Small birhombatotetracontoctachoron

Small birhombatotetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Sabric
Coxeter diagram	ao3ob4bo3oa&#zc (1/2 < b:a < 2+√2)
Elements
Cells	288 tetragonal disphenoids, 384 triangular antipodiums, 144 square antiprisms, 48 cuboctahedra
Faces	192+384 triangles, 1152+1152 isosceles triangles, 288 squares
Edges	576+1152+1152
Vertices	576
Vertex figure	Apex-truncated augmented triangular prism
Measures (based on two small rhombated icositetrachora of edge length 1)
Edge lengths	Lacing edges (1152): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	Base edge 1 (576): 1
	Base edge 2 (1152): 1
Circumradius	${\sqrt {4+2{\sqrt {2}}}}\approx 2.61313$
Central density	1
Related polytopes
Army	Sabric
Regiment	Sabric
Dual	Small biorthotetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The small birhombatotetracontoctachoron or sabric is a convex isogonal polychoron that consists of 48 cuboctahedra, 144 square antiprisms, 384 triangular antipodiums, and 288 tetragonal disphenoids. 1 cuboctahedron, 2 square antiprisms, 4 triangular antipodiums, and 2 tetragonal disphenoids join at each vertex.

It is one of a total of five distinct polychora (including two transitional cases) that can be obtained as the convex hull of two opposite small rhombated icositetrachora. In this case, the ratio between the edges of the small rhombated icositetrachoron a3o4b3o is between b:a = 1/2 (producing the rectified small prismatotetracontoctachoron) and b:a = 2+√2 (producing the rectified tetracontoctachoron). This includes the convex hull of two uniform small rhombated icositetrachora. The lacing edges generally have length ${\sqrt {(2-{\sqrt {2}})a^{2}-(6-4{\sqrt {2}})ab+(6-4{\sqrt {2}})b^{2}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {2+{\sqrt {2}}}{2}}}$ ≈ 1:1.30656.

External links

• Klitzing, Richard. "sabric".