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. Most complex polytopes are 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".

Complex $n$-space has an obvious mapping onto Euclidean 2$n$-space by unfolding each complex dimension into two real dimensions. In complex 1-space, this unfolding is known as an Argand diagram. While distances and translations work identically in complex $n$-space and Euclidean 2$n$-space, rotations and scaling do not, and isometries are more restrictive.

The isometries in $$\mathbb{C}^n$$ are precisely the transformations of the form $$\mathbf{x} \mapsto \mathbf{A}\mathbf{x} + \mathbf{b}$$ where $$\mathbf{A}$$ is a unitary matrix and $$\mathbf{b} \in \mathbb{C}^n$$.

Regularity for complex polytopes is, as usual, all flags being identical under the polytope's symmetry group.

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 rotation/uniform scaling/translation 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 $n$-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.