.pngCDel branch 11.pngCDel label-ii.png Truncated complex polygon i-3-i-3-i-3-.png
p[q]r δp,r
2
= p{q}r
CDel pnode 1.pngCDel q.pngCDel rnode.png p{}
3[12]2 δ3,2
2
= 3{12}2
CDel 3node 1.pngCDel 12.pngCDel node.png 3{} r{3,6} Complex apeirogon 3-12-2.png Same as CDel 3node 1.pngCDel 6.pngCDel 3node 1.png Truncated complex polygon 3-6-3.png
δ2,3
2
= 2{12}3
CDel node 1.pngCDel 12.pngCDel 3node.png {} {6,3} Complex apeirogon 2-12-3.png
3[6]3 δ3,3
2
= 3{6}3
CDel 3node 1.pngCDel 6.pngCDel 3node.png 3{} {3,6} Complex apeirogon 3-6-3.png Same as CDel node h.pngCDel 12.pngCDel 3node.png
4[8]2 δ4,2
2
= 4{8}2
CDel 4node 1.pngCDel 8.pngCDel node.png 4{} {4,4} Complex apeirogon 4-8-2.png Same as CDel 4node 1.pngCDel 4.pngCDel 4node 1.png Truncated complex polygon 4-4-4.png
δ2,4
2
= 2{8}4
CDel node 1.pngCDel 8.pngCDel 4node.png {} {4,4} Complex apeirogon 2-8-4.png
4[4]4 δ4,4
2
= 4{4}4
CDel 4node 1.pngCDel 4.pngCDel 4node.png 4{} {4,4} Complex apeirogon 4-4-4.png Same as CDel node h.pngCDel 8.pngCDel 4node.png
6[6]2 δ6,2
2
= 6{6}2
CDel 6node 1.pngCDel 6.pngCDel node.png 6{} r{3,6} Complex apeirogon 6-6-2.png Same as CDel 6node 1.pngCDel 3.pngCDel 6node 1.png
δ2,6
2
= 2{6}6
CDel node 1.pngCDel 6.pngCDel 6node.png {} {3,6} Complex apeirogon 2-6-6.png
6[4]3 δ6,3
2
= 6{4}3
CDel 6node 1.pngCDel 4.pngCDel 3node.png 6{} {6,3} Complex apeirogon 6-4-3.png
δ3,6
2
= 3{4}6
CDel 3node 1.pngCDel 4.pngCDel 6node.png 3{} {3,6} Complex apeirogon 3-4-6.png
6[3]6 δ6,6
2
= 6{3}6
CDel 6node 1.pngCDel 3.pngCDel 6node.png 6{} {3,6} Complex apeirogon 6-3-6.png Same as CDel node h.pngCDel 6.pngCDel 6node.png

Regular complex apeirohedra[edit]

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.

CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel qnode.png is the same as CDel pnode 1.pngCDel 3split1-44.pngCDel branch.pngCDel labelq.png, as well as CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png, for p,r=2,3,4,6. Also CDel pnode 1.pngCDel 4.pngCDel pnode.pngCDel 4.pngCDel node.png = CDel pnode.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel pnode.png.[36]

