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.
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 2space (also sometimes called the complex plane) is thus a fourdimensional space over the reals, and so on in higher dimensions.
A complex npolytope in complex nspace is the analogue of a real npolytope in real nspace.
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]}
(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 4edges 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. 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 B_{4} 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.
A real 1dimensional 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 1polytope 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 1dimensional 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 1polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane.
A regular real 1dimensional polytope is represented by an empty Schläfli symbol {}, or CoxeterDynkin diagram . The dot or node of the CoxeterDynkin 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 1dimensional polytope in has CoxeterDynkin 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 0polytope, 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 1polytope with p vertices is e^{2πi/p} = cos(2π/p) + i sin(2π/p). When p = 2, the generator is e^{πi} = –1, the same as a point reflection in the real plane.
In higher complex polytopes, 1polytopes form pedges. A 2edge is similar to an ordinary real edge, in that it contains two vertices, but need not exist on a real line.
While 1polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons _{p}{4}_{2}, are limited to 5edge (pentagonal edges) elements, and infinite regular apeirogons also include 6edge (hexagonal edges) elements.
Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p_{1}edges, with a p_{2}set as vertex figure and overall symmetry group of order g, we denote the polygon as p_{1}(g)p_{2}.
The number of vertices V is then g/p_{2} and the number of edges E is g/p_{1}.
The complex polygon illustrated above has eight square edges (p_{1}=4) and sixteen vertices (p_{2}=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
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 R_{1}, R_{2}, where: R_{1}^{p1} = R_{2}^{p2} = I. If q is even, (R_{2}R_{1})^{q/2} = (R_{1}R_{2})^{q/2}. If q is odd, (R_{2}R_{1})^{(q−1)/2}R_{2} = (R_{1}R_{2})^{(q−1)/2}R_{1}. When q is odd, p_{1}=p_{2}.
For _{4}[4]_{2} has R_{1}^{4} = R_{2}^{2} = I, (R_{2}R_{1})^{2} = (R_{1}R_{2})^{2}.
For _{3}[5]_{3} has R_{1}^{3} = R_{2}^{3} = I, (R_{2}R_{1})^{2}R_{2} = (R_{1}R_{2})^{2}R_{1}.
Coxeter also generalised the use of CoxeterDynkin 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.
Coxeter enumerated this list of regular complex polygons in . A regular complex polygon, _{p}{q}_{r} or , has pedges, and rgonal vertex figures. _{p}{q}_{r} is a finite polytope if (p+r)q>pr(q2).
Its symmetry is written as _{p}[q]_{r}, called a Shephard group, analogous to a Coxeter group, while also allowing unitary reflections.
For nonstarry groups, the order of the group _{p}[q]_{r} can be computed as .^{[9]}
The Coxeter number for _{p}[q]_{r} is , so the group order can also be computed as . A regular complex polygon can be drawn in orthogonal projection with hgonal symmetry.
The rank 2 solutions that generate complex polygons are:
Group  G_{3}=G(q,1,1)  G_{2}=G(p,1,2)  G_{4}  G_{6}  G_{5}  G_{8}  G_{14}  G_{9}  G_{10}  G_{20}  G_{16}  G_{21}  G_{17}  G_{18} 

_{2}[q]_{2}, q=3,4...  _{p}[4]_{2}, p=2,3...  _{3}[3]_{3}  _{3}[6]_{2}  _{3}[4]_{3}  _{4}[3]_{4}  _{3}[8]_{2}  _{4}[6]_{2}  _{4}[4]_{3}  _{3}[5]_{3}  _{5}[3]_{5}  _{3}[10]_{2}  _{5}[6]_{2}  _{5}[4]_{3}  
Order  2q  2p^{2}  24  48  72  96  144  192  288  360  600  720  1200  1800 
h  q  2p  6  12  24  30  60 
Excluded solutions with odd q and unequal p and r are: _{6}[3]_{2}, _{6}[3]_{3}, _{9}[3]_{3}, _{12}[3]_{3}, ..., _{5}[5]_{2}, _{6}[5]_{2}, _{8}[5]_{2}, _{9}[5]_{2}, _{4}[7]_{2}, _{9}[5]_{2}, _{3}[9]_{2}, and _{3}[11]_{2}.
Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .
The dual polygon of _{p}{q}_{r} is _{r}{q}_{p}. A polygon of the form _{p}{q}_{p} is selfdual. 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.
The group p[q]r, , can be represented by two matrices:^{[11]}
Name  R_{1} 
R_{2} 

Order  p  r 
Matrix 


With


 



Coxeter enumerated the complex polygons in Table III of Regular Complex Polytopes.^{[12]}
Group  Order  Coxeter number 
Polygon  Vertices  Edges  Notes  

G(q,q,2) _{2}[q]_{2} = [q] q=2,3,4,... 
2q  q  _{2}{q}_{2}  q  q  {}  Real regular polygons Same as Same as if q even 
Group  Order  Coxeter number 
Polygon  Vertices  Edges  Notes  

