Prismatic transitional biomnitruncatodecachoron

Prismatic transitional biomnitruncatodecachoron
File:Prismatic transitional biomnitruncatodecachoron.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Petbotid
Coxeter diagram	ad3bc3cb3da&#ze (d = a+b/2-c/2)
Elements
Cells	20 ditrigonal trapezoprisms, 20 orthoaligned truncated triangular prisms, 10 great rhombitetratetrahedra
Faces	120 isosceles trapezoids, 60 rectangles, 40+40 ditrigons, 30 ditetragons
Edges	120+120+120+120
Vertices	240
Vertex figure	Irregular tetrahedron
Measures (edge length 1)
Central density	1
Related polytopes
Dual	Prismatic transitional intersected biomnistellatodecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The prismatic transitional biomnitruncatodecachoron or petbotid is a convex isogonal polychoron that consists of 10 great rhombitetratetrahedra, 20 orthoaligned truncated triangular prisms, and 20 ditrigonal trapezoprisms. 1 great rhombitetratetrahedron, 2 orthoaligned truncated triangular prisms, and 1 ditrigonal 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 disprismatopentapentachora (that is, variants of the great prismatodecachoron with single A4 symmetry). If the great disprismatopentapentachora are of the form a3b3c3d, then this form occurs when the ditrigonal prisms a3b d and a c3d have the same circumradius, which happens if d = a+b/2-c/2. The lacing edges generally have length ${\sqrt {\frac {3(a-d)^{2}+2(b-c)^{2}+2(a-d)(b-c)}{5}}}$.

This polychoron can be alternated into a prismatic transitional omnisnub bidecachoron, which is also nonuniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {2+{\sqrt {2}}}{2}}$ ≈ 1:1.70711.

External links

• Bowers, Jonathan. "Pennic and Decaic Isogonals".