Hexagonal double antiprismoid

Hexagonal double antiprismoid
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Hidiap
Elements
Cells	288 sphenoids, 144 tetragonal disphenoids, 24 hexagonal antiprisms
Faces	144+288+576 isosceles triangles, 24 hexagons
Edges	144+288+288
Vertices	144
Vertex figure	Sphenocorona
Measures (for variant with unit uniform hexagonal antiprisms)
Edge lengths	Base edges of antiprisms (144): 1
	Side edges of antiprisms (288): 1
	Lacing edges of disphenoids (288): ${\sqrt {4+2{\sqrt {3}}-{\sqrt {19+11{\sqrt {3}}}}}}\approx 1.13817$
Circumradius	${\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\approx 1.93185$
Central density	1
Related polytopes
Army	Hidiap
Regiment	Hidiap
Dual	Hexagonal double antitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12)+≀S2×2, order 576
Convex	Yes
Nature	Tame

The hexagonal double antiprismoid or hidiap 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.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {11+4{\sqrt {3}}-{\sqrt {41+24{\sqrt {3}}}}}{8}}}$ ≈ 1:1.05128. For this variant the edges of the hexagons of the inscribed duoantiprisms have ratio 1:${\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}}$ ≈ 1:1.54171. A variant with uniform hexagonal antiprisms also exists; this variant is based on a duoantiprism with hexagons with edge length ratio 1:${\sqrt {1+{\sqrt {3}}}}$ ≈ 1:1.65289.

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:

• $\left(0,\,\pm 1,\,0,\,\pm {\sqrt {1+{\sqrt {3}}}}\right),$
• $\left(0,\,\pm 1,\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\sqrt {1+{\sqrt {3}}}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}}\right),$
• $\left(\pm 1,\,0,\,\pm {\sqrt {1+{\sqrt {3}}}},\,0\right),$
• $\left(\pm 1,\,0,\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\sqrt {1+{\sqrt {3}}}},\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}}\right),$
• $\left(\pm {\sqrt {1+{\sqrt {3}}}},\,0,\,0,\,\pm 1\right),$
• $\left(\pm {\sqrt {1+{\sqrt {3}}}},\,0,\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,0,\,\pm 1\right),$
• $\left(\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\sqrt {1+{\sqrt {3}}}},\,\pm 1,\,0\right),$
• $\left(0,\,\pm {\sqrt {1+{\sqrt {3}}}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right),$
• $\left(\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm 1,\,0\right),$
• $\left(\pm {\frac {\sqrt {3+3{\sqrt {3}}}}{2}},\,\pm {\frac {\sqrt {1+{\sqrt {3}}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right).$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by:

• $\left(0,\,\pm 1,\,0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}}\right),$
• $\left(0,\,\pm 1,\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}}\right),$
• $\left(\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}}\right),$
• $\left(\pm 1,\,0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,0\right),$
• $\left(\pm 1,\,0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,0\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,0,\,0,\,\pm 1\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,0,\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,0,\,\pm 1\right),$
• $\left(\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,\pm {\frac {\sqrt {3}}{2}},\,\pm {\frac {1}{2}}\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,\pm 1,\,0\right),$
• $\left(0,\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right),$
• $\left(\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm 1,\,0\right),$
• $\left(\pm {\frac {3+2{\sqrt {3}}+{\sqrt {12{\sqrt {3}}-3}}}{8}},\,\pm {\frac {2+{\sqrt {3}}+{\sqrt {4{\sqrt {3}}-1}}}{8}},\,\pm {\frac {1}{2}},\,\pm {\frac {\sqrt {3}}{2}}\right).$