Great transitional hexadecafold tetraantiprismatoswirlchoron

Great transitional hexadecafold tetraantiprismatoswirlchoron
File:Great transitional hexadecafold tetraantiprismatoswirlchoron.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	128 phyllic disphenoids, 128 rhombic disphenoids, 128 bilaterally-symmetric notches, 32 square gyroprisms
Faces	256+256+256+256 scalene triangles, 32 squares
Edges	128+128+128+128+256
Vertices	128
Vertex figure	12-vertex polyhedron with 5 tetragons 10 triangles
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Great transitional tetraantitegmatoswirlic hecatonicosoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	((I2(8)×A1)/2)●I2(16), order 256
Convex	Yes
Nature	Tame

The great transitional hexadecafold tetraantiprismatoswirlchoron is an isogonal polychoron with 32 square gyroprisms, 128 mirror-symmetric notches, 128 rhombic disphenoids, and 128 phyllic disphenoids. 2 square gyroprisms, 5 bilaterally-symmetric notches, 4 rhombic disphenoids, and 4 phyllic disphenoids join at each vertex. It is the second in an infinite family of isogonal square antiprismatic swirlchora, the others being the small hexadecafold tetraantiprismatoswirlchoron, great hexadecafold tetraantiprismatoswirlchoron and small transitional hexadecafold tetraantiprismatoswirlchoron.

The ratio between the longest and shortest edges is 1:a ≈ 1:2.24110, where a is the largest positive real root of 1217a¹⁶-14888a¹⁴+69880a¹²-169568a¹⁰+235472a⁸-190976a⁶+86272a⁴-18432a²+1024.

Vertex coordinates

Coordinates for the vertices of a great transitional hexadecafold pentaantiprismatoswirlchoron, centered at the origin, are given by, along with their 90°, 180°, and 270° rotations in the xy axis of:

• ±(a*sin(kπ/8), a*cos(kπ/8), b*cos(kπ/8), b*sin(kπ/8)),
• ±(b*sin((k+2)π/8), b*cos((k+2)π/8), a*cos(kπ/8), a*sin(kπ/8)),

where b/a = (2+√2+2√2+√2+√8√2-2-4√4-2√2)/4 and k is an integer from 0 to 7.