Complex polytope

A complex polytope is a generalized polytope-like object whose containing space is $n$-dimensional $$\mathbb{C}^n$$. Complex coordinate space is an extension of Euclidean space $$\mathbb{R}^n$$ where each dimension has a real and imaginary axis. Complex polytopes are in general not actually polytopes in the traditional sense, since they violate dyadicity, but are rather a kind of incidence geometry together with a realization. Another major difference from real polytopes is that complex polytopes don't enclose points and don't have interiors.

Foundations
Be aware that complex 1-space is sometimes called the "complex line" and other times the "complex plane".

To define isometries and therefore symmetries, complex $n$-space requires a distance metric $$d(\mathbf{x}, \mathbf{y}) = ||x - y||$$ which comes from the norm $$||\mathbf{x}|| = \sqrt{\textstyle \sum_i |x_i|^2}$$ (which in turn follows from the standard inner product typically defined from $$\mathbb{C}^n$$). By this definition, complex $n$-space as a metric space is isomorphic to 2$n$-dimensional Euclidean space, and symmetries and isometries are analogous. Unlike in Euclidean space, in complex $n$-space isometries are not necessarily affine transformations. For example, the transformation $$z \mapsto \bar{z}$$ in $$\mathbb{C}^1$$ is an isometry that is not an affine transformation.

Complex 1-polytopes
Complex 1-polytopes may have two vertices or more, so dyadicity is already violated. Their vertices are simply located at distinct points in $$\mathbb{C}^1$$.

Real 1-polytopes enclose an interval of points as a consequence of the ordering of the real numbers. But as the complex numbers don't have an analogue of this ordering, it isn't meaningful to speak of the interior of a complex 1-polytope, nor a complex polytope of any higher rank.

There is exactly one regular complex 1-polytope with $n$ vertices for each $n$ &ge; 2, and its vertices are located at the $n$th roots of unity (or any combination of isometries and uniform scalings thereof), forming the vertices of a regular polygon in its Argand diagram.

Complex polygons
Complex polygons have vertices located in $$\mathbb{C}^2$$. While real edges connect two points in a line, a complex edge of degree $n$ (a.k.a. an $n$-edge) connects $n$ &ge; 2 points that live in an affine complex subspace of dimension 1. This is the complex space equivalent of the planarity condition for Euclidean polytopes. Each point must be adjacent to 2 or more edges, and the number of edges is also called its degree.

The regular complex polygons were completely characterized in 1991 by Coxeter. All the edges must have the same degree and the same goes for the vertices, so regular complex polygons are abstractly a type of configuration. Clearly, all edges of a regular complex polygon are themselves regular and congruent to each other.