Rank 3
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
2[3]2[4] {4}2{3}2 CDel infinnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png {} {4}2 Same as {}×{}×{} or CDel infinnode 1.pngCDel 2c.pngCDel infinnode 1.pngCDel 2c.pngCDel infinnode 1.png
Real representation {4,3,4}
p[4]2[4]r p{4}2{4}r            
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel qnode.png
p2 2pq p{} r2 p{4}2 2{q}r Same as CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png, p,r=2,3,4,6
[4,4] δ2,2
3
= {4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 4 8 {} 4 {4} {∞} Real square tiling
Same as CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png or CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png or CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node 1.png
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
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 3node.png
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 3node.png
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 4node.png
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 4node.png
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 6node.png
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 3node.png
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 6node.png
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 6node.png
9
4
9
16
4
16
36
4
36
9
36
12
12
18
16
16
32
24
24
36
36
72
3{}
{}
3{}
4{}
{}
4{}
6{}
{}
6{}
3{}
6{}
4
9
9
4
16
16
4
36
9
36
36
3{4}2
{4}
3{4}2
4{4}2
{4}
4{4}2
6{4}2
{4}
6{4}2
3{4}2
6{4}2
p{q}r Same as CDel 3node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel 3node 1.pngCDel 12.pngCDel node.png or CDel 3node 1.pngCDel 6.pngCDel 3node 1.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node 1.png or CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 3node 1.png
Same as CDel node 1.pngCDel 12.pngCDel 3node.pngCDel 2.pngCDel node 1.pngCDel 12.pngCDel 3node.png
Same as CDel 3node 1.pngCDel 6.pngCDel 3node.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node.png
Same as CDel 4node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel 4node 1.pngCDel 8.pngCDel node.png or CDel 4node 1.pngCDel 4.pngCDel 4node 1.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node 1.png or CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 4node 1.png
Same as CDel node 1.pngCDel 8.pngCDel 4node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel 4node.png
Same as CDel 4node 1.pngCDel 4.pngCDel 4node.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node.png
Same as CDel 6node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel 6node 1.pngCDel 6.pngCDel node.png or CDel 6node 1.pngCDel 3.pngCDel 6node 1.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node 1.png or CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 6node 1.png
Same as CDel node 1.pngCDel 6.pngCDel 6node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel 6node.png
Same as CDel 6node 1.pngCDel 4.pngCDel 3node.pngCDel 2.pngCDel 6node 1.pngCDel 4.pngCDel 3node.png
Same as CDel 3node 1.pngCDel 4.pngCDel 6node.pngCDel 2.pngCDel 3node 1.pngCDel 4.pngCDel 6node.png
Same as CDel 6node 1.pngCDel 3.pngCDel 6node.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node.png
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
2[4]r[4]2 2{4}r{4}2            
CDel node 1.pngCDel 4.pngCDel rnode.pngCDel 4.pngCDel node.png
2 {} 2 p{4}2' 2{4}r Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel rnode.png and CDel rnode.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel rnode.png, r=2,3,4,6
[4,4] {4,4} CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 2 4 {} 2 {4} {∞} Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png and CDel node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel node.png
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
CDel node 1.pngCDel 4.pngCDel 3node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel 4node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 4.pngCDel 6node.pngCDel 4.pngCDel node.png
2 9
16
36
{} 2 2{4}3
2{4}4
2{4}6
2{q}r Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 3node.png and CDel 3node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel 3node.png
Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 4node.png and CDel 4node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel 4node.png
Same as CDel node h.pngCDel 4.pngCDel node.pngCDel 4.pngCDel 6node.png and CDel 6node.pngCDel 4.pngCDel node 1.pngCDel 4.pngCDel 6node.png[37]
Space Group Apeirohedron Vertex Edge Face van Oss
apeirogon
Notes
2[6]2[3]2
= [6,3]
{3,6}            
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png
1 3 {} 2 {3} {∞} Real triangular tiling
{6,3} CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png 2 3 {} 1 {6} none Real hexagonal tiling
3[4]3[3]3 3{3}3{4}3 CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel 3node.png 1 8 3{} 3 3{3}3 3{4}6 Same as CDel 3node 1.pngCDel 3split1.pngCDel branch.pngCDel label-33.png
3{4}3{3}3 CDel 3node 1.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.png 3 8 3{} 2 3{4}3 3{12}2
4[3]4[3]4 4{3}4{3}4 CDel 4node 1.pngCDel 3.pngCDel 4node.pngCDel 3.pngCDel 4node.png 1 6 4{} 1 4{3}4 4{4}4 Self-dual, same as CDel node h.pngCDel 4.pngCDel 4node.pngCDel 3.pngCDel 4node.png
4[3]4[4]2 4{3}4{4}2 CDel 4node 1.pngCDel 3.pngCDel 4node.pngCDel 4.pngCDel node.png 1 12 4{} 3 4{3}4 2{8}4 Same as CDel 4node.pngCDel 3.pngCDel 4node 1.pngCDel 3.pngCDel 4node.png
2{4}4{3}4 CDel node 1.pngCDel 4.pngCDel 4node.pngCDel 3.pngCDel 4node.png 3 12 {} 1 2{4}4 4{4}4

Regular complex 3-apeirotopes[edit]

There are 16 regular complex apeirotopes in . 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: CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png = CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png. The first case is the cubic honeycomb.

