Triangular double antiprismoid

The triangular double antiprismoid is a convex isogonal polychoron and the second member of the double antiprismoids that consists of 12 triangular antiprisms, 36 tetragonal disphenoids and 72 sphenoids obtained as the convex hull of two orthogonal triangular-triangular duoantiprisms. However, it cannot be made uniform. It is the first in an infinite family of isogonal triangular antiprismatic swirlchora.

If the triangular antiprisms are regular octahedra, the triangular double antiprismoid can be vertex-inscribed into a bitetracontoctachoron.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt{14+2√17}$ ≈ 1:1.17915.

Vertex coordinates
Coordinates for the vertices of a triangular double antiprismoid, assuming that the octahedra are regular of edge length 1, centered at the origin, are given by:
 * (0, $\sqrt{3}$/3, 0, $\sqrt{6}$/3),
 * (0, –$\sqrt{3}$/3, 0, –$\sqrt{6}$/3),
 * (0, $\sqrt{3}$/3, ±$\sqrt{2}$/2, –$\sqrt{6}$/6),
 * (0, –$\sqrt{3}$/3, ±$\sqrt{2}$/2, $\sqrt{6}$/6),
 * (±1/2, –$\sqrt{3}$/6, 0, $\sqrt{6}$/3),
 * (±1/2, $\sqrt{3}$/6, 0, –$\sqrt{6}$/3),
 * (±1/2, $\sqrt{3}$/6, ±$\sqrt{2}$/2, $\sqrt{6}$/6),
 * (±1/2, –$\sqrt{3}$/6, ±$\sqrt{2}$/2, –$\sqrt{6}$/6),
 * (0, $\sqrt{6}$/3, 0, –$\sqrt{3}$/3),
 * (0, –$\sqrt{6}$/3, 0, $\sqrt{3}$/3),
 * (0, $\sqrt{6}$/3, ±1/2, $\sqrt{3}$/6),
 * (0, –$\sqrt{6}$/3, ±1/2, –$\sqrt{3}$/6),
 * (±$\sqrt{2}$/2, $\sqrt{6}$/6, 0, $\sqrt{3}$/3),
 * (±$\sqrt{2}$/2, –$\sqrt{6}$/6, 0, –$\sqrt{3}$/3),
 * (±$\sqrt{2}$/2, $\sqrt{6}$/6, ±1/2, –$\sqrt{3}$/6),
 * (±$\sqrt{2}$/2, –$\sqrt{6}$/6, ±1/2, $\sqrt{3}$/6).

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * (0, $\sqrt{3}$/3, 0, $\sqrt{78+18√17}$/12),
 * (0, –$\sqrt{3}$/3, 0, –$\sqrt{78+18√17}$/12),
 * (0, $\sqrt{3}$/3, ±(3+$\sqrt{17}$)/8, –$\sqrt{78+18√17}$/24),
 * (0, –$\sqrt{3}$/3, ±(3+$\sqrt{17}$)/8, $\sqrt{78+18√17}$/24),
 * (±1/2, –$\sqrt{3}$/6, 0, $\sqrt{78+18√17}$/12),
 * (±1/2, $\sqrt{3}$/6, 0, –$\sqrt{78+18√17}$/12),
 * (±1/2, $\sqrt{3}$/6, ±(3+$\sqrt{17}$)/8, $\sqrt{78+18√17}$/24),
 * (±1/2, –$\sqrt{3}$/6, ±(3+$\sqrt{17}$)/8, –$\sqrt{78+18√17}$/24),
 * (0, $\sqrt{78+18√17}$/12, 0, –$\sqrt{3}$/3),
 * (0, –$\sqrt{78+18√17}$/12, 0, $\sqrt{3}$/3),
 * (0, $\sqrt{78+18√17}$/12, ±1/2, $\sqrt{3}$/6),
 * (0, –$\sqrt{78+18√17}$/12, ±1/2, –$\sqrt{3}$/6),
 * (±(3+$\sqrt{17}$)/8, $\sqrt{78+18√17}$/24, 0, $\sqrt{3}$/3),
 * (±(3+$\sqrt{17}$)/8, –$\sqrt{78+18√17}$/24, 0, –$\sqrt{3}$/3),
 * (±(3+$\sqrt{17}$)/8, $\sqrt{78+18√17}$/24, ±1/2, –$\sqrt{3}$/6),
 * (±(3+$\sqrt{17}$)/8, –$\sqrt{78+18√17}$/24, ±1/2, $\sqrt{3}$/6).