Hexagonal duoexpandoprism

The hexagonal duoexpandoprism or hiddep is a convex isogonal polychoron and the fourth member of the duoexpandoprism family. It consists of 24 hexagonal prisms of two kinds, 36 rectangular trapezoprisms, 72 wedges, and 36 tetragonal disphenoids. Each vertex joins 2 hexagonal prisms, 1 tetragonal disphenoid, 3 wedges, and 2 rectangular trapezoprisms. It can be obtained as the convex hull of two orthogonal hexagonal-dodecagonal duoprisms, or more generally hexagonal-dihexagonal duoprisms, and a subset of its variations can be constructed by expanding the cells of the hexagonal duoprism outward. However, it cannot be made uniform.

This is one of a total of five polychora that can be obtained as the convex hull of two orthogonal hexagonal-dihexagonal duoprisms. To produce variants of this polychoron, if the polychoron is written as ao3bc oa3cb&#zy, c must be in the range $$c < b+\frac{a\sqrt3}{2}$$. It generally has circumradius $$\sqrt{a^2+b^2+ab\sqrt3+c^2}$$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$2+\sqrt3-\sqrt{3+2\sqrt3}$$ ≈ 1:1.18959.

Vertex coordinates
The vertices of a hexagonal duoexpandoprism, constructed as the convex hull of two orthogonal hexagonal-dodecagonal duoprisms of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,±1,\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),$$
 * $$\left(0,\,±1,\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),$$
 * $$\left(0,\,±1,\,±\frac{2+\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac12,\,±\frac{2+\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,0,\,±1\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt3}{2},\,0,\,±1\right),$$
 * $$\left(±\frac{2+\sqrt3}{2},\,±\frac12,\,0,\,±1\right),$$
 * $$\left(±\frac{1+\sqrt3}{2},\,±\frac{1+\sqrt3}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac12,\,±\frac{2+\sqrt3}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{2+\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt3}{2},\,±\frac12\right).$$