Complex Polytopes

In 1954 Shephard found some extension of polytopes from real space to the unitary complex one. Considering the mapping of the Argand plane of 1-dimensional complex space onto the 2-dimensional real space, there are possibilities to not only considering the second root of unity (resulting in a mirror, then fully representable in the 1-dimensional real sub-space), but higher roots of unity as well. These higher roots of unity, resulting in the vertices of a real-space regular n-gon, then will represent complex n-edges, i.e. edges with n incident vertices.

Using this type of complex edges, one likewise can build up polytopes in complex space, i.e. abstract polytopes, as one nowadays would speak, having any next-"dimensional" element within the next complex dimension. So, by the above mapping from complex d-spaces to real 2d-spaces, this results in an appropriate subset of elements of an appropriate real polytope in doubled up dimensional space. For sure, only the sub-elements of even-dimensional sub-spaces are to be considered, and therefrom generally also only some special subset.

In 1974 Coxeter came up with his second famous book, "Regular Complex Polytopes", where he took over this theme. He also changed a bit Shephard's original notation. Now we usually denote these polytopes in kind of an extension of the Schläfli symbols. Consider generators R_i of that complex space group, that is, representable accordingly by complex unitary matrices, which are subject to the following equations

Ripi = 1
Ri * Ri+k = Ri+k * Ri    (k>1)
Ri * Ri+1 * Ri * ... = Ri+1 * Ri * Ri+1 * ...    (qi alternating factors each)

Since Coxeter the thus derived regular complex polytopes are being denoted as p₀{q₁}p₁...p_d-2{q_d-1}p_d-1 .

In the special case of p₀ = ... = p_d-1 = 2 the roots of unity each range within the 1-dimensional real subspace of the corresponding 1-dimensional complex space (real number line of the Argand plane). Each being a mere mirror reflection. Therefore we come back to some d-dimensional real polytope, in fact the one being described by the (usual) Schläfli symbols {q₁,...,q_d-1}. (In fact, this incidence was why Coxeter changed Shephard's original notation, which used parantheses instead of curly brackets and gave order numbers instead of factor counts q_i.)

It shall be noted here additionally that the edge-order numbers p_i in general are not independent. In fact the above definition of these numbers by means of the generators implies that when q_i is odd then p_i-1 = p_i is required.

regular complex polytopes   (up)

n{2}n  (n,n) *)

n{4}2  (n2,2n)
2{4}n  (2n,n2)

3{3}3  (8,8)

3{4}3  (24,24)

3{5}3  (120,120)

3{6}2  (24,16)
2{6}3  (16,24)

3{8}2  (72,48)
2{8}3  (48,72)

3{10}2 (360,240)
2{10}3 (240,360)

4{3}4  (24,24)

4{4}3  (96,72)
3{4}4  (72,96)

4{6}2  (96,48)
2{6}4  (48,96)

5{3}5  (120,120)

5{4}3  (600,360)
3{4}5  (360,600)

5{6}2  (600,240)
2{6}5  (240,600)

*) degenerate

n{4}2{3}2  (n3,3n)
2{3}2{4}n  (3n,n3)

3{3}3{3}3  (27,27)

3{3}3{4}2  (72,54)
2{4}3{3}3  (54,72)

n{4}2{3}2{3}2  (n4,4n)
2{3}2{3}2{4}n  (4n,n4)

3{3}3{3}3{3}3  (240,240)

n{4}2{3}2...2{3}2  (nd,nd)
2{3}2...2{3}2{4}n  (nd,nd)

As an aside, clearly any complex unitary matrix of dimension d is also representable as real orthogonal matrix of dimension 2d. But conversely, not every real orthogonal matrix of dimension 2d is representable as a complex unitary matrix of dimension d. Therefore several additional abstract polytopes can be found, which are derivable likewise from sub-structures of "parent"-polytopes, even such which do use even dimensional elements only, but which are not "complex polytopes" in that sense, that they are subject to complex space groups, i.e. unitary ones. Examples here are what else could be denoted 3{8}4 and 3{12}4.