In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.
A complex polytope may be understood as a collection of complex points, lines, planes, and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on.
Precise definitions exist only for the regular complex polytopes, which are configurations. The regular complex polytopes have been completely characterized, and can be described using a symbolic notation developed by Coxeter.
Some complex polytopes which are not fully regular have also been described.
Definitions and introduction The complex line 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) may be considered as a four-dimensional space over the reals. Likewise, complex n-space may be viewed as a real 2n-dimensional vector space.
A complex n-polytope in complex n-space is the analogue of a real n-polytope in real n-space. However, 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:Peter Orlik, Victor Reiner, Anne V. Shepler. The sign representation for Shephard groups. Mathematische Annalen. March 2002, Volume 322, Issue 3, pp. 477–492. DOI:10.1007/s002080200001 for every , if is a flat in P of dimension i and is a flat in P of dimension k such that then there are at least two flats G in P of dimension j such that ; for every such that , if 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 of flats of P (with 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.Shephard (1952), p. 83
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, 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. In the middle image, each edge is represented as a real line and the four vertices in each line can be more clearly seen. A perspective sketch representing the 16 vertex points as large black dots and the 8 4-edges as bounded squares within each edge. The green path represents the octagonal perimeter of the left hand image.
A complex polytope exists in the complex space of equivalent dimension. For example, the vertices of a complex polygon are points in the complex plane (a plane in which each point has two complex numbers as its coordinates, not to be confused with the Argand plane of complex numbers), and the edges are complex lines existing as (affine) subspaces of the plane and intersecting at the vertices. Thus, as a one-dimensional complex space, an edge can be given its own coordinate system, within which the points of the edge are each represented by a single complex number.
In a regular complex polytope the vertices incident on the edge are arranged symmetrically about their centroid, which is often used as the origin of the edge's coordinate system (in the real case the centroid is just the midpoint of the edge). The symmetry arises from a complex reflection about the centroid; this reflection will leave the magnitude of any vertex unchanged, but change its argument by a fixed amount, moving it to the coordinates of the next vertex in order. So we may assume (after a suitable choice of scale) that the vertices on the edge satisfy the equation where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin.
Three real projections of regular complex polygon 4{4}2 are illustrated above, with edges a, b, c, d, e, f, g, h. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices and each vertex lies on two edges, hence each edge meets four other edges. In the first diagram, each edge is represented by a square. The sides of the square are not parts of the polygon but are drawn purely to help visually relate the four vertices. The edges are laid out symmetrically. (Note that the diagram looks similar to the B4 Coxeter plane projection of the tesseract, but it is structurally different).
The middle diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a real line, and each meeting point of two lines is a vertex. The connectivity between the various edges is clear to see.
The last diagram gives a flavour of the structure projected into three dimensions: the two cubes of vertices are in fact the same size but are seen in perspective at different distances away in the fourth dimension.
Regular complex one-dimensional polytopes
A real 1-dimensional polytope exists as a closed segment in the real line , 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.
Unlike points on the real line, points on the complex line have no natural ordering. Thus, unlike real polytopes, no interior can be defined.Shephard, 1952; "It is from considerations such as these that we derive the notion of the interior of a polytope, and it will be seen that in unitary space where the numbers cannot be so ordered such a concept of interior is impossible. [Para break] Hence ... we have to consider unitary polytopes as configurations." Despite this, complex 1-polytopes are often drawn, as here, as a bounded regular polygon in the Argand plane.
A regular real 1-dimensional polytope is represented by an empty Schläfli symbol {}, or Coxeter-Dynkin diagram . The dot or node of the Coxeter-Dynkin diagram itself represents a reflection generator while the circle around the node means the generator point is not on the reflection, so its reflective image is a distinct point from itself. By extension, a regular complex 1-dimensional polytope in has Coxeter-Dynkin diagram , for any positive integer p, 2 or greater, containing p vertices. p can be suppressed if it is 2. It can also be represented by an empty Schläfli symbol p{}, }p{, {}p, or p{2}1. The 1 is a notational placeholder, representing a nonexistent reflection, or a period 1 identity generator. (A 0-polytope, real or complex is a point, and is represented as } {, or 1{2}1.)
The symmetry is denoted by the Coxeter diagram , and can alternatively be described in Coxeter notation as p[], []p or ]p[, p1 or pp. The symmetry is isomorphic to the cyclic group, order p. The subgroups of p[] are any whole divisor d, d[], where d≥2.
A unitary operator generator for is seen as a rotation by 2π/p radians counter clockwise, and a edge is created by sequential applications of a single unitary reflection. A unitary reflection generator for a 1-polytope with p vertices is . When p = 2, the generator is eπi = –1, the same as a point reflection in the real plane.
In higher complex polytopes, 1-polytopes form p-edges. A 2-edge is similar to an ordinary real edge, in that it contains two vertices, but need not exist on a real line.
Regular complex polygons
While 1-polytopes can have unlimited p, finite regular complex polygons, excluding the double prism polygons p{4}2, are limited to 5-edge (pentagonal edges) elements, and infinite regular apeirogons also include 6-edge (hexagonal edges) elements.
Notations
Shephard's modified Schläfli notation Shephard originally devised a modified form of Schläfli's notation for regular polytopes. For a polygon bounded by p1-edges, with a p2-set as vertex figure and overall symmetry group of order g, we denote the polygon as p1(g)p2.
The number of vertices V is then g/p2 and the number of edges E is g/p1.
The complex polygon illustrated above has eight square edges (p1=4) and sixteen vertices (p2=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.
Coxeter's revised modified Schläfli notation A more modern notation p1{q}p2 is due to Coxeter, 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 42 has R14 = R22 = I, (R2R1)2 = (R1R2)2.
For 33 has R13 = R23 = I, (R2R1)2R2 = (R1R2)2R1.
Coxeter-Dynkin diagrams Coxeter also generalised the use of Coxeter-Dynkin diagrams to complex polytopes, for example the complex polygon p{q}r is represented by and the equivalent symmetry group, p[q]r, is a ringless diagram . The nodes p and r represent mirrors producing p and r images in the plane. Unlabeled nodes in a diagram have implicit 2 labels. For example, a real regular polygon is 2{q}2 or {q} or .
One limitation, nodes connected by odd branch orders must have identical node orders. If they do not, the group will create "starry" polygons, with overlapping element. So and are ordinary, while is starry.
12 Irreducible Shephard groups
Coxeter enumerated this list of regular complex polygons in . A regular complex polygon, p{q}r or , has p-edges, and r-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 unitary reflections.
For nonstarry groups, the order of the group p[q]r can be computed as .Lehrer & Taylor 2009, p. 87
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: GroupG3=G(q,1,1) G2=G(p,1,2)G4G6G5G8G14G9G10G20G16G21G17G182[q]2, q=3,4...p2, p=2,3...333233443242433355325253 Order2q2p22448729614419228836060072012001800hq2p612243060
Excluded solutions with odd q and unequal p and r are: 62, 63, 93, 123, ..., 52, 62, 82, 92, 42, 92, 32, and 32.
Other whole q with unequal p and r, create starry groups with overlapping fundamental domains: , , , , , and .