Small bicantitruncatotetracontoctachoron

Small bicantitruncatotetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Sobcotic
Coxeter diagram	ao3bc4cb3oa&#zd (c < b+a/2)
Elements
Cells	288 tetragonal disphenoids, 192 triangular prisms, 576 wedges, 144 ditetragonal trapezoprisms, 48 truncated cubes
Faces	384 triangles, 1152 isosceles triangles, 576 rectangles, 1152 isosceles trapezoids, 288 ditetragons
Edges	576+576+1152+1152
Vertices	1152
Vertex figure	Isosceles triangular antiprism
Measures (based on two great rhombated icositetrachora of edge length 1)
Edge lengths	Lacing edges (1152): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	Remaining edges (576+576+1152): 1
Circumradius	${\sqrt {10+6{\sqrt {2}}}}\approx 4.29945$
Central density	1
Related polytopes
Army	Sobcotic
Regiment	Sobcotic
Dual	Small bimetatetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The small bicantitruncatotetracontoctachoron or sobcotic, also known as the runcinated tetracontoctachoron or runcinated bitetracontoctachoron, is a convex isogonal polychoron that consists of 48 truncated cubes, 144 ditetragonal trapezoprisms, 192 triangular prisms, 576 wedges, and 288 tetragonal disphenoids. 1 truncated cube, 2 ditetragonal trapezoprisms, 1 triangular prism, 3 wedges, and 1 tetragonal disphenoid join at each vertex.

It is one of a total of five distinct polychora (including two transitional cases) that can be obtained as the convex hull of two opposite great rhombated icositetrachora. In this case, if the great rhombated icositetrachora are of the form a3b4c3o, then c must be less than b+a/2 (producing the transitional bicantitruncatotetracontoctachoron in the limiting case). This includes the convex hull of two uniform great rhombated icositetrachora. The lacing edges generally have length ${\sqrt {(2-{\sqrt {2}})a^{2}+(6-4{\sqrt {2}})a(b-c)+(6-4{\sqrt {2}})(b-c)^{2}}}$.

The cases where b = c (producing uniform truncated cubic cells) can also be considered the result of expanding the cells of the tetracontoctachoron or its dual bitetracontoctachoron outward and filling in the gaps.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {18-3{\sqrt {2}}+{\sqrt {410+266{\sqrt {2}}}}}{34}}$ ≈ 1:1.22930.

External links

• Klitzing, Richard. "sobcotic".