Small biprismatorhombatotetracontoctachoron

Small biprismatorhombatotetracontoctachoron
(OFF file)
Rank	4
Type	Isogonal
Notation
Bowers style acronym	Sabiparc
Coxeter diagram	ac3bo4ob3ca&#zd (c < a+(2-√2)b/2
Elements
Cells	192 triangular prisms, 576 wedges, 288 rectangular trapezoprisms, 144 square antiprisms, 48 small rhombicuboctahedra
Faces	384 triangles, 1152 isosceles triangles, 288 squares, 576+576 rectangles, 1152 isosceles trapezoids
Edges	576+1152+1152+1152
Vertices	1152
Vertex figure	Triangular-isosceles trapezoidal wedge
Measures (based on two prismatorhombated icositetrachora of edge length 1)
Edge lengths	Lacing edges (1152): $2-{\sqrt {2}}\approx 0.58579$
	Remaining edges (576+1152+1152): 1
Circumradius	${\sqrt {8+3{\sqrt {2}}}}\approx 3.49895$
Central density	1
Related polytopes
Army	Sabiparc
Regiment	Sabiparc
Dual	Small bicrystallorthotetracontoctachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	F4×2, order 2304
Convex	Yes
Nature	Tame

The small biprismatorhombatotetracontoctachoron or sabiparc is a convex isogonal polychoron that consists of 48 small rhombicuboctahedra, 144 square antiprisms, 288 rectangular trapezoprisms, 192 triangular prisms, and 576 wedges. 1 small rhombicuboctahedron, 1 square antiprism, 2 rectangular trapezoprisms, 1 triangular prism, and 3 wedges 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 prismatorhombated icositetrachora. In this case, if the prismatorhombated icositetrachora are of the form a3b4o3c, then c must be less than $a+{\frac {2-{\sqrt {2}}}{2}}b$ (producing the transitional biprismatorhombatotetracontoctachoron in the limiting case). This includes the convex hull of two uniform prismatorhombated icositetrachora. The lacing edges generally have length ${\sqrt {(6-4{\sqrt {2}})b^{2}+(6-4{\sqrt {2}})a(b-c)+(6-4{\sqrt {2}})(b-c)^{2}+(2-{\sqrt {2}})(a-c)^{2}}}$.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {38+3{\sqrt {2}}+{\sqrt {842+786{\sqrt {2}}}}}{62}}$ ≈ 1:1.39422.