G(p,1,2) _{p}[4]_{2} p=2,3,4,... 
2p^{2}  2p  p(2p^{2})2  _{p}{4}_{2}  
p^{2}  2p  _{p}{}  same as _{p}{}×_{p}{} or representation as pp duoprism 
2(2p^{2})p  _{2}{4}_{p}  2p  p^{2}  {}  representation as pp duopyramid  
G(2,1,2) _{2}[4]_{2} = [4] 
8  4  _{2}{4}_{2} = {4}  4  4  {}  same as {}×{} or Real square  
G(3,1,2) _{3}[4]_{2} 
18  6  6(18)2  _{3}{4}_{2}  9  6  _{3}{}  same as _{3}{}×_{3}{} or representation as 33 duoprism  
2(18)3  _{2}{4}_{3}  6  9  {}  representation as 33 duopyramid  
G(4,1,2) _{4}[4]_{2} 
32  8  8(32)2  _{4}{4}_{2}  16  8  _{4}{}  same as _{4}{}×_{4}{} or representation as 44 duoprism or {4,3,3}  
2(32)4  _{2}{4}_{4}  8  16  {}  representation as 44 duopyramid or {3,3,4}  
G(5,1,2) _{5}[4]_{2} 
50  25  5(50)2  _{5}{4}_{2}  25  10  _{5}{}  same as _{5}{}×_{5}{} or representation as 55 duoprism  
2(50)5  _{2}{4}_{5}  10  25  {}  representation as 55 duopyramid  
G(6,1,2) _{6}[4]_{2} 
72  36  6(72)2  _{6}{4}_{2}  36  12  _{6}{}  same as _{6}{}×_{6}{} or representation as 66 duoprism  
2(72)6  _{2}{4}_{6}  12  36  {}  representation as 66 duopyramid  
G_{4}=G(1,1,2) _{3}[3]_{3} <2,3,3> 
24  6  3(24)3  _{3}{3}_{3}  8  8  _{3}{}  Möbius–Kantor configuration selfdual, same as representation as {3,3,4}  
G_{6} _{3}[6]_{2} 
48  12  3(48)2  _{3}{6}_{2}  24  16  _{3}{}  same as  
_{3}{3}_{2}  starry polygon  
2(48)3  _{2}{6}_{3}  16  24  {}  
_{2}{3}_{3}  starry polygon  
G_{5} _{3}[4]_{3} 
72  12  3(72)3  _{3}{4}_{3}  24  24  _{3}{}  selfdual, same as representation as {3,4,3}  
G_{8} _{4}[3]_{4} 
96  12  4(96)4  _{4}{3}_{4}  24  24  _{4}{}  selfdual, same as representation as {3,4,3}  
G_{14} _{3}[8]_{2} 
144  24  3(144)2  _{3}{8}_{2}  72  48  _{3}{}  same as  
_{3}{8/3}_{2}  starry polygon, same as  
2(144)3  _{2}{8}_{3}  48  72  {}  
_{2}{8/3}_{3}  starry polygon  
G_{9} _{4}[6]_{2} 
192  24  4(192)2  _{4}{6}_{2}  96  48  _{4}{}  same as  
2(192)4  _{2}{6}_{4}  48  96  {}  
_{4}{3}_{2}  96  48  {}  starry polygon  
_{2}{3}_{4}  48  96  {}  starry polygon  
G_{10} _{4}[4]_{3} 
288  24  4(288)3  _{4}{4}_{3}  96  72  _{4}{}  
12  _{4}{8/3}_{3}  starry polygon  
24  3(288)4  _{3}{4}_{4}  72  96  _{3}{}  
12  _{3}{8/3}_{4}  starry polygon  
G_{20} _{3}[5]_{3} 
360  30  3(360)3  _{3}{5}_{3}  120  120  _{3}{}  selfdual, same as representation as {3,3,5}  
_{3}{5/2}_{3}  selfdual, starry polygon  
G_{16} _{5}[3]_{5} 
600  30  5(600)5  _{5}{3}_{5}  120  120  _{5}{}  selfdual, same as representation as {3,3,5}  
10  _{5}{5/2}_{5}  selfdual, starry polygon  
G_{21} _{3}[10]_{2} 
720  60  3(720)2  _{3}{10}_{2}  360  240  _{3}{}  same as  
_{3}{5}_{2}  starry polygon  
_{3}{10/3}_{2}  starry polygon, same as  
_{3}{5/2}_{2}  starry polygon  
2(720)3  _{2}{10}_{3}  240  360  {}  
_{2}{5}_{3}  starry polygon  
_{2}{10/3}_{3}  starry polygon  
_{2}{5/2}_{3}  starry polygon  
G_{17} _{5}[6]_{2} 
1200  60  5(1200)2  _{5}{6}_{2}  600  240  _{5}{}  same as  
20  _{5}{5}_{2}  starry polygon  
20  _{5}{10/3}_{2}  starry polygon  
60  _{5}{3}_{2}  starry polygon  
60  2(1200)5  _{2}{6}_{5}  240  600  {}  
20  _{2}{5}_{5}  starry polygon  
20  _{2}{10/3}_{5}  starry polygon  
60  _{2}{3}_{5}  starry polygon  
G_{18} _{5}[4]_{3} 
1800  60  5(1800)3  _{5}{4}_{3}  600  360  _{5}{}  
15  _{5}{10/3}_{3}  starry polygon  
30  _{5}{3}_{3}  starry polygon  
30  _{5}{5/2}_{3}  starry polygon  
60  3(1800)5  _{3}{4}_{5}  360  600  _{3}{}  
15  _{3}{10/3}_{5}  starry polygon  
30  _{3}{3}_{5}  starry polygon  
30  _{3}{5/2}_{5}  starry polygon 
Polygons of the form _{p}{2r}_{q} can be visualized by q color sets of pedge. Each pedge is seen as a regular polygon, while there are no faces.
Polygons of the form _{2}{4}_{q} are called generalized orthoplexes. They share vertices with the 4D qq duopyramids, vertices connected by 2edges.
_{2}{4}_{2}, , with 4 vertices, and 4 edges
_{2}{4}_{3}, , with 6 vertices, and 9 edges^{[13]}
_{2}{4}_{4}, , with 8 vertices, and 16 edges
_{2}{4}_{5}, , with 10 vertices, and 25 edges
_{2}{4}_{6}, , with 12 vertices, and 36 edges
_{2}{4}_{7}, , with 14 vertices, and 49 edges
_{2}{4}_{8}, , with 16 vertices, and 64 edges
_{2}{4}_{9}, , with 18 vertices, and 81 edges
_{2}{4}_{10}, , with 20 vertices, and 100 edges
Polygons of the form _{p}{4}_{2} are called generalized hypercubes (squares for polygons). They share vertices with the 4D pp duoprisms, vertices connected by pedges. Vertices are drawn in green, and pedges are drawn in alternate colors, red and blue. The perspective is distorted slightly for odd dimensions to move overlapping vertices from the center.
_{2}{4}_{2}, or , with 4 vertices, and 4 2edges
_{3}{4}_{2}, or , with 9 vertices, and 6 (triangular) 3edges^{[13]}
_{4}{4}_{2}, or , with 16 vertices, and 8 (square) 4edges
_{5}{4}_{2}, or , with 25 vertices, and 10 (pentagonal) 5edges
_{6}{4}_{2}, or , with 36 vertices, and 12 (hexagonal) 6edges
_{7}{4}_{2}, or , with 49 vertices, and 14 (heptagonal)7edges
_{8}{4}_{2}, or , with 64 vertices, and 16 (octagonal) 8edges
_{9}{4}_{2}, or , with 81 vertices, and 18 (enneagonal) 9edges
_{10}{4}_{2}, or , with 100 vertices, and 20 (decagonal) 10edges
_{3}{4}_{2}, or with 9 vertices, 6 3edges in 2 sets of colors
_{2}{4}_{3}, with 6 vertices, 9 edges in 3 sets
_{4}{4}_{2}, or with 16 vertices, 8 4edges in 2 sets of colors and filled square 4edges
_{5}{4}_{2}, or with 25 vertices, 10 5edges in 2 sets of colors
_{3}{6}_{2}, or , with 24 vertices in black, and 16 3edges colored in 2 sets of 3edges in red and blue^{[14]}
_{3}{8}_{2}, or , with 72 vertices in black, and 48 3edges colored in 2 sets of 3edges in red and blue^{[15]}
Polygons of the form _{p}{r}_{p} have equal number of vertices and edges. They are also selfdual.
_{3}{3}_{3}, or , with 8 vertices in black, and 8 3edges colored in 2 sets of 3edges in red and blue^{[16]}
_{3}{4}_{3}, or , with 24 vertices and 24 3edges shown in 3 sets of colors, one set filled^{[17]}
_{4}{3}_{4}, or , with 24 vertices and 24 4edges shown in 4 sets of colors^{[17]}
_{3}{5}_{3}, or , with 120 vertices and 120 3edges^{[18]}
_{5}{3}_{5}, or , with 120 vertices and 120 5edges^{[19]}
In general, a regular complex polytope is represented by Coxeter as _{p}{z_{1}}_{q}{z_{2}}_{r}{z_{3}}_{s}… or Coxeter diagram …, having symmetry _{p}[z_{1}]_{q}[z_{2}]_{r}[z_{3}]_{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 γ^{p}
_{n} = _{p}{4}_{2}{3}_{2}…_{2}{3}_{2} and diagram …. Its symmetry group has diagram _{p}[4]_{2}[3]_{2}…_{2}[3]_{2}; in the Shephard–Todd classification, this is the group G(p, 1, n) generalizing the signed permutation matrices. Its dual regular polytope, the "generalized cross polytope", is represented by the symbol β^{p}
_{n} = _{2}{3}_{2}{3}_{2}…_{2}{4}_{p} and diagram ….^{[21]}
A 1dimensional regular complex polytope in is represented as , having p vertices, with its real representation a regular polygon, {p}. Coxeter also gives it symbol γ^{p}
_{1} or β^{p}
_{1} as 1dimensional generalized hypercube or cross polytope. Its symmetry is _{p}[] or , a cyclic group of order p. In a higher polytope, _{p}{} or represents a pedge element, with a 2edge, {} or , representing an ordinary real edge between two vertices.^{[21]}
A dual complex polytope is constructed by exchanging k and (n1k)elements of an npolytope. For example, a dual complex polygon has vertices centered on each edge, and new edges are centered at the old vertices. A vvalence vertex creates a new vedge, and eedges become evalence vertices.^{[22]} The dual of a regular complex polytope has a reversed symbol. Regular complex polytopes with symmetric symbols, i.e. _{p}{q}_{p}, _{p}{q}_{r}{q}_{p}, _{p}{q}_{r}{s}_{r}{q}_{p}, etc. are self dual.
Coxeter enumerated this list of nonstarry regular complex polyhedra in , including the 5 platonic solids in .^{[23]}
A regular complex polyhedron, _{p}{n_{1}}_{q}{n_{2}}_{r} or , has faces, edges, and vertex figures.
A complex regular polyhedron _{p}{n_{1}}_{q}{n_{2}}_{r} requires both g_{1} = order(_{p}[n_{1}]_{q}) and g_{2} = order(_{q}[n_{2}]_{r}) be finite.
Given g = order(_{p}[n_{1}]_{q}[n_{2}]_{r}), the number of vertices is g/g_{2}, and the number of faces is g/g_{1}. 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  α_{3} = _{2}{3}_{2}{3}_{2} = {3,3} 
4  6  {}  4  {3}  {3}  none  Real tetrahedron Same as  
G_{23} _{2}[3]_{2}[5]_{2} = [3,5] 
120  10  _{2}{3}_{2}{5}_{2} = {3,5}  12  30  {}  20  {3}  {5}  none  Real icosahedron  
_{2}{5}_{2}{3}_{2} = {5,3}  20  30  {}  12  {5}  {3}  none  Real dodecahedron  
G(2,1,3) _{2}[3]_{2}[4]_{2} = [3,4] 
48  6  β^{2} _{3} = β_{3} = {3,4} 
6  12  {}  8  {3}  {4}  {4}  Real octahedron Same as {}+{}+{}, order 8 Same as , order 24  
γ^{2} _{3} = γ_{3} = {4,3} 
8  12  {}  6  {4}  {3}  none  Real cube Same as {}×{}×{} or  
G(p,1,3) _{2}[3]_{2}[4]_{p} p=2,3,4,... 
6p^{3}  3p  β^{p} _{3} = _{2}{3}_{2}{4}_{p} 