Rank 4
Space Group 3-apeirotope Vertex Edge Face Cell van Oss
apeirogon
Notes
p[4]2[3]2[4]r δp,r
3
= p{4}2{3}2{4}r
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png
p{} p{4}2 p{4}2{3}2 p{q}r Same as CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png
2[4]2[3]2[4]2
=[4,3,4]
δ2,2
3
= 2{4}2{3}2{4}2
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{} {4} {4,3} Cubic honeycomb
Same as CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png or CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node 1.png or CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node 1.png
3[4]2[3]2[4]2 δ3,2
3
= 3{4}2{3}2{4}2
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
3{} 3{4}2 3{4}2{3}2 Same as CDel 3node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel 3node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel 3node 1.pngCDel 12.pngCDel node.png or CDel 3node 1.pngCDel 12.pngCDel node.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node 1.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node 1.png or CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node 1.png
δ2,3
3
= 2{4}2{3}2{4}3
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
{} {4} {4,3} Same as CDel node 1.pngCDel 12.pngCDel 3node.pngCDel 2.pngCDel node 1.pngCDel 12.pngCDel 3node.pngCDel 2.pngCDel node 1.pngCDel 12.pngCDel 3node.png
3[4]2[3]2[4]3 δ3,3
3
= 3{4}2{3}2{4}3
CDel 3node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
3{} 3{4}2 3{4}2{3}2 Same as CDel 3node 1.pngCDel 6.pngCDel 3node.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node.pngCDel 2.pngCDel 3node 1.pngCDel 6.pngCDel 3node.png
4[4]2[3]2[4]2 δ4,2
3
= 4{4}2{3}2{4}2
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
4{} 4{4}2 4{4}2{3}2 Same as CDel 4node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel 4node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel 4node 1.pngCDel 8.pngCDel node.png or CDel 4node 1.pngCDel 8.pngCDel node.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node 1.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node 1.png or CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node 1.png
δ2,4
3
= 2{4}2{3}2{4}4
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
{} {4} {4,3} Same as CDel node 1.pngCDel 8.pngCDel 4node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel 4node.pngCDel 2.pngCDel node 1.pngCDel 8.pngCDel 4node.png
4[4]2[3]2[4]4 δ4,4
3
= 4{4}2{3}2{4}4
CDel 4node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 4node.png
4{} 4{4}2 4{4}2{3}2 Same as CDel 4node 1.pngCDel 4.pngCDel 4node.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node.pngCDel 2.pngCDel 4node 1.pngCDel 4.pngCDel 4node.png
6[4]2[3]2[4]2 δ6,2
3
= 6{4}2{3}2{4}2
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
6{} 6{4}2 6{4}2{3}2 Same as CDel 6node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel 6node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel 6node 1.pngCDel 6.pngCDel node.png or CDel 6node 1.pngCDel 3.pngCDel 6node 1.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node 1.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node 1.png or CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node 1.png
δ2,6
3
= 2{4}2{3}2{4}6
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
{} {4} {4,3} Same as CDel node 1.pngCDel 6.pngCDel 6node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel 6node.pngCDel 2.pngCDel node 1.pngCDel 6.pngCDel 6node.png
6[4]2[3]2[4]3 δ6,3
3
= 6{4}2{3}2{4}3
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
6{} 6{4}2 6{4}2{3}2 Same as CDel 6node 1.pngCDel 4.pngCDel 3node.pngCDel 2.pngCDel 6node 1.pngCDel 4.pngCDel 3node.pngCDel 2.pngCDel 6node 1.pngCDel 4.pngCDel 3node.png
δ3,6
3
= 3{4}2{3}2{4}6
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.png
3{} 3{4}2 3{4}2{3}2 Same as CDel 6node 1.pngCDel 4.pngCDel 3node.pngCDel 2.pngCDel 6node 1.pngCDel 4.pngCDel 3node.pngCDel 2.pngCDel 6node 1.pngCDel 4.pngCDel 3node.png
6[4]2[3]2[4]6 δ6,6
3
= 6{4}2{3}2{4}6
CDel 6node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 6node.png
6{} 6{4}2 6{4}2{3}2 Same as CDel 6node 1.pngCDel 3.pngCDel 6node.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node.pngCDel 2.pngCDel 6node 1.pngCDel 3.pngCDel 6node.png
Rank 4, exceptional cases
Space Group 3-apeirotope Vertex Edge Face Cell van Oss
apeirogon
Notes
2[4]3[3]3[3]3 3{3}3{3}3{4}2
CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png
1 24 3{} 27 3{3}3 2 3{3}3{3}3 3{4}6 Same as CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel split1.pngCDel nodes.pngCDel label-33.png
2{4}3{3}3{3}3
CDel node 1.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
2 27 {} 24 2{4}3 1 2{4}3{3}3 2{12}3
2[3]2[4]3[3]3 2{3}2{4}3{3}3
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel 3node.pngCDel 3.pngCDel 3node.png
1 27 {} 72 2{3}2 8 2{3}2{4}3 2{6}6
3{3}3{4}2{3}2
CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
8 72 3{} 27 3{3}3 1 3{3}3{4}2 3{6}3 Same as CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel split1.pngCDel nodes.pngCDel label-33.png or CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png

