# Small rhombated faceted hexacosichoron

Small rhombated faceted hexacosichoron
Rank4
TypeUniform
Notation
Bowers style acronymSirfix
Coxeter diagramo5/2x5o3x ()
Elements
Cells720 pentagrammic prisms, 120 dodecadodecahedra, 120 small rhombicosidodecahedra
Faces1200 triangles, 3600 squares, 1440 pentagons, 1440 pentagrams
Edges3600+7200
Vertices3600
Vertex figureWedge, edge lengths (1+5)/2 (two edges of base rectangle), (5–1)/2 (other 2 base edges), 2 (side edges), and 1 (top edge)
Measures (edge length 1)
Circumradius${\displaystyle {\sqrt {10+4{\sqrt {5}}}}\approx 4.35250}$
Hypervolume${\displaystyle 425{\frac {13+6{\sqrt {5}}}{2}}\approx 5613.48667}$
Dichoral anglesDid–5/2–stip: 162°
Srid–4–stip: ${\displaystyle \arccos \left(-{\sqrt {\frac {5+{\sqrt {5}}}{10}}}\right)\approx 148.28253^{\circ }}$
Srid–5–did: 144°
Srid–3–srid: 120°
Related polytopes
ArmySemi-uniform Srix, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (icosidodecahedra), 1 (main triangles)
RegimentSirfix
ConjugateQuasirhombated great faceted hexacosichoron
Convex coreHecatonicosachoron
Abstract & topological properties
Flag count129600
Euler characteristic–480
OrientableYes
Properties
SymmetryH4, order 14400
ConvexNo
NatureTame

The small rhombated faceted hexacosichoron, or sirfix, is a nonconvex uniform polychoron that consists of 720 pentagrammic prisms, 120 dodecadodecahedra, and 120 small rhombicosidodecahedra. 1 dodecadodecahedron, 2 pentagrammic prisms, and 2 small rhombicosidodecahedra join at each vertex. it can be obtained by cantellating the faceted hexacosichoron.

## Vertex coordinates

Coordinates for the vertices of a small rhombated faceted hexacosichoron of edge length 1 are given by all permutations of:

• ${\displaystyle \left(0,\,0,\,\pm 1,\,\pm (2+{\sqrt {5}})\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {1}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}}\right)}$,

together with all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+2{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(0,\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm (2+{\sqrt {5}}),\,\pm {\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm 3{\frac {3+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {5+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {3+2{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm {\frac {4+{\sqrt {5}}}{2}}\right)}$,
• ${\displaystyle \left(\pm 1,\,\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {7+3{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}},\,\pm {\frac {3+{\sqrt {5}}}{2}},\,\pm 3{\frac {1+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,\pm {\frac {5+3{\sqrt {5}}}{4}},\,\pm {\frac {2+{\sqrt {5}}}{2}}\right)}$.

## Related polychora

The small rhombated faceted hexacosichoron is the colonel of a regiment of 7 members. Its other members include the small retrosphenoverted trishecatonicosachoron, rhombic small hecatonicosihecatonicosachoron, small pseudorhombic hecatonicosihecatonicosachoron, grand rhombic dishecatonicosachoron, small dishecatonicosintercepted dishecatonicosachoron, and hecatonicosintercepted prismatodishecatonicosachoron.