3p  3p^{2}  {}  p^{3}  {3}  _{2}{4}_{p}  _{2}{4}_{p}  Generalized octahedron Same as _{p}{}+_{p}{}+_{p}{}, order p^{3} Same as , order 6p^{2}  
γ^{p} _{3} = _{p}{4}_{2}{3}_{2} 
p^{3}  3p^{2}  _{p}{}  3p  _{p}{4}_{2}  {3}  none  Generalized cube Same as _{p}{}×_{p}{}×_{p}{} or  
G(3,1,3) _{2}[3]_{2}[4]_{3} 
162  9  β^{3} _{3} = _{2}{3}_{2}{4}_{3} 
9  27  {}  27  {3}  _{2}{4}_{3}  _{2}{4}_{3}  Same as _{3}{}+_{3}{}+_{3}{}, order 27 Same as , order 54  
γ^{3} _{3} = _{3}{4}_{2}{3}_{2} 
27  27  _{3}{}  9  _{3}{4}_{2}  {3}  none  Same as _{3}{}×_{3}{}×_{3}{} or  
G(4,1,3) _{2}[3]_{2}[4]_{4} 
384  12  β^{4} _{3} = _{2}{3}_{2}{4}_{4} 
12  48  {}  64  {3}  _{2}{4}_{4}  _{2}{4}_{4}  Same as _{4}{}+_{4}{}+_{4}{}, order 64 Same as , order 96  
γ^{4} _{3} = _{4}{4}_{2}{3}_{2} 
64  48  _{4}{}  12  _{4}{4}_{2}  {3}  none  Same as _{4}{}×_{4}{}×_{4}{} or  
G(5,1,3) _{2}[3]_{2}[4]_{5} 
750  15  β^{5} _{3} = _{2}{3}_{2}{4}_{5} 
15  75  {}  125  {3}  _{2}{4}_{5}  _{2}{4}_{5}  Same as _{5}{}+_{5}{}+_{5}{}, order 125 Same as , order 150  
γ^{5} _{3} = _{5}{4}_{2}{3}_{2} 
125  75  _{5}{}  15  _{5}{4}_{2}  {3}  none  Same as _{5}{}×_{5}{}×_{5}{} or  
G(6,1,3) _{2}[3]_{2}[4]_{6} 
1296  18  β^{6} _{3} = _{2}{3}_{2}{4}_{6} 
36  108  {}  216  {3}  _{2}{4}_{6}  _{2}{4}_{6}  Same as _{6}{}+_{6}{}+_{6}{}, order 216 Same as , order 216  
γ^{6} _{3} = _{6}{4}_{2}{3}_{2} 
216  108  _{6}{}  18  _{6}{4}_{2}  {3}  none  Same as _{6}{}×_{6}{}×_{6}{} or  
G_{25} _{3}[3]_{3}[3]_{3} 
648  9  _{3}{3}_{3}{3}_{3}  27  72  _{3}{}  27  _{3}{3}_{3}  _{3}{3}_{3}  _{3}{4}_{2}  Same as . representation as 2_{21} Hessian polyhedron  
G_{26} _{2}[4]_{3}[3]_{3} 
1296  18  _{2}{4}_{3}{3}_{3}  54  216  {}  72  _{2}{4}_{3}  _{3}{3}_{3}  {6}  
_{3}{3}_{3}{4}_{2}  72  216  _{3}{}  54  _{3}{3}_{3}  _{3}{4}_{2}  _{3}{4}_{3}  Same as ^{[24]} representation as 1_{22} 
Real {3,3}, or has 4 vertices, 6 edges, and 4 faces
_{3}{3}_{3}{3}_{3}, or , has 27 vertices, 72 3edges, and 27 faces, with one face highlighted blue.^{[25]}
_{2}{4}_{3}{3}_{3}, has 54 vertices, 216 simple edges, and 72 faces, with one face highlighted blue.^{[26]}
_{3}{3}_{3}{4}_{2}, or , has 72 vertices, 216 3edges, and 54 vertices, with one face highlighted blue.^{[27]}
Generalized octahedra have a regular construction as and quasiregular form as . All elements are simplexes.
Real {3,4}, or , with 6 vertices, 12 edges, and 8 faces
_{2}{3}_{2}{4}_{3}, or , with 9 vertices, 27 edges, and 27 faces
_{2}{3}_{2}{4}_{4}, or , with 12 vertices, 48 edges, and 64 faces
_{2}{3}_{2}{4}_{5}, or , with 15 vertices, 75 edges, and 125 faces
_{2}{3}_{2}{4}_{6}, or , with 18 vertices, 108 edges, and 216 faces
_{2}{3}_{2}{4}_{7}, or , with 21 vertices, 147 edges, and 343 faces
_{2}{3}_{2}{4}_{8}, or , with 24 vertices, 192 edges, and 512 faces
_{2}{3}_{2}{4}_{9}, or , with 27 vertices, 243 edges, and 729 faces
_{2}{3}_{2}{4}_{10}, or , with 30 vertices, 300 edges, and 1000 faces
Generalized cubes have a regular construction as and prismatic construction as , a product of three pgonal 1polytopes. Elements are lower dimensional generalized cubes.
Real {4,3}, or has 8 vertices, 12 edges, and 6 faces
_{3}{4}_{2}{3}_{2}, or has 27 vertices, 27 3edges, and 9 faces^{[25]}
_{4}{4}_{2}{3}_{2}, or , with 64 vertices, 48 edges, and 12 faces
_{5}{4}_{2}{3}_{2}, or , with 125 vertices, 75 edges, and 15 faces
_{6}{4}_{2}{3}_{2}, or , with 216 vertices, 108 edges, and 18 faces
_{7}{4}_{2}{3}_{2}, or , with 343 vertices, 147 edges, and 21 faces
_{8}{4}_{2}{3}_{2}, or , with 512 vertices, 192 edges, and 24 faces
_{9}{4}_{2}{3}_{2}, or , with 729 vertices, 243 edges, and 27 faces
_{10}{4}_{2}{3}_{2}, or , with 1000 vertices, 300 edges, and 30 faces
Coxeter enumerated this list of nonstarry regular complex 4polytopes in , including the 6 convex regular 4polytopes in .^{[23]}
Space  Group  Order  Coxeter number 
Polytope  Vertices  Edges  Faces  Cells  Van Oss polygon 
Notes 

