Truncated small prismatotetracontoctachoron

Truncated small prismatotetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Tispic
Elements
Cells	144 square antiprisms, 192 truncated triangular prisms, 48 truncated octahedra
Faces	1152 isosceles triangles, 288 squares, 384 ditrigons, 288 ditetragons
Edges	576+1152+1152
Vertices	1152
Vertex figure	Isosceles trapezoidal pyramid
Measures (variant with hexagons of edge length 1)
Edge lengths	Edges of hexagons (576+1152): 1
	Lacing edges (1152): ${\sqrt {2}}\approx 1.41421$
Circumradius	${\sqrt {16+9{\sqrt {2}}}}\approx 5.35984$
Central density	1
Related polytopes
Army	Tispic
Regiment	Tispic
Dual	Octakis square-antitegmatic hecatontetracontatetrachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The truncated small prismatotetracontoctachoron or tispic is a convex isogonal polychoron that consists of 48 truncated octahedra, 192 truncated triangular prisms, and 144 square antiprisms. 1 square antiprism, 1 truncated octahedron, and 3 truncated triangular prisms join at each vertex. It can be formed by truncating the small prismatotetracontoctachoron.

It can also be formed as the convex hull of 2 oppositely oriented semi-uniform variants of the prismatorhombated icositetrachoron, where if the prismatorhombated icositetrachora are of the form a3b4o3c, then c = a+2b. It is one of five polychora (including two transitional cases) formed from two prismatorhombated icositetrachora, and is the transitional point between the medial biprismatorhombatotetracontoctachoron and great biprismatorhombatotetracontoctachoron.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {2}}$ ≈ 1:1.41421. This variant uses regular hexagons as faces.

Vertex coordinates

The vertices of a truncated small prismatotetracontoctachoron of edge length 1, centered at the origin, are given by:

• $\left(0,\,\pm {\frac {\sqrt {2}}{2}},\,\pm {\sqrt {2}},\,\pm 3{\frac {2+{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {1+3{\sqrt {2}}}{2}},\,\pm {\frac {5+3{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {3}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}}\right),$
• $\left(\pm 1,\,\pm 1,\,\pm {\frac {2+3{\sqrt {2}}}{2}},\,\pm {\frac {4+3{\sqrt {2}}}{2}}\right),$
• $\left(\pm {\frac {3}{2}},\,\pm {\frac {3+{\sqrt {2}}}{2}},\,\pm {\frac {3+2{\sqrt {2}}}{2}},\,\pm 3{\frac {1+{\sqrt {2}}}{2}}\right).$

External links

• Klitzing, Richard. "tispic".