Tetrahedral transitional biomnitruncatodecachoron

Tetrahedral transitional biomnitruncatodecachoron
File:Tetrahedral transitional biomnitruncatodecachoron.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Tetbotid
Coxeter diagram	ad3bc3cb3da&#ze (d = a+b/3-c/3)
Elements
Cells	30 rectangular trapezoprisms, 20 ditrigonal prisms, 10 orthoaligned ditruncated tetrahedra
Faces	120 isosceles trapezoids, 60+60 rectangles, 40 ditrigons, 20 dihexagons
Edges	120+120+120+120
Vertices	240
Vertex figure	Irregular tetrahedron
Measures (edge length 1)
Central density	1
Related polytopes
Army	Tetbotid
Regiment	Tetbotid
Dual	Tetrahedral transitional intersected biomnistellatodecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A4×2, order 240
Convex	Yes
Nature	Tame

The tetrahedral transitional biomnitruncatodecachoron or tetbotid is a convex isogonal polychoron that consists of 10 orthoaligned ditruncated tetrahedra, 20 ditrigonal prisms, and 30 rectangular trapezoprisms. 2 orthoaligned ditruncated tetrahedra, 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 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 great rhombitetratetrahedra a3b3c and b3c3d have the same circumradius, which happens if d = a+b/3-c/3. 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 tetrahedral transitional omnisnub bidecachoron, which is also nonuniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {3+{\sqrt {3}}}{3}}$ ≈ 1:1.57735.

External links

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