Great hexagonal hexecontahedron

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Great hexagonal hexecontahedron
Rank3
TypeUniform dual
Notation
Coxeter diagramp5/3p5/2p3*a
Elements
Faces60 irregular hexagons
Edges60+60+60
Vertices20+24+60
Vertex figure20+60 triangles, 24 pentagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {2}}{4}}\approx 0.35356}$
Dihedral angle90°
Central density10
Number of external pieces240
Related polytopes
DualGreat snub dodecicosidodecahedron
ConjugateGreat hexagonal hexecontahedron
Convex coreNon-Catalan deltoidal hexecontahedron
Abstract & topological properties
Flag count720
Euler characteristic–16
OrientableYes
Properties
SymmetryH3+, order 60
ConvexNo
NatureTame

The great hexagonal hexecontahedron is a uniform dual polyhedron. It consists of 60 irregular hexagons, each with two short, two medium, and two long edges.

If its dual, the great snub dodecicosidodecahedron, has an edge length of 1, then the hexagon faces have short edge length ${\displaystyle {\frac {\sqrt {2\left({\sqrt {5}}-1-2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}}\approx 0.18177}$, medium edge length ${\displaystyle {\frac {\sqrt {2\left({\sqrt {5}}-1+2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}}\approx 0.52533}$, and long edge length ${\displaystyle {\frac {\sqrt {2}}{2}}\approx 0.70711}$. The hexagons have one interior angle of ${\displaystyle \arccos \left(-\phi ^{-1}\right)\approx 128.17271^{\circ }}$, one of ${\displaystyle 360^{\circ }-\arccos \left(\phi ^{-1}\right)\approx 231.82792^{\circ }}$, and four of 90°, where ${\displaystyle \phi }$ is the golden ratio.

The great hexagonal hexecontahedron and the cube are the only finite non-degenerate isohedral polyhedra with right dihedral angles.

Vertex coordinates

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

• ${\displaystyle \left(\pm {\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,\pm {\frac {{\sqrt {10}}+{\sqrt {2}}}{8}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {\sqrt {{\sqrt {5}}-1}}{4}},\,0\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}},\,\pm {\frac {\sqrt {2}}{4}}\right),}$

as well as all even permutations and even sign changes of:

• ${\displaystyle \left({\frac {\sqrt {2\left({\sqrt {5}}-2\right)}}{4}},\,{\frac {\sqrt {2\left(1-2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}},\,{\frac {\sqrt {2\left(4-{\sqrt {5}}+2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}}\right),}$
• ${\displaystyle \left({\frac {{\sqrt {10}}-{\sqrt {2}}}{8}},\,{\frac {{\sqrt {10}}-3{\sqrt {2}}}{8}},\,{\frac {\sqrt {{\sqrt {5}}-1}}{2}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {2\left(4-{\sqrt {5}}-2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}},\,{\frac {\sqrt {2\left({\sqrt {5}}-2\right)}}{4}},\,{\frac {\sqrt {2\left(1+2{\sqrt {{\sqrt {5}}-2}}\right)}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {{\sqrt {5}}-1+2{\sqrt {2\left(5{\sqrt {5}}-1\right)}}}}{4}},\,-{\frac {\sqrt {2\left(3-{\sqrt {5}}-{\sqrt {2\left(5{\sqrt {5}}-1\right)}}\right)}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right),}$
• ${\displaystyle \left({\frac {\sqrt {{\sqrt {5}}-1-2{\sqrt {2\left(5{\sqrt {5}}-1\right)}}}}{4}},\,{\frac {\sqrt {2\left(3-{\sqrt {5}}+{\sqrt {2\left(5{\sqrt {5}}-1\right)}}\right)}}{4}},\,{\frac {1+{\sqrt {5}}}{4}}\right).}$