Triacontahexafold octaswirlchoron

Triacontahexafold octaswirlchoron
File:Triacontahexafold octaswirlchoron.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	864 phyllic disphenoids, 288 triangular gyroprisms
Faces	1728 scalene triangles, 864 isosceles triangles, 288 triangles
Edges	216+864+864
Vertices	216
Vertex figure	Polyhedron with 8 tetragons and 16 triangles
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Cubiswirlic diacosihexadecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3●I2(36), order 1728
Convex	Yes
Nature	Tame

The triacontahexafold octaswirlchoron is an isogonal polychoron with 288 triangular gyroprisms, 864 phyllic disphenoids, and 216 vertices. 8 triangular gyroprisms and 16 phyllic disphenoids join at each vertex. It is the ninth in an infinite family of isogonal octahedral swirlchora.

The ratio between the longest and shortest edges is 1:${\frac {\sqrt {6-2{\sqrt {3}}}}{4\sin {\frac {\pi }{36}}}}$ ≈ 1:4.56783.

Vertex coordinates

Coordinates for the vertices of a triacontahexafold octaswirlchoron of circumradius 1, centered at the origin, are given by all permutations and sign changes of:

• ±(0, 0, sin(kπ/18), cos(kπ/18)),
• ±(sin(kπ/18), cos(kπ/18), 0, 0),

along with 90°, 180° and 270° rotations in the xy axis of:

• ±(sin((k+1/2)π/18)/√2, cos((k+1/2)π/18)/√2, cos((k+1/2)π/18)/√2, sin((k+1/2)π/18)/√2),

where k is an integer from 0 to 17.