# Small biprismatorhombatotetracontoctachoron

Small biprismatorhombatotetracontoctachoron
Rank4
TypeIsogonal
Notation
Bowers style acronymSabiparc
Coxeter diagramac3bo4ob3ca&#zd (c < a+(2-2)b/2
Elements
Cells192 triangular prisms, 576 wedges, 288 rectangular trapezoprisms, 144 square antiprisms, 48 small rhombicuboctahedra
Faces384 triangles, 1152 isosceles triangles, 288 squares, 576+576 rectangles, 1152 isosceles trapezoids
Edges576+1152+1152+1152
Vertices1152
Vertex figureTriangular-isosceles trapezoidal wedge
Measures (based on two prismatorhombated icositetrachora of edge length 1)
Edge lengthsLacing edges (1152): ${\displaystyle 2-{\sqrt {2}}\approx 0.58579}$
Remaining edges (576+1152+1152): 1
Circumradius${\displaystyle {\sqrt {8+3{\sqrt {2}}}}\approx 3.49895}$
Central density1
Related polytopes
ArmySabiparc
RegimentSabiparc
DualSmall bicrystallorthotetracontoctachoron
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryF4×2, order 2304
ConvexYes
NatureTame

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 ${\displaystyle a+{\frac {2-{\sqrt {2}}}{2}}b}$ (producing the transitional biprismatorhombatotetracontoctachoron in the limiting case). This includes the convex hull of two uniform prismatorhombated icositetrachora. The lacing edges generally have length ${\displaystyle {\sqrt {(6-4{\sqrt {2}})b^{2}+(6-4{\sqrt {2}})a(b-c)+(6-4{\sqrt {2}})(b-c)^{2}+(2-{\sqrt {2}})(a-c)^{2}}}}$.

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