Hexagonal double gyroantiprismoid

The hexagonal double gyroantiprismoid is a convex isogonal polychoron and the third member of the double gyroantiprismoid family. It consists of 24 hexagonal antiprisms, 72 tetragonal disphenoids, 144 rhombic disphenoids, and 288 sphenoids. 2 hexagonal antiprisms, 2 tetragonal disphenoids, 4 rhombic disphenoids, and 8 sphenoids join at each vertex. However, it cannot be made uniform. It is the second in an infinite family of isogonal hexagonal prismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{5-2\sqrt3+\sqrt{14-8\sqrt3}}$$ ≈ 1:1.38378.

Vertex coordinates
The vertices of a hexagonal double gyroantiprismoid, assuming that the hexagonal antiprisms are regular of edge length 1, centered at the origin, are given by:
 * $$\left(0,\,±1,\,0,\,±\sqrt{1+\sqrt3}\right),$$
 * $$\left(0,\,±1,\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,±\sqrt{1+\sqrt3}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2}\right),$$
 * $$\left(±1,\,0,\,±\sqrt{1+\sqrt3},\,0\right),$$
 * $$\left(±1,\,0,\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\sqrt{1+\sqrt3},\,0\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2}\right),$$
 * $$\left(0,\,±\sqrt{1+\sqrt3},\,0,\,±1\right),$$
 * $$\left(0,\,±\sqrt{1+\sqrt3},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,0,\,±1\right),$$
 * $$\left(±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\sqrt{1+\sqrt3},\,0,\,±1,\,0\right),$$
 * $$\left(±\sqrt{1+\sqrt3},\,0,\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±1,\,0\right),$$
 * $$\left(±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac12,\,±\frac{\sqrt3}{2}\right).$$