Regular complex 4-apeirotopes[edit]

There are 15 regular complex apeirotopes in . 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: CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png = CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png. The first case is the 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
p[4]2[3]2[3]2[4]r δp,r
4
= p{4}2{3}2{3}2{4}r
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png
p{} p{4}2 p{4}2{3}2 p{4}2{3}2{3}2 p{q}r Same as CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png
2[4]2[3]2[3]2[4]2 δ2,2
4
= {4,3,3,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{} {4} {4,3} {4,3,3} {∞} Tesseractic honeycomb
Same as CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png
2[4]2[3]2[3]2[3]2
=[3,4,3,3]
{3,3,4,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
1 12 {} 32 {3} 24 {3,3} 3 {3,3,4} Real 16-cell honeycomb
Same as CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel split1.pngCDel nodes.png
{3,4,3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
3 24 {} 32 {3} 12 {3,4} 1 {3,4,3} Real 24-cell honeycomb
Same as CDel nodes.pngCDel split2.pngCDel node 1.pngCDel split1.pngCDel nodes.png or CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
3[3]3[3]3[3]3[3]3 3{3}3{3}3{3}3{3}3
CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png
1 80 3{} 270 3{3}3 80 3{3}3{3}3 1 3{3}3{3}3{3}3 3{4}6 representation 521

Regular complex 5-apeirotopes and higher[edit]

There are only 12 regular complex apeirotopes in 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: CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png ... CDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png = CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png ... CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png. The first case is the real hypercube honeycomb.

Rank 6
Space Group 5-apeirotopes Vertices Edge Face Cell 4-face 5-face van Oss
apeirogon
Notes
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
CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel rnode.png
p{} p{4}2 p{4}2{3}2 p{4}2{3}2{3}2 p{4}2{3}2{3}2{3}2 p{q}r Same as CDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.pngCDel 2.pngCDel pnode 1.pngCDel q.pngCDel rnode.png
2[4]2[3]2[3]2[3]2[4]2
=[4,3,3,3,4]
δ2,2
5
= {4,3,3,3,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
{} {4} {4,3} {4,3,3} {4,3,3,3} {∞} 5-cubic honeycomb
Same as CDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.pngCDel 2.pngCDel node 1.pngCDel infin.pngCDel node.png

van Oss polygon[edit]

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 , or unitary plane ) 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[edit]

Product complex polytopes[edit]

Example product complex polytope
Complex polygon 2x5 stereographic3.png
Complex product polygon CDel node 1.pngCDel 2.pngCDel 5node 1.png or {}×5{} has 10 vertices connected by 5 2-edges and 2 5-edges, with its real representation as a 3-dimensional pentagonal prism.
Dual complex polygon 2x5 perspective.png
The dual polygon,{}+5{} has 7 vertices centered on the edges of the original, connected by 10 edges. Its real representation is a pentagonal bipyramid.

Some complex polytopes can be represented as Cartesian products. These product polytopes are not strictly regular since they'll have more than one facet type, but some can represent lower symmetry of regular forms if all the orthogonal polytopes are identical. For example, the product p{}×p{} or CDel pnode 1.pngCDel 2.pngCDel pnode 1.png of two 1-dimensional polytopes is the same as the regular p{4}2 or CDel pnode 1.pngCDel 4.pngCDel node.png. More general products, like p{}×q{} have real representations as the 4-dimensional p-q duoprisms. The dual of a product polytope can be written as a sum p{}+q{} and have real representations as the 4-dimensional p-q duopyramid. The p{}+p{} can have its symmetry doubled as a regular complex polytope 2{4}p or CDel node 1.pngCDel 4.pngCDel pnode.png.

