Hexagonal double antiprismoid

{{Infobox polytope The hexagonal double antiprismoid orhidiap is a convex isogonal polychoron and the fifth member of the double antiprismoid family. It consists of 24 hexagonal antiprisms, 144 tetragonal disphenoids, and 288 sphenoids. 2 hexagonal antiprisms, 4 tetragonal disphenoids, and 8 sphenoids join at each vertex. It can be obtained as the convex hull of two orthogonal hexagonal-hexagonal duoantiprisms or by alternating the dodecagonal ditetragoltriate. However, it cannot be made uniform. It is the first in an infinite family of isogonal hexagonal antiprismatic swirlchora.
 * type=Isogonal
 * img=auto
 * off=auto
 * dim = 4
 * obsa = Hidiap
 * cells = 144 tetragonal disphenoids, 288 sphenoids, 24 hexagonal antiprisms
 * faces = 144+288+576 isosceles triangles, 24 hexagons
 * edges = 144+288+288
 * vertices = 144
 * verf = Sphenocorona
 * symmetry = I2(12)≀S2+, order 576
 * army=Hidiap
 * reg=Hidiap
 * custom_measure = (for variant with unit uniform hexagonal antiprisms)
 * el = Base edges of antiprisms (144): 1
 * el2 = Side edges of antiprisms (288): 1
 * el = Lacing edges of disphenoids (288): $$\sqrt{4+2\sqrt3-\sqrt{19+11\sqrt3}}} ≈ 1.13817$$
 * circum = $$\frac{\sqrt2+\sqrt6}{2} ≈ 1.93185$$
 * dual = Hexagonal double trapezohedroid
 * conv = Yes
 * orientable=Yes
 * nat=Tame}}

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{11+4\sqrt3-\sqrt{41+24\sqrt3}}{8}}$$ ≈ 1:1.05128. FOr this variant the edges of the hexagons of the inscribed duoantiprisms have ratio 1:$$\sqrt{1+\sqrt3}$$ ≈ 1:1.65289. A variant with uniform hexagonal antiprisms also exists; this variant is based on a duoantiprism with based on hexagons with edge length ratio 1:$$\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4}$$ ≈ 1:1.54171.

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:
 * $$\left(0,\,±1,\,0,\,±\sqrt{1+\sqrt3}\right),$$
 * $$\left(0,\,±1,\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,±\sqrt{1+\sqrt3}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2}\right),$$
 * $$\left(±1,\,0,\,±\sqrt{1+\sqrt3},\,0\right),$$
 * $$\left(±1,\,0,\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\sqrt{1+\sqrt3},\,0\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2}\right),$$
 * $$\left(±\sqrt{1+\sqrt3},\,0,\,0,\,±1\right),$$
 * $$\left(±\sqrt{1+\sqrt3},\,0,\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2},\,0,\,±1\right),$$
 * $$\left(±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(0,\,±\sqrt{1+\sqrt3},\,±1,\,0\right),$$
 * $$\left(0,\,±\sqrt{1+\sqrt3},\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,±1,\,0\right),$$
 * $$\left(±\frac{\sqrt{3+3\sqrt3}}{2},\,±\frac{\sqrt{1+\sqrt3}}{2},\,±\frac12,\,±\frac{\sqrt3}{2}\right).$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:
 * $$\left(0,\,±1,\,0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4}\right),$$
 * $$\left(0,\,±1,\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4}\right),$$
 * $$\left(±\frac{\sqrt3}{2},\,±\frac12,\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8}\right),$$
 * $$\left(±1,\,0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,0\right),$$
 * $$\left(±1,\,0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8}\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,0\right),$$
 * $$\left(±\frac12,\,±\frac{\sqrt3}{2},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8}\right),$$
 * $$\left(±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,0,\,0,\,±1\right),$$
 * $$\left(±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,0,\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,0,\,±1\right),$$
 * $$\left(±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,±\frac{\sqrt3}{2},\,±\frac12\right),$$
 * $$\left(0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,±1,\,0\right),$$
 * $$\left(0,\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{4},\,±\frac12,\,±\frac{\sqrt3}{2}\right),$$
 * $$\left(±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±1,\,0\right),$$
 * $$\left(±\frac{3+2\sqrt3+\sqrt{12\sqrt3-3}}{8},\,±\frac{2+\sqrt3+\sqrt{4\sqrt3-1}}{8},\,±\frac12,\,±\frac{\sqrt3}{2}\right).$$