Bitruncatotetracontoctachoron

Bitruncatotetracontoctachoron
File:Bitruncatotetracontoctachoron.png (OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Bitec
Coxeter diagram	xox3o4ox3ox&#zy
Elements
Cells	288 tetragonal disphenoids, 576 digonal disphenoids, 48 cubes, 144 square antiprisms
Faces	1152+1152 isosceles triangles, 288 squares
Edges	192+576+1152
Vertices	384
Vertex figure	Hexakis triangular cupola
Measures (based on two truncated icositetrachora of edge length 1)
Edge lengths	Edges of truncated icositetrachora (192+576): 1
	Lacing edges (1152): ${\sqrt {14-9{\sqrt {2}}}}\approx 1.12786$
Circumradius	${\sqrt {7}}\approx 2.64575$
Central density	1
Related polytopes
Army	Bitec
Regiment	Bitec
Dual	Biapiculatotetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The bitruncatotetracontoctachoron or bitec is a convex isogonal polychoron that consists of 48 cubes, 144 square antiprisms, 288 tetragonal disphenoids, and 576 digonal disphenoids. 1 cube, 3 square antiprisms, 3 tetragonal disphenoids, and 6 digonal disphenoids join at each vertex. It can be obtained as the convex hull of two oppositely oriented truncated icositetrachora.

The bitruncatotetracontoctachoron contains the vertices of a square-octagonal prismantiprismoid and the square double prismantiprismoid.

This polychoron generally has one degree of variation. If the edge length of the truncated icositetrachora are a (those surrounded by truncated octahedra) and b (of cubes), its lacing edges have length ${\sqrt {2a^{2}+6b^{2}+6ab-(a^{2}+4b^{2}+4ab){\sqrt {2}}}}$ and it has circumradius ${\sqrt {a^{2}+3b^{2}+3ab}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {14-9{\sqrt {2}}}}$ ≈ 1:1.12786.

Vertex coordinates

The vertices of a bitruncatotetracontoctachoron, assuming that the square antiprisms are uniform of edge length 1, centered at the origin, are given by all permutations of:

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

An alternate set of coordinates, based on two uniform truncated icositetrachora of edge length 1, centered at the origin, are given by all permutations of:

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

External links

• Klitzing, Richard. "bitec".