Square double gyroantiprismoid

Square double gyroantiprismoid
File:Square double gyroantiprismoid.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	128 sphenoids, 64 rhombic disphenoids, 32 tetragonal disphenoids, 16 square antiprisms
Faces	256 scalene triangles, 128+128 isosceles triangles, 16 squares
Edges	32+64+128+128
Vertices	64
Vertex figure	Octakis digonal-octagonal gyrowedge
Measures (for variant with unit uniform square antiprisms)
Edge lengths	Disphenoid bases (32): ${\sqrt {1+{\sqrt {2}}}}-1\approx 0.55377$
	Edges of squares (64): 1
	Side edges of antiprisms (128): 1
	Lacing edges (128): ${\sqrt {2+{\sqrt {2}}+{\sqrt {2+2{\sqrt {2}}}}}}\approx 1.10311$
Circumradius	${\sqrt {\frac {2+{\sqrt {2}}}{2}}}\approx 1.30656$
Central density	1
Related polytopes
Dual	Square double gyroantitegmoid
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(8)≀S2)/2, order 256
Convex	Yes
Nature	Tame

The square double gyroantiprismoid is a convex isogonal polychoron and the third member of the double gyroantiprismoid family. It consists of 16 square antiprisms, 32 tetragonal disphenoids, 64 rhombic disphenoids, and 128 sphenoids. 2 square 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 square prismatic swirlchora.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {5-2{\sqrt {2}}}}$ ≈ 1:1.47363.

Vertex coordinates

Coordinates for the vertices of a square double gyroantiprismoid, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by:

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