Similarly, a complex polyhedron can be constructed as a triple product: p{}×p{}×p{} or CDel pnode 1.pngCDel 2c.pngCDel pnode 1.pngCDel 2c.pngCDel pnode 1.png is the same as the regular generalized cube, p{4}2{3}2 or CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, as well as product p{4}2×p{} or CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 2.pngCDel pnode 1.png.[40]

Quasiregular polygons[edit]

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

Example quasiregular polygons
p[q]r 2[4]2 3[4]2 4[4]2 5[4]2 6[4]2 7[4]2 8[4]2 3[3]3 3[4]3
Regular
CDel pnode 1.pngCDel q.pngCDel rnode.png
2-generalized-2-cube.svg
CDel node 1.pngCDel 4.pngCDel node.png
4 2-edges
3-generalized-2-cube skew.svg
CDel 3node 1.pngCDel 4.pngCDel node.png
9 3-edges
4-generalized-2-cube.svg
CDel 4node 1.pngCDel 4.pngCDel node.png
16 4-edges
5-generalized-2-cube skew.svg
CDel 5node 1.pngCDel 4.pngCDel node.png
25 5-edges
6-generalized-2-cube.svg
CDel 6node 1.pngCDel 4.pngCDel node.png
36 6-edges
7-generalized-2-cube skew.svg
CDel 7node 1.pngCDel 4.pngCDel node.png
49 8-edges
8-generalized-2-cube.svg
CDel 8node 1.pngCDel 4.pngCDel node.png
64 8-edges
Complex polygon 3-3-3.png
CDel 3node 1.pngCDel 3.pngCDel 3node.png
Complex polygon 3-4-3.png
CDel 3node 1.pngCDel 4.pngCDel 3node.png
Quasiregular
CDel pnode 1.pngCDel q.pngCDel rnode 1.png
Truncated 2-generalized-square.svg
CDel node 1.pngCDel 4.pngCDel node 1.png = CDel node 1.pngCDel 8.pngCDel node.png
4+4 2-edges
Truncated 3-generalized-square skew.svg
CDel 3node 1.pngCDel 4.pngCDel node 1.png
6 2-edges
9 3-edges
Truncated 4-generalized-square.svg
CDel 4node 1.pngCDel 4.pngCDel node 1.png
8 2-edges
16 4-edges
Truncated 5-generalized-square skew.svg
CDel 5node 1.pngCDel 4.pngCDel node 1.png
10 2-edges
25 5-edges
Truncated 6-generalized-square.svg
CDel 6node 1.pngCDel 4.pngCDel node 1.png
12 2-edges
36 6-edges
Truncated 7-generalized-square skew.svg
CDel 7node 1.pngCDel 4.pngCDel node 1.png
14 2-edges
49 7-edges
Truncated 8-generalized-square.svg
CDel 8node 1.pngCDel 4.pngCDel node 1.png
16 2-edges
64 8-edges
Complex polygon 3-6-2.png
CDel 3node 1.pngCDel 3.pngCDel 3node 1.png = CDel 3node 1.pngCDel 6.pngCDel node.png
Complex polygon 3-8-2.png
CDel 3node 1.pngCDel 4.pngCDel 3node 1.png = CDel 3node 1.pngCDel 8.pngCDel node.png
Regular
CDel pnode 1.pngCDel q.pngCDel rnode.png
2-generalized-2-orthoplex.svg
CDel node 1.pngCDel 4.pngCDel node.png
4 2-edges
3-generalized-2-orthoplex skew.svg
CDel node 1.pngCDel 4.pngCDel 3node.png
6 2-edges
3-generalized-2-orthoplex.svg
CDel node 1.pngCDel 4.pngCDel 4node.png
8 2-edges
5-generalized-2-orthoplex skew.svg
CDel node 1.pngCDel 4.pngCDel 5node.png
10 2-edges
6-generalized-2-orthoplex.svg
CDel node 1.pngCDel 4.pngCDel 6node.png
12 2-edges
7-generalized-2-orthoplex skew.svg
CDel node 1.pngCDel 4.pngCDel 7node.png
14 2-edges
8-generalized-2-orthoplex.svg
CDel node 1.pngCDel 4.pngCDel 8node.png
16 2-edges
Complex polygon 3-3-3.png
CDel 3node 1.pngCDel 3.pngCDel 3node.png
Complex polygon 3-4-3.png
CDel 3node 1.pngCDel 4.pngCDel 3node.png

