Definitions

Complex polytope

A complex polytope is one which exists in a complex Hilbert space, where each real dimension is accompanied by an imaginary one.

The "imaginary" number $i$ is defined as the square root of −1. A complex number, say $\left(a + ib\right)$ lies in a complex plane, which may be represented as a real Argand diagram. An n-dimensional unitary space comprises n such complex planes, all orthogonal to each other.

For example a complex polygon exists in the unitary plane of two real dimensions $x$ and $y,$ and two imaginary dimensions $ix$ and $iy.$ (Note however that in the case of polygons, the term 'complex polygon' also has other meanings.)

Characteristics

In complex Hilbert spaces, a "line" has both real and imaginary parts, and so may be represented as an Argand diagram of its complex plane. An edge of an ordinary polytope is bounded by two points, or vertices, at each end, and the edge is the segment in between. If we allow these vertices to have complex coordinates and place a third vertex on the unitary line (i.e. in the Argand diagram), there is no location that can be identified as "between" the first two. A complex polytope is an extension to the idea of a real polytope, allowing many such vertex points on a given unitary line.

In the unitary plane, two such lines generally meet at a single point, and we say that the two edges of the polytope meet at a common vertex. Thus a complex polygon comprises many unitary edges, such that a given edge may meet more than two other edges at various vertices.

Any given pair of edges meet at only one point, which may or may not be a vertex of the polytope according to whether the edges are adjacent or not.

Because there is no idea of "between", there is in general no idea of inside vs. outside, or indeed any sense of an enclosing boundary. For this reason, complex polytopes are most conveniently treated mathematically as configurations.

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 - constructing his book along the lines of a Bruckner symphony.

In the Argand diagram, of the edge of a regular complex polytope, the vertex points lie at the vertices of a regular polygon.

Two representations 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 (and coincidentally the diagram looks the same as a common projection of the hypercube). 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.

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.

References

• Shephard, G.C.; Regular complex polytopes, Proc. London math. Soc. Series 3, Vol 2, (1952), pp 82-97.
• Coxeter, H. S. M., Regular Complex Polytopes, Cambridge University Press, (1974).

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