G(1,1,4) _{2}[3]_{2}[3]_{2}[3]_{2} = [3,3,3] 
120  5  α_{4} = _{2}{3}_{2}{3}_{2}{3}_{2} = {3,3,3} 
5  10 {} 
10 {3} 
5 {3,3} 
none  Real 5cell (simplex)  
G_{28} _{2}[3]_{2}[4]_{2}[3]_{2} = [3,4,3] 
1152  12  _{2}{3}_{2}{4}_{2}{3}_{2} = {3,4,3} 
24  96 {} 
96 {3} 
24 {3,4} 
{6}  Real 24cell  
G_{30} _{2}[3]_{2}[3]_{2}[5]_{2} = [3,3,5] 
14400  30  _{2}{3}_{2}{3}_{2}{5}_{2} = {3,3,5} 
120  720 {} 
1200 {3} 
600 {3,3} 
{10}  Real 600cell  
_{2}{5}_{2}{3}_{2}{3}_{2} = {5,3,3} 
600  1200 {} 
720 {5} 
120 {5,3} 
Real 120cell  
G(2,1,4) _{2}[3]_{2}[3]_{2}[4]_{p} =[3,3,4] 
384  8  β^{2} _{4} = β_{4} = {3,3,4} 
8  24 {} 
32 {3} 
16 {3,3} 
{4}  Real 16cell Same as , order 192  
γ^{2} _{4} = γ_{4} = {4,3,3} 
16  32 {} 
24 {4} 
8 {4,3} 
none  Real tesseract Same as {}^{4} or , order 16  
G(p,1,4) _{2}[3]_{2}[3]_{2}[4]_{p} p=2,3,4,... 
24p^{4}  4p  β^{p} _{4} = _{2}{3}_{2}{3}_{2}{4}_{p} 
4p  6p^{2} {} 
4p^{3} {3} 
p^{4} {3,3} 
_{2}{4}_{p}  Generalized 4orthoplex Same as , order 24p^{3}  
γ^{p} _{4} = _{p}{4}_{2}{3}_{2}{3}_{2} 
p^{4}  4p^{3} _{p}{} 
6p^{2} _{p}{4}_{2} 
4p _{p}{4}_{2}{3}_{2} 
none  Generalized tesseract Same as _{p}{}^{4} or , order p^{4}  
G(3,1,4) _{2}[3]_{2}[3]_{2}[4]_{3} 
1944  12  β^{3} _{4} = _{2}{3}_{2}{3}_{2}{4}_{3} 
12  54 {} 
108 {3} 
81 {3,3} 
_{2}{4}_{3}  Generalized 4orthoplex Same as , order 648  
γ^{3} _{4} = _{3}{4}_{2}{3}_{2}{3}_{2} 
81  108 _{3}{} 
54 _{3}{4}_{2} 
12 _{3}{4}_{2}{3}_{2} 
none  Same as _{3}{}^{4} or , order 81  
G(4,1,4) _{2}[3]_{2}[3]_{2}[4]_{4} 
6144  16  β^{4} _{4} = _{2}{3}_{2}{3}_{2}{4}_{4} 
16  96 {} 
256 {3} 
64 {3,3} 
_{2}{4}_{4}  Same as , order 1536  
γ^{4} _{4} = _{4}{4}_{2}{3}_{2}{3}_{2} 
256  256 _{4}{} 
96 _{4}{4}_{2} 
16 _{4}{4}_{2}{3}_{2} 
none  Same as _{4}{}^{4} or , order 256  
G(5,1,4) _{2}[3]_{2}[3]_{2}[4]_{5} 
15000  20  β^{5} _{4} = _{2}{3}_{2}{3}_{2}{4}_{5} 
20  150 {} 
500 {3} 
625 {3,3} 
_{2}{4}_{5}  Same as , order 3000  
γ^{5} _{4} = _{5}{4}_{2}{3}_{2}{3}_{2} 
625  500 _{5}{} 
150 _{5}{4}_{2} 
20 _{5}{4}_{2}{3}_{2} 
none  Same as _{5}{}^{4} or , order 625  
G(6,1,4) _{2}[3]_{2}[3]_{2}[4]_{6} 
31104  24  β^{6} _{4} = _{2}{3}_{2}{3}_{2}{4}_{6} 
24  216 {} 
864 {3} 
1296 {3,3} 
_{2}{4}_{6}  Same as , order 5184  
γ^{6} _{4} = _{6}{4}_{2}{3}_{2}{3}_{2} 
1296  864 _{6}{} 
216 _{6}{4}_{2} 
24 _{6}{4}_{2}{3}_{2} 
none  Same as _{6}{}^{4} or , order 1296  
G_{32} _{3}[3]_{3}[3]_{3}[3]_{3} 
155520  30  _{3}{3}_{3}{3}_{3}{3}_{3} 
240  2160 _{3}{} 
2160 _{3}{3}_{3} 
240 _{3}{3}_{3}{3}_{3} 
_{3}{4}_{3}  Witting polytope representation as 4_{21} 
Real {3,3,3}, , had 5 vertices, 10 edges, 10 {3} faces, and 5 {3,3} cells
Real {3,4,3}, , had 24 vertices, 96 edges, 96 {3} faces, and 24 {3,4} cells
Real {5,3,3}, , had 600 vertices, 1200 edges, 720 {5} faces, and 120 {5,3} cells
Real {3,3,5}, , had 120 vertices, 720 edges, 1200 {3} faces, and 600 {3,3} cells
Witting polytope, , has 240 vertices, 2160 3edges, 2160 3{3}3 faces, and 240 3{3}3{3}3 cells
Generalized 4orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
Real {3,3,4}, or , with 8 vertices, 24 edges, 32 faces, and 16 cells
_{2}{3}_{2}{3}_{2}{4}_{3}, or , with 12 vertices, 54 edges, 108 faces, and 81 cells
_{2}{3}_{2}{3}_{2}{4}_{4}, or , with 16 vertices, 96 edges, 256 faces, and 256 cells
_{2}{3}_{2}{3}_{2}{4}_{5}, or , with 20 vertices, 150 edges, 500 faces, and 625 cells
_{2}{3}_{2}{3}_{2}{4}_{6}, or , with 24 vertices, 216 edges, 864 faces, and 1296 cells
_{2}{3}_{2}{3}_{2}{4}_{7}, or , with 28 vertices, 294 edges, 1372 faces, and 2401 cells
_{2}{3}_{2}{3}_{2}{4}_{8}, or , with 32 vertices, 384 edges, 2048 faces, and 4096 cells
_{2}{3}_{2}{3}_{2}{4}_{9}, or , with 36 vertices, 486 edges, 2916 faces, and 6561 cells
_{2}{3}_{2}{3}_{2}{4}_{10}, or , with 40 vertices, 600 edges, 4000 faces, and 10000 cells
Generalized tesseracts have a regular construction as and prismatic construction as , a product of four pgonal 1polytopes. Elements are lower dimensional generalized cubes.
Real {4,3,3}, or , with 16 vertices, 32 edges, 24 faces, and 8 cells
_{3}{4}_{2}{3}_{2}{3}_{2}, or , with 81 vertices, 108 edges, 54 faces, and 12 cells
_{4}{4}_{2}{3}_{2}{3}_{2}, or , with 256 vertices, 96 edges, 96 faces, and 16 cells
_{5}{4}_{2}{3}_{2}{3}_{2}, or , with 625 vertices, 500 edges, 150 faces, and 20 cells
_{6}{4}_{2}{3}_{2}{3}_{2}, or , with 1296 vertices, 864 edges, 216 faces, and 24 cells
_{7}{4}_{2}{3}_{2}{3}_{2}, or , with 2401 vertices, 1372 edges, 294 faces, and 28 cells
_{8}{4}_{2}{3}_{2}{3}_{2}, or , with 4096 vertices, 2048 edges, 384 faces, and 32 cells
_{9}{4}_{2}{3}_{2}{3}_{2}, or , with 6561 vertices, 2916 edges, 486 faces, and 36 cells
_{10}{4}_{2}{3}_{2}{3}_{2}, or , with 10000 vertices, 4000 edges, 600 faces, and 40 cells
Regular complex 5polytopes in or higher exist in three families, the real simplexes and the generalized hypercube, and orthoplex.
Space  Group  Order  Polytope  Vertices  Edges  Faces  Cells  4faces  Van Oss polygon 
Notes 