Quasiregular apeirogons[edit]

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: CDel pnode 1.pngCDel q.pngCDel pnode 1.png = CDel pnode 1.pngCDel 2x.pngCDel q.pngCDel node.png

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

Quasiregular polyhedra[edit]

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

Like real polytopes, a complex quasiregular polyhedron can be constructed as a rectification (a complete truncation) of a regular polyhedron. Vertices are created mid-edge of the regular polyhedron and faces of the regular polyhedron and its dual are positioned alternating across common edges.

For example, a p-generalized cube, CDel pnode 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, has p3 vertices, 3p2 edges, and 3p p-generalized square faces, while the p-generalized octahedron, CDel pnode.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, has 3p vertices, 3p2 edges and p3 triangular faces. The middle quasiregular form p-generalized cuboctahedron, CDel pnode.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png, has 3p2 vertices, 3p3 edges, and 3p+p3 faces.

Also the rectification of the Hessian polyhedron CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 3.pngCDel 3node.png, is CDel 3node.pngCDel 3.pngCDel 3node 1.pngCDel 3.pngCDel 3node.png, a quasiregular form sharing the geometry of the regular complex polyhedron CDel 3node 1.pngCDel 3.pngCDel 3node.pngCDel 4.pngCDel node.png.

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

Other complex polytopes with unitary reflections of period two[edit]

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 CDel node 1.pngCDel 3split1.pngCDel branch.png 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 CDel node.pngCDel psplit1.pngCDel branch.png is defined by 3 unitary reflections, R1, R2, R3, all order 2: R12 = R12 = R32 = (R1R2)3 = (R2R3)3 = (R3R1)3 = (R1R2R3R1)p = 1. The period p can be seen as a double rotation in real .

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 CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, with octahedral symmetry CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png order 48, and subgroup dihedral symmetry CDel node.pngCDel 3.pngCDel node.png order 6, so the number of vertices of a cube is 48/6=8. Facets are constructed by removing one node furthest from the ringed node, for example CDel node 1.pngCDel 4.pngCDel node.png for the cube. Vertex figures are generated by removing a ringed node and ringing one or more connected nodes, and CDel node 1.pngCDel 3.pngCDel node.png for the cube.

Coxeter represents these groups by the following symbols. Some groups have the same order, but a different structure, defining the same vertex arrangement in complex polytopes, but different edges and higher elements, like CDel node.pngCDel psplit1.pngCDel branch.png and CDel node.pngCDel 3split1.pngCDel branch.pngCDel labelp.png with p≠3.[43]

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

Coxeter calls some of these complex polyhedra almost regular because they have regular facets and vertex figures. The first is a lower symmetry form of the generalized cross-polytope in . 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 .

Some almost regular complex polyhedra[44]
Space Group Order Coxeter
symbols
Vertices Edges Faces Vertex
figure
Notes
[1 1 1p]3
CDel node.pngCDel 3split1.pngCDel branch.pngCDel labelp.png
p=2,3,4...
6p2 (1 1 11p)3
CDel node 1.pngCDel 3split1.pngCDel branch.pngCDel labelp.png
3p 3p2 {3} {2p} Shephard symbol (1 1; 11)p
same as βp
3
= CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel pnode.png
(11 1 1p)3
CDel node.pngCDel 3split1.pngCDel branch 10l.pngCDel labelp.png
p2 {3} {6} Shephard symbol (11 1; 1)p
1/p γp
3
[1 1 12]3
CDel node.pngCDel split1.pngCDel nodes.png
24 (1 1 112)3
CDel node 1.pngCDel split1.pngCDel nodes.png
6 12 8 {3} {4} Same as β2
3
= CDel node 1.png