# Small icosacronic hexecontahedron

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Small icosacronic hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramm5/2o3m3*a
Elements
Faces60 kites
Edges60+60
Vertices12+20+20
Vertex figure20 triangles, 12 pentagrams, 20 hexagons
Measures (edge length 1)
Inradius${\displaystyle 3{\frac {\sqrt {305\left(9+2{\sqrt {5}}\right)}}{122}}\approx 1.57627}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {44+3{\sqrt {5}}}{61}}\right)\approx 146.23066^{\circ }}$
Central density2
Number of external pieces120
Related polytopes
DualSmall icosicosidodecahedron
ConjugateGreat icosacronic hexecontahedron
Convex coreNon-Catalan deltoidal hexecontahedron
Abstract & topological properties
Flag count480
Euler characteristic–8
OrientableYes
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

The small icosacronic hexecontahedron is a uniform dual polyhedron. It consists of 60 kites.

If its dual, the small icosicosidodecahedron, has an edge length of 1, then the short edges of the kites will measure ${\displaystyle {\frac {\sqrt {30\left(65-19{\sqrt {5}}\right)}}{22}}\approx 1.18133}$, and the long edges will be ${\displaystyle {\frac {\sqrt {30\left(85-{\sqrt {5}}\right)}}{38}}\approx 1.31129}$. ​The kite faces will have length ${\displaystyle 3{\frac {\sqrt {10\left(3517-585{\sqrt {5}}\right)}}{418}}\approx 1.06668}$, and width ${\displaystyle {\sqrt {5}}\approx 2.23607}$. ​The kites have two interior angles of ${\displaystyle \arccos \left({\frac {15-{\sqrt {5}}}{20}}\right)\approx 50.34252^{\circ }}$, one of ${\displaystyle \arccos \left(-{\frac {5+19{\sqrt {5}}}{60}}\right)\approx 142.31856^{\circ }}$, and one of ${\displaystyle \arccos \left(-{\frac {25+{\sqrt {5}}}{60}}\right)\approx 116.99640^{\circ }}$.

## Vertex coordinates

A small icosacronic hexecontahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• ${\displaystyle \left(\pm {\frac {5-{\sqrt {5}}}{4}},\,\pm {\frac {5+{\sqrt {5}}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm 3{\frac {5+7{\sqrt {5}}}{44}},\,\pm 3{\frac {15-{\sqrt {5}}}{44}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {\sqrt {5}}{2}},\,\pm {\frac {\sqrt {5}}{2}}\right),}$
• ${\displaystyle \left(\pm 3{\frac {9{\sqrt {5}}-5}{76}},\,\pm 3{\frac {15+11{\sqrt {5}}}{76}},\,0\right),}$
• ${\displaystyle \left(\pm 3{\frac {10+{\sqrt {5}}}{38}},\,\pm 3{\frac {10+{\sqrt {5}}}{38}},\,\pm 3{\frac {10+{\sqrt {5}}}{38}}\right).}$