G(1,1,5) = [3,3,3,3] 
720  α_{5} = {3,3,3,3} 
6  15 {} 
20 {3} 
15 {3,3} 
6 {3,3,3} 
none  Real 5simplex  
G(2,1,5) =[3,3,3,4] 
3840  β^{2} _{5} = β_{5} = {3,3,3,4} 
10  40 {} 
80 {3} 
80 {3,3} 
32 {3,3,3} 
{4}  Real 5orthoplex Same as , order 1920  
γ^{2} _{5} = γ_{5} = {4,3,3,3} 
32  80 {} 
80 {4} 
40 {4,3} 
10 {4,3,3} 
none  Real 5cube Same as {}^{5} or , order 32  
G(p,1,5) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{p} 
120p^{5}  β^{p} _{5} = _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{p} 
5p  10p^{2} {} 
10p^{3} {3} 
5p^{4} {3,3} 
p^{5} {3,3,3} 
_{2}{4}_{p}  Generalized 5orthoplex Same as , order 120p^{4}  
γ^{p} _{5} = _{p}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
p^{5}  5p^{4} _{p}{} 
10p^{3} _{p}{4}_{2} 
10p^{2} _{p}{4}_{2}{3}_{2} 
5p _{p}{4}_{2}{3}_{2}{3}_{2} 
none  Generalized 5cube Same as _{p}{}^{5} or , order p^{5}  
G(3,1,5) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{3} 
29160  β^{3} _{5} = _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{3} 
15  90 {} 
270 {3} 
405 {3,3} 
243 {3,3,3} 
_{2}{4}_{3}  Same as , order 9720  
γ^{3} _{5} = _{3}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
243  405 _{3}{} 
270 _{3}{4}_{2} 
90 _{3}{4}_{2}{3}_{2} 
15 _{3}{4}_{2}{3}_{2}{3}_{2} 
none  Same as _{3}{}^{5} or , order 243  
G(4,1,5) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{4} 
122880  β^{4} _{5} = _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{4} 
20  160 {} 
640 {3} 
1280 {3,3} 
1024 {3,3,3} 
_{2}{4}_{4}  Same as , order 30720  
γ^{4} _{5} = _{4}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
1024  1280 _{4}{} 
640 _{4}{4}_{2} 
160 _{4}{4}_{2}{3}_{2} 
20 _{4}{4}_{2}{3}_{2}{3}_{2} 
none  Same as _{4}{}^{5} or , order 1024  
G(5,1,5) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{5} 
375000  β^{5} _{5} = _{2}{3}_{2}{3}_{2}{3}_{2}{5}_{5} 
25  250 {} 
1250 {3} 
3125 {3,3} 
3125 {3,3,3} 
_{2}{5}_{5}  Same as , order 75000  
γ^{5} _{5} = _{5}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
3125  3125 _{5}{} 
1250 _{5}{5}_{2} 
250 _{5}{5}_{2}{3}_{2} 
25 _{5}{4}_{2}{3}_{2}{3}_{2} 
none  Same as _{5}{}^{5} or , order 3125  
G(6,1,5) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{6} 
933210  β^{6} _{5} = _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{6} 
30  360 {} 
2160 {3} 
6480 {3,3} 
7776 {3,3,3} 
_{2}{4}_{6}  Same as , order 155520  
γ^{6} _{5} = _{6}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
7776  6480 _{6}{} 
2160 _{6}{4}_{2} 
360 _{6}{4}_{2}{3}_{2} 
30 _{6}{4}_{2}{3}_{2}{3}_{2} 
none  Same as _{6}{}^{5} or , order 7776 
Generalized 5orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
Real {3,3,3,4}, , with 10 vertices, 40 edges, 80 faces, 80 cells, and 32 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{3}, , with 15 vertices, 90 edges, 270 faces, 405 cells, and 243 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{4}, , with 20 vertices, 160 edges, 640 faces, 1280 cells, and 1024 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{5}, , with 25 vertices, 250 edges, 1250 faces, 3125 cells, and 3125 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{6}, , with 30 vertices, 360 edges, 2160 faces, 6480 cells, 7776 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{7}, , with 35 vertices, 490 edges, 3430 faces, 12005 cells, 16807 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{8}, , with 40 vertices, 640 edges, 5120 faces, 20480 cells, 32768 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{9}, , with 45 vertices, 810 edges, 7290 faces, 32805 cells, 59049 4faces
_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{10}, , with 50 vertices, 1000 edges, 10000 faces, 50000 cells, 100000 4faces
Generalized 5cubes have a regular construction as and prismatic construction as , a product of five pgonal 1polytopes. Elements are lower dimensional generalized cubes.
Real {4,3,3,3}, , with 32 vertices, 80 edges, 80 faces, 40 cells, and 10 4faces
_{3}{4}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 243 vertices, 405 edges, 270 faces, 90 cells, and 15 4faces
_{4}{4}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 1024 vertices, 1280 edges, 640 faces, 160 cells, and 20 4faces
_{5}{4}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 3125 vertices, 3125 edges, 1250 faces, 250 cells, and 25 4faces
_{6}{4}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 7776 vertices, 6480 edges, 2160 faces, 360 cells, and 30 4faces
Space  Group  Order  Polytope  Vertices  Edges  Faces  Cells  4faces  5faces  Van Oss polygon 
Notes 

