Hexagonal double antiprismoid

The hexagonal double antiprismoid is a convex isogonal polychoron and the third member of the double antiprismoids that consists of 24 hexagonal antiprisms, 144 tetragonal disphenoids and 288 sphenoids obtained as the convex hull of two orthogonal hexagonal-hexagonal duoantiprisms. However, it cannot be made uniform. Together with its dual, it is the second in an infinite family of hexagonal antiprismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt{22+8√3-2√41+24√3}$/4 ≈ 1:1.05128.

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

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * (0, ±1, 0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4),
 * (0, ±1, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8),
 * (±$\sqrt{3}$/2, ±1/2, 0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4),
 * (±$\sqrt{3}$/2, ±1/2, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8),
 * (±1, 0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, 0),
 * (±1, 0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8),
 * (±1/2, ±$\sqrt{3}$/2, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, 0),
 * (±1/2, ±$\sqrt{3}$/2, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8),
 * (±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, 0, 0, ±1),
 * (±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, 0, ±$\sqrt{3}$/2, ±1/2),
 * (±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, 0, ±1),
 * (±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, ±$\sqrt{3}$/2, ±1/2),
 * (0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, ±1, 0),
 * (0, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/4, ±1/2, ±$\sqrt{3}$/2),
 * (±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±1, 0),
 * (±(3+2$\sqrt{3}$+$\sqrt{12√3-3}$)/8, ±(2+$\sqrt{3}$+$\sqrt{4√3-1}$)/8, ±1/2, ±$\sqrt{3}$/2).