Triangular double gyroantiprismoid

Triangular double gyroantiprismoid
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	72 sphenoids, 36 rhombic disphenoids, 18 tetragonal disphenoids, 12 triangular antiprisms
Faces	144 scalene triangles, 72+72 isosceles triangles, 12 triangles
Edges	18+36+72+72
Vertices	36
Vertex figure	Octakis digonal-octagonal gyrowedge
Measures (for variant with unit uniform triangular antiprisms)
Edge lengths	Disphenoid bases (18): ${\frac {2{\sqrt {3}}-{\sqrt {6}}}{3}}\approx 0.33820$
	Edges of base triangles (36): 1
	Side edges of antiprisms (72): 1
	Lacing edges (72): ${\frac {\sqrt {18-6{\sqrt {2}}}}{3}}\approx 1.02820$
Circumradius	1
Central density	1
Related polytopes
Dual	Triangular double gyroantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(G2≀S2)/2, order 144
Convex	Yes
Nature	Tame

The triangular double gyroantiprismoid is a convex isogonal polychoron and the second member of the double gyroantiprismoid family. It consists of 12 triangular antiprisms, 18 tetragonal disphenoids, 36 rhombic disphenoids, and 72 sphenoids. 2 triangular 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 triangular prismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {15+3{\sqrt {6}}}}{3}}$ ≈ 1:1.57581.

Vertex coordinates

The vertices of a triangular double gyroantiprismoid, assuming that the octahedra are regular of edge length 1, centered at the origin, are given by:

• $\left(0,\,{\frac {\sqrt {3}}{3}},\,0,\,{\frac {\sqrt {6}}{3}}\right),$
• $\left(0,\,-{\frac {\sqrt {3}}{3}},\,0,\,-{\frac {\sqrt {6}}{3}}\right),$
• $\left(0,\,{\frac {\sqrt {3}}{3}},\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt {6}}{6}}\right),$
• $\left(0,\,-{\frac {\sqrt {3}}{3}},\,\pm {\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {6}}{6}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,0,\,{\frac {\sqrt {6}}{3}}\right),$
• $\left(\pm {\frac {1}{2}},\,{\frac {\sqrt {3}}{6}},\,0,\,-{\frac {\sqrt {6}}{3}}\right),$
• $\left(\pm {\frac {1}{2}},\,{\frac {\sqrt {3}}{6}},\,\pm {\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {6}}{6}}\right),$
• $\left(\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}},\,\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt {6}}{6}}\right),$
• $\left(0,\,{\frac {\sqrt {6}}{3}},\,0,\,{\frac {\sqrt {3}}{3}}\right),$
• $\left(0,\,-{\frac {\sqrt {6}}{3}},\,0,\,-{\frac {\sqrt {3}}{3}}\right),$
• $\left(0,\,{\frac {\sqrt {6}}{3}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}}\right),$
• $\left(0,\,-{\frac {\sqrt {6}}{3}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {3}}{6}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt {6}}{6}},\,0,\,{\frac {\sqrt {3}}{3}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {6}}{6}},\,0,\,-{\frac {\sqrt {3}}{3}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,{\frac {\sqrt {6}}{6}},\,\pm {\frac {1}{2}},\,{\frac {\sqrt {3}}{6}}\right),$
• $\left(\pm {\frac {\sqrt {2}}{2}},\,-{\frac {\sqrt {6}}{6}},\,\pm {\frac {1}{2}},\,-{\frac {\sqrt {3}}{6}}\right).$