G(1,1,6) = [3,3,3,3,3] 
720  α_{6} = {3,3,3,3,3} 
7  21 {} 
35 {3} 
35 {3,3} 
21 {3,3,3} 
7 {3,3,3,3} 
none  Real 6simplex  
G(2,1,6) [3,3,3,4] 
46080  β^{2} _{6} = β_{6} = {3,3,3,4} 
12  60 {} 
160 {3} 
240 {3,3} 
192 {3,3,3} 
64 {3,3,3,3} 
{4}  Real 6orthoplex Same as , order 23040  
γ^{2} _{6} = γ_{6} = {4,3,3,3} 
64  192 {} 
240 {4} 
160 {4,3} 
60 {4,3,3} 
12 {4,3,3,3} 
none  Real 6cube Same as {}^{6} or , order 64  
G(p,1,6) _{2}[3]_{2}[3]_{2}[3]_{2}[4]_{p} 
720p^{6}  β^{p} _{6} = _{2}{3}_{2}{3}_{2}{3}_{2}{4}_{p} 
6p  15p^{2} {} 
20p^{3} {3} 
15p^{4} {3,3} 
6p^{5} {3,3,3} 
p^{6} {3,3,3,3} 
_{2}{4}_{p}  Generalized 6orthoplex Same as , order 720p^{5}  
γ^{p} _{6} = _{p}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
p^{6}  6p^{5} _{p}{} 
15p^{4} _{p}{4}_{2} 
20p^{3} _{p}{4}_{2}{3}_{2} 
15p^{2} _{p}{4}_{2}{3}_{2}{3}_{2} 
6p _{p}{4}_{2}{3}_{2}{3}_{2}{3}_{2} 
none  Generalized 6cube Same as _{p}{}^{6} or , order p^{6} 
Generalized 6orthoplexes have a regular construction as and quasiregular form as . All elements are simplexes.
Real {3,3,3,3,4}, , with 12 vertices, 60 edges, 160 faces, 240 cells, 192 4faces, and 64 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{3}, , with 18 vertices, 135 edges, 540 faces, 1215 cells, 1458 4faces, and 729 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{4}, , with 24 vertices, 240 edges, 1280 faces, 3840 cells, 6144 4faces, and 4096 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{5}, , with 30 vertices, 375 edges, 2500 faces, 9375 cells, 18750 4faces, and 15625 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{6}, , with 36 vertices, 540 edges, 4320 faces, 19440 cells, 46656 4faces, and 46656 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{7}, , with 42 vertices, 735 edges, 6860 faces, 36015 cells, 100842 4faces, 117649 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{8}, , with 48 vertices, 960 edges, 10240 faces, 61440 cells, 196608 4faces, 262144 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{9}, , with 54 vertices, 1215 edges, 14580 faces, 98415 cells, 354294 4faces, 531441 5faces
_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{10}, , with 60 vertices, 1500 edges, 20000 faces, 150000 cells, 600000 4faces, 1000000 5faces
Generalized 6cubes have a regular construction as and prismatic construction as , a product of six pgonal 1polytopes. Elements are lower dimensional generalized cubes.
Real {3,3,3,3,3,4}, , with 64 vertices, 192 edges, 240 faces, 160 cells, 60 4faces, and 12 5faces
_{3}{4}_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 729 vertices, 1458 edges, 1215 faces, 540 cells, 135 4faces, and 18 5faces
_{4}{4}_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 4096 vertices, 6144 edges, 3840 faces, 1280 cells, 240 4faces, and 24 5faces
_{5}{4}_{2}{3}_{2}{3}_{2}{3}_{2}{3}_{2}, , with 15625 vertices, 18750 edges, 9375 faces, 2500 cells, 375 4faces, and 30 5faces
Coxeter enumerated this list of nonstarry regular complex apeirotopes or honeycombs.^{[28]}
For each dimension there are 12 apeirotopes symbolized as δ^{p,r}
_{n+1} exists in any dimensions , or if p=q=2. Coxeter calls these generalized cubic honeycombs for n>2.^{[29]}
Each has proportional element counts given as:
The only regular complex 1polytope is _{∞}{}, or . Its real representation is an apeirogon, {∞}, or .
Rank 2 complex apeirogons have symmetry _{p}[q]_{r}, where 1/p + 2/q + 1/r = 1. Coxeter expresses them as δ^{p,r}
_{2} where q is constrained to satisfy q = 2/(1 – (p + r)/pr).^{[30]}
There are 8 solutions:
_{2}[∞]_{2}  _{3}[12]_{2}  _{4}[8]_{2}  _{6}[6]_{2}  _{3}[6]_{3}  _{6}[4]_{3}  _{4}[4]_{4}  _{6}[3]_{6} 
There are two excluded solutions odd q and unequal p and r: _{10}[5]_{2} and _{12}[3]_{4}, or and .
A regular complex apeirogon _{p}{q}_{r} has pedges and rgonal vertex figures. The dual apeirogon of _{p}{q}_{r} is _{r}{q}_{p}. An apeirogon of the form _{p}{q}_{p} is selfdual. Groups of the form _{p}[2q]_{2} have a half symmetry _{p}[q]_{p}, so a regular apeirogon is the same as quasiregular .^{[31]}
Apeirogons can be represented on the Argand plane share four different vertex arrangements. Apeirogons of the form _{2}{q}_{r} have a vertex arrangement as {q/2,p}. The form _{p}{q}_{2} have vertex arrangement as r{p,q/2}. Apeirogons of the form _{p}{4}_{r} have vertex arrangements {p,r}.
Including affine nodes, and , there are 3 more infinite solutions: _{∞}[2]_{∞}, _{∞}[4]_{2}, _{∞}[3]_{3}, and , , and . The first is an index 2 subgroup of the second. The vertices of these apeirogons exist in .
Space  Group  Apeirogon  Edge  rep.^{[32]}  Picture  Notes  

