Square tetraswirlprism

Square tetraswirlprism
File:Square tetraswirlprism.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	128 phyllic disphenoids, 64 rhombic disphenoids, 32 square gyroprisms
Faces	256+256 scalene triangles, 32 squares
Edges	64+64+128+128
Vertices	64
Vertex figure	12-vertex polyhedron with 4 tetragons and 12 triangles
Measures (based on square duoprisms of edge length 1)
Edge lengths	Short side edges (64): ${\sqrt {2-{\sqrt {2+{\sqrt {2}}}}}}\approx 0.39018$
	Medium side edges (64): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	Long side edges (128): ${\sqrt {2-{\sqrt {\frac {2+{\sqrt {2}}}{2}}}}}\approx 0.83273$
	Edges of squares (128): 1
Circumradius	1
Central density	1
Related polytopes
Dual	Square tetraswirltegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(16)≀S2)+/4, order 256
Convex	Yes
Nature	Tame

The square tetraswirlprism is a convex isogonal polychoron and member of the duoprismatic swirlprism family that consists of 32 square gyroprisms, 64 rhombic disphenoids, and 128 phyllic disphenoids. 4 square gyroprisms, 4 rhombic disphenoids and 8 phyllic disphenoids join at each vertex. It can be obtained as a subsymmetrical faceting of the hexadecagonal duoprism. It is the fourth in an infinite family of isogonal square dihedral swirlchora and also the seventh in an infinite family of isogonal digonal prismatic swirlchora, the other being the digonal double octaswirlprism.

The ratio between the longest and shortest edges is 1:${\frac {\sqrt {8+4{\sqrt {2}}+2{\sqrt {20+14{\sqrt {2}}}}}}{2}}$ ≈ 1:2.56292.

Vertex coordinates

Coordinates for the vertices of a square tetraswirlprism constructed as the convex hull of four square duoprisms of edge length 1, are given as Cartesian products of the vertices of square S₁:

• S₁ × S₁,
• S₂ × S₂ (S₁ rotated 22.5 degrees),
• S₃ × S₃ (S₁ rotated 45 degrees),
• S₄ × S₄ (S₁ rotated 67.5 degrees).