Cubic transitional biomnitruncatotetracontoctachoron

Cubic transitional biomnitruncatotetracontoctachoron
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Catbotic
Coxeter diagram	ad3bc4cb3da&#ze (d = a+√2b/3-√2c/3)
Elements
Cells	288 rectangular trapezoprisms, 192 ditrigonal prisms, 48 orthoaligned ditruncated cubes
Faces	1152 isosceles trapezoids, 576+576 rectangles, 384 ditrigons, 144 dioctagons
Edges	1152+1152+1152+1152
Vertices	2304
Vertex figure	Irregular tetrahedron
Measures (edge length 1)
Central density	1
Related polytopes
Army	Catbotic
Regiment	Catbotic
Dual	Cubic transitional intersected biomnistellatotetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The cubic transitional biomnitruncatotetracontoctachoron or catbotic, also known as the octahedral transitional biomnitruncatotetracontoctachoron, is a convex isogonal polychoron that consists of 48 orthoaligned ditruncated cubes, 192 ditrigonal prisms, and 288 rectangular trapezoprisms. 2 orthoaligned ditruncated cubes, 1 ditrigonal prism, and 1 rectangular trapezoprism join at each vertex.

It is one of a total of four distinct polychora (including two transitional cases) that can be obtained as the convex hull of two opposite great disprismatoicositetricositetrachora (that is, variants of the great prismatotetracontoctachoron with single F4 symmetry). If the great disprismatoicositetricositetrachora are of the form a3b4c3d, then this form occurs when the great rhombicuboctahedra a3b4c and b4c3d have the same circumradius, which happens if d = $a+{\frac {\sqrt {2}}{3}}b-{\frac {\sqrt {2}}{3}}c$. The lacing edges generally have length ${\sqrt {(2-{\sqrt {2}})(a-d)^{2}+(6-4{\sqrt {2}})(b-c)^{2}+(6-4{\sqrt {2}})(a-d)(b-c)}}$.

This polychoron can be alternated into a cubic transitional omnisnub bitetracontoctachoron, which is also nonuniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {2+{\sqrt {4-2{\sqrt {2}}}}}{2}}$ ≈ 1:1.54120.