_{2}[∞]_{2} = [∞]  δ^{2,2} _{2} = {∞} 

{}  Real apeirogon Same as  
/  _{∞}[4]_{2}  _{∞}{4}_{2}  _{∞}{}  {4,4}  Same as  
_{∞}[3]_{3}  _{∞}{3}_{3}  _{∞}{}  {3,6}  Same as  
_{p}[q]_{r}  δ^{p,r} _{2} = _{p}{q}_{r} 
_{p}{}  
_{3}[12]_{2}  δ^{3,2} _{2} = _{3}{12}_{2} 
_{3}{}  r{3,6}  Same as  
δ^{2,3} _{2} = _{2}{12}_{3} 
{}  {6,3}  
_{3}[6]_{3}  δ^{3,3} _{2} = _{3}{6}_{3} 
_{3}{}  {3,6}  Same as  
_{4}[8]_{2}  δ^{4,2} _{2} = _{4}{8}_{2} 
_{4}{}  {4,4}  Same as  
δ^{2,4} _{2} = _{2}{8}_{4} 
{}  {4,4}  
_{4}[4]_{4}  δ^{4,4} _{2} = _{4}{4}_{4} 
_{4}{}  {4,4}  Same as  
_{6}[6]_{2}  δ^{6,2} _{2} = _{6}{6}_{2} 
_{6}{}  r{3,6}  Same as  
δ^{2,6} _{2} = _{2}{6}_{6} 
{}  {3,6}  
_{6}[4]_{3}  δ^{6,3} _{2} = _{6}{4}_{3} 
_{6}{}  {6,3}  
δ^{3,6} _{2} = _{3}{4}_{6} 
_{3}{}  {3,6}  
_{6}[3]_{6}  δ^{6,6} _{2} = _{6}{3}_{6} 
_{6}{}  {3,6}  Same as 
There are 22 regular complex apeirohedra, of the form _{p}{a}_{q}{b}_{r}. 8 are selfdual (p=r and a=b), while 14 exist as dual polytope pairs. Three are entirely real (p=q=r=2).
Coxeter symbolizes 12 of them as δ^{p,r}
_{3} or _{p}{4}_{2}{4}_{r} is the regular form of the product apeirotope δ^{p,r}
_{2} × δ^{p,r}
_{2} or _{p}{q}_{r} × _{p}{q}_{r}, where q is determined from p and r.
is the same as , as well as , for p,r=2,3,4,6. Also = .^{[33]}
Space  Group  Apeirohedron  Vertex  Edge  Face  van Oss apeirogon 
Notes  

_{2}[3]_{2}[4]_{∞}  _{∞}{4}_{2}{3}_{2}  _{∞}{}  _{∞}{4}_{2}  Same as _{∞}{}×_{∞}{}×_{∞}{} or Real representation {4,3,4}  
_{p}[4]_{2}[4]_{r}  _{p}{4}_{2}{4}_{r}  
p^{2}  2pq  _{p}{}  r^{2}  _{p}{4}_{2}  _{2}{q}_{r}  Same as , p,r=2,3,4,6  
[4,4]  δ^{2,2} _{3} = {4,4} 
4  8  {}  4  {4}  {∞}  Real square tiling Same as or or  
_{3}[4]_{2}[4]_{2} _{3}[4]_{2}[4]_{3} _{4}[4]_{2}[4]_{2} _{4}[4]_{2}[4]_{4} _{6}[4]_{2}[4]_{2} _{6}[4]_{2}[4]_{3} _{6}[4]_{2}[4]_{6} 
_{3}{4}_{2}{4}_{2} _{2}{4}_{2}{4}_{3} _{3}{4}_{2}{4}_{3} _{4}{4}_{2}{4}_{2} _{2}{4}_{2}{4}_{4} _{4}{4}_{2}{4}_{4} _{6}{4}_{2}{4}_{2} _{2}{4}_{2}{4}_{6} _{6}{4}_{2}{4}_{3} _{3}{4}_{2}{4}_{6} _{6}{4}_{2}{4}_{6} 
9 4 9 16 4 16 36 4 36 9 36 
12 12 18 16 16 32 24 24 36 36 72 
_{3}{} {} _{3}{} _{4}{} {} _{4}{} _{6}{} {} _{6}{} _{3}{} _{6}{} 
4 9 9 4 16 16 4 36 9 36 36 
_{3}{4}_{2} {4} _{3}{4}_{2} _{4}{4}_{2} {4} _{4}{4}_{2} _{6}{4}_{2} {4} _{6}{4}_{2} _{3}{4}_{2} _{6}{4}_{2} 
_{p}{q}_{r}  Same as or or Same as Same as Same as or or Same as Same as Same as or or Same as Same as Same as Same as 
Space  Group  Apeirohedron  Vertex  Edge  Face  van Oss apeirogon 
Notes  

_{2}[4]_{r}[4]_{2}  _{2}{4}_{r}{4}_{2}  
2  {}  2  _{p}{4}_{2'}  _{2}{4}_{r}  Same as and , r=2,3,4,6  
[4,4]  {4,4}  2  4  {}  2  {4}  {∞}  Same as and  
_{2}[4]_{3}[4]_{2} _{2}[4]_{4}[4]_{2} _{2}[4]_{6}[4]_{2} 
_{2}{4}_{3}{4}_{2} _{2}{4}_{4}{4}_{2} _{2}{4}_{6}{4}_{2} 
2  9 16 36 
{}  2  _{2}{4}_{3} _{2}{4}_{4} _{2}{4}_{6} 
_{2}{q}_{r}  Same as and Same as and Same as and ^{[34]} 
Space  Group  Apeirohedron  Vertex  Edge  Face  van Oss apeirogon 
Notes  

_{2}[6]_{2}[3]_{2} = [6,3] 
{3,6}  
1  3  {}  2  {3}  {∞}  Real triangular tiling  
{6,3}  2  3  {}  1  {6}  none  Real hexagonal tiling  
_{3}[4]_{3}[3]_{3}  _{3}{3}_{3}{4}_{3}  1  8  _{3}{}  3  _{3}{3}_{3}  _{3}{4}_{6}  Same as  
_{3}{4}_{3}{3}_{3}  3  8  _{3}{}  2  _{3}{4}_{3}  _{3}{12}_{2}  
_{4}[3]_{4}[3]_{4}  _{4}{3}_{4}{3}_{4}  1  6  _{4}{}  1  _{4}{3}_{4}  _{4}{4}_{4}  Selfdual, same as  
_{4}[3]_{4}[4]_{2}  _{4}{3}_{4}{4}_{2}  1  12  _{4}{}  3  _{4}{3}_{4}  _{2}{8}_{4}  Same as  
_{2}{4}_{4}{3}_{4}  3  12  {}  1  _{2}{4}_{4}  _{4}{4}_{4} 
There are 16 regular complex apeirotopes in . Coxeter expresses 12 of them by δ^{p,r}
_{3} where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: = . The first case is the cubic honeycomb.
Space  Group  3apeirotope  Vertex  Edge  Face  Cell  van Oss apeirogon 
Notes 

