Complex polytope

In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

On a real line, two points bound a segment. This defines an edge with two bounding vertices. For a real polytope it is not possible to have a third vertex associated with an edge because one of them would then lie between the other two. On the complex line, which may be represented as an Argand diagram, points are not ordered and there is no idea of "between", so more than two vertex points may be associated with a given edge.

Also, a real polygon has just two sides at each vertex, such that the boundary forms a closed loop. A real polyhedron has two faces at each edge such that the boundary forms a closed surface. A polychoron has two cells at each wall, and so on. These loops and surfaces have no analogy in complex spaces, for example a set of complex lines and points may form a closed chain of connections, but this chain does not bound a polygon. Thus, more than two elements meeting in one place may be allowed.

Since bounding does not occur, we cannot think of a complex edge as a line segment, but as the whole line. Similarly, we cannot think of a bounded polygonal face but must accept the whole plane.

Thus, a complex polytope may be understood as an arrangement of connected points, lines, planes and so on, where every point is the junction of multiple lines, every line of multiple planes, and so on. Likewise, each line must contain multiple points, each plane multiple lines, and so on. Hence, unlike the case of convex real polytopes, convexity does not exist as a notion and simply listing the coordinates of the vertices does not uniquely determine the higher-dimensional faces of a complex polytope.

Regular complex polytopes

The only complex polytopes to have been systematically studied are the regular ones. Shephard (1952) discovered them, and Coxeter (1974) developed the idea extensively. Shephard treated his figures as configurations from the start, while Coxeter only found it necessary to do so from Chapter 12 onwards.

Recall that a configuration consists of a collection of points, lines, and higher-dimensional objects, and an incidence relation connecting objects of different dimensions. A complex polytope must be realized in the appropriate power of $\mathbb{C}$; so that the vertices of a complex polygon would be points of the "complex plane" $\mathbb{C}^2$, and the edges would be one-complex-dimensional [affine] subspaces of the "plane". Thus, an edge can be given a coordinate system consisting of a single complex number.

For a regular complex polytope, the vertices incident on the edge must be arranged symmetrically about a "center", which we will use as the origin of the edge's coordinate system; in the real case the center is simply the midpoint of the edge. The symmetry is a "reflection" about the center, which extends to a symmetry of the whole polytope; 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 $x^p - 1 = 0$ where p is the number of incident vertices. Thus, in the Argand diagram of the edge, the vertex points lie at the vertices of a regular polygon centered on the origin.

Similarly Two real projections of the same regular complex octagon with edges a,b,c,d,e,f,g,h are illustrated. It has 16 vertices, which for clarity have not been individually marked. Each edge has four vertices at which it meets another edge, 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 - this is important to understand - but are drawn in purely to help visually relate the four vertices. The edges are laid out symmetrically (coincidentally the diagram looks the same as a common projection of the tesseract, but in the case of the complex octagon the diamond shapes which can be traced are not parts of the structure). The second diagram abandons octagonal symmetry in favour of clarity. Each edge is shown as a line, and each meeting point on the line is a vertex on that edge. The connectivity between the various edges is clear to see.

Modified Schläfli notation

Shephard's 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 octagon illustrated has eight 4-edges (p₁=4) and sixteen 2-vertices (p₂=2). From this we can work out that g = 32, giving the modified Schläfli symbol 4(32)2.

Coxeter's notation

The modern notation p₁{q}p₂ is due to Coxeter, and is based on group theory. The nodes p₁ and p₂ represent mirrors producing p₁ and p₂ images in the plane. In group theory, this might be represented (for the example left) as AAAA = BB = 1. q represents the number of alternate reflections in the two mirrors that become equal to its opposite, i.e. for q=4, ABAB = BABA. When q is odd, then p₁ = p₂, e.g. 3{5}3 means AAA = BBB = 1; ABABA = BABAB.

The example octagon is represented as 4{4}2, which belongs to symmetry group AAAA = BB = 1, ABAB = BABA.

Real conjugates

In the ordinary, or real plane, we can construct a visible figure as the real conjugate of some complex polygon. Likewise in ordinary space, we can construct a visible figure as the real conjugate of some complex polyhedron.

To obtain the real conjugate, we discard the imaginary part of any coordinate. For example, the complex point (a + ib) has real conjugate a.

The real conjugate of a complex edge is a line with the vertex points distributed along it (not generally evenly spaced). The second of the two octagon projections above shows the real conjugates of the sides.

References

• Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, (1974).
• Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244,
• Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82–97.