_{p}[4]_{2}[3]_{2}[4]_{r}  δ^{p,r} _{3} = _{p}{4}_{2}{3}_{2}{4}_{r} 
_{p}{}  _{p}{4}_{2}  _{p}{4}_{2}{3}_{2}  _{p}{q}_{r}  Same as  
_{2}[4]_{2}[3]_{2}[4]_{2} =[4,3,4] 
δ^{2,2} _{3} = _{2}{4}_{2}{3}_{2}{4}_{2} 
{}  {4}  {4,3}  Cubic honeycomb Same as or or  
_{3}[4]_{2}[3]_{2}[4]_{2}  δ^{3,2} _{3} = _{3}{4}_{2}{3}_{2}{4}_{2} 
_{3}{}  _{3}{4}_{2}  _{3}{4}_{2}{3}_{2}  Same as or or  
δ^{2,3} _{3} = _{2}{4}_{2}{3}_{2}{4}_{3} 
{}  {4}  {4,3}  Same as  
_{3}[4]_{2}[3]_{2}[4]_{3}  δ^{3,3} _{3} = _{3}{4}_{2}{3}_{2}{4}_{3} 
_{3}{}  _{3}{4}_{2}  _{3}{4}_{2}{3}_{2}  Same as  
_{4}[4]_{2}[3]_{2}[4]_{2}  δ^{4,2} _{3} = _{4}{4}_{2}{3}_{2}{4}_{2} 
_{4}{}  _{4}{4}_{2}  _{4}{4}_{2}{3}_{2}  Same as or or  
δ^{2,4} _{3} = _{2}{4}_{2}{3}_{2}{4}_{4} 
{}  {4}  {4,3}  Same as  
_{4}[4]_{2}[3]_{2}[4]_{4}  δ^{4,4} _{3} = _{4}{4}_{2}{3}_{2}{4}_{4} 
_{4}{}  _{4}{4}_{2}  _{4}{4}_{2}{3}_{2}  Same as  
_{6}[4]_{2}[3]_{2}[4]_{2}  δ^{6,2} _{3} = _{6}{4}_{2}{3}_{2}{4}_{2} 
_{6}{}  _{6}{4}_{2}  _{6}{4}_{2}{3}_{2}  Same as or or  
δ^{2,6} _{3} = _{2}{4}_{2}{3}_{2}{4}_{6} 
{}  {4}  {4,3}  Same as  
_{6}[4]_{2}[3]_{2}[4]_{3}  δ^{6,3} _{3} = _{6}{4}_{2}{3}_{2}{4}_{3} 
_{6}{}  _{6}{4}_{2}  _{6}{4}_{2}{3}_{2}  Same as  
δ^{3,6} _{3} = _{3}{4}_{2}{3}_{2}{4}_{6} 
_{3}{}  _{3}{4}_{2}  _{3}{4}_{2}{3}_{2}  Same as  
_{6}[4]_{2}[3]_{2}[4]_{6}  δ^{6,6} _{3} = _{6}{4}_{2}{3}_{2}{4}_{6} 
_{6}{}  _{6}{4}_{2}  _{6}{4}_{2}{3}_{2}  Same as 
Space  Group  3apeirotope  Vertex  Edge  Face  Cell  van Oss apeirogon 
Notes 

_{2}[4]_{3}[3]_{3}[3]_{3}  _{3}{3}_{3}{3}_{3}{4}_{2} 
1  24 _{3}{}  27 _{3}{3}_{3}  2 _{3}{3}_{3}{3}_{3}  _{3}{4}_{6}  Same as  
_{2}{4}_{3}{3}_{3}{3}_{3} 
2  27 {}  24 _{2}{4}_{3}  1 _{2}{4}_{3}{3}_{3}  _{2}{12}_{3}  
_{2}[3]_{2}[4]_{3}[3]_{3}  _{2}{3}_{2}{4}_{3}{3}_{3} 
1  27 {}  72 _{2}{3}_{2}  8 _{2}{3}_{2}{4}_{3}  _{2}{6}_{6}  
_{3}{3}_{3}{4}_{2}{3}_{2} 
8  72 _{3}{}  27 _{3}{3}_{3}  1 _{3}{3}_{3}{4}_{2}  _{3}{6}_{3}  Same as or 
There are 15 regular complex apeirotopes in . Coxeter expresses 12 of them by δ^{p,r}
_{4} where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed as product apeirotopes: = . The first case is the tesseractic honeycomb. The 16cell honeycomb and 24cell honeycomb are real solutions. The last solution is generated has Witting polytope elements.
Space  Group  4apeirotope  Vertex  Edge  Face  Cell  4face  van Oss apeirogon 
Notes 

_{p}[4]_{2}[3]_{2}[3]_{2}[4]_{r}  δ^{p,r} _{4} = _{p}{4}_{2}{3}_{2}{3}_{2}{4}_{r} 
_{p}{}  _{p}{4}_{2}  _{p}{4}_{2}{3}_{2}  _{p}{4}_{2}{3}_{2}{3}_{2}  _{p}{q}_{r}  Same as  
_{2}[4]_{2}[3]_{2}[3]_{2}[4]_{2}  δ^{2,2} _{4} = {4,3,3,3} 
{}  {4}  {4,3}  {4,3,3}  {∞}  Tesseractic honeycomb Same as  
_{2}[3]_{2}[4]_{2}[3]_{2}[3]_{2} =[3,4,3,3] 
{3,3,4,3} 
1  12 {}  32 {3}  24 {3,3}  3 {3,3,4}  Real 16cell honeycomb Same as  
{3,4,3,3} 
3  24 {}  32 {3}  12 {3,4}  1 {3,4,3}  Real 24cell honeycomb Same as or  
_{3}[3]_{3}[3]_{3}[3]_{3}[3]_{3}  _{3}{3}_{3}{3}_{3}{3}_{3}{3}_{3} 
1  80 _{3}{}  270 _{3}{3}_{3}  80 _{3}{3}_{3}{3}_{3}  1 _{3}{3}_{3}{3}_{3}{3}_{3}  _{3}{4}_{6}  representation 5_{21} 
There are only 12 regular complex apeirotopes in or higher,^{[35]} expressed δ^{p,r}
_{n} where q is constrained to satisfy q = 2/(1 – (p + r)/pr). These can also be decomposed a product of n apeirogons: ... = ... . The first case is the real hypercube honeycomb.
Space  Group  5apeirotopes  Vertices  Edge  Face  Cell  4face  5face  van Oss apeirogon 
Notes 

_{p}[4]_{2}[3]_{2}[3]_{2}[3]_{2}[4]_{r}  δ^{p,r} _{5} = _{p}{4}_{2}{3}_{2}{3}_{2}{3}_{2}{4}_{r} 
_{p}{}  _{p}{4}_{2}  _{p}{4}_{2}{3}_{2}  _{p}{4}_{2}{3}_{2}{3}_{2}  _{p}{4}_{2}{3}_{2}{3}_{2}{3}_{2}  _{p}{q}_{r}  Same as  
_{2}[4]_{2}[3]_{2}[3]_{2}[3]_{2}[4]_{2} =[4,3,3,3,4] 
δ^{2,2} _{5} = {4,3,3,3,4} 
{}  {4}  {4,3}  {4,3,3}  {4,3,3,3}  {∞}  5cubic honeycomb Same as 
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 edgetocenter 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.^{[36]}
If it exists, the van Oss polygon of regular complex polytope of the form _{p}{q}_{r}{s}_{t}... has pedges.
Complex product polygon or {}×_{5}{} has 10 vertices connected by 5 2edges and 2 5edges, with its real representation as a 3dimensional 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 1dimensional polytopes is the same as the regular _{p}{4}_{2} or . More general products, like _{p}{}×_{q}{} have real representations as the 4dimensional pq duoprisms. The dual of a product polytope can be written as a sum _{p}{}+_{q}{} and have real representations as the 4dimensional pq duopyramid. The _{p}{}+_{p}{} can have its symmetry doubled as a regular complex polytope _{2}{4}_{p} or .
Similarly, a 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 .^{[37]}
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 pedges of the regular form.
_{p}[q]_{r}  _{2}[4]_{2}  _{3}[4]_{2}  _{4}[4]_{2}  _{5}[4]_{2}  _{6}[4]_{2}  _{7}[4]_{2}  _{8}[4]_{2}  _{3}[3]_{3}  _{3}[4]_{3} 

Regular 
4 2edges 
9 3edges 
16 4edges 