Second noble unihexagrammic hexecontahedron

Second noble unihexagrammic hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	60 unicursal hexagrams
Edges	60+120
Vertices	60
Vertex figure	Mirror-symmetric hexagon
Measures (edge lengths $6{\sqrt {5}}-10$, $4{\sqrt {15-6{\sqrt {5}}}}$)
Edge length ratio	${\sqrt {\frac {3\left(5+{\sqrt {5}}\right)}{10}}}\approx 1.47337$
Circumradius	${\sqrt {74-30{\sqrt {5}}}}\approx 2.63020$
Related polytopes
Army	Semi-uniform Srid, edge lengths $4{\sqrt {5}}-8$ (pentagons) and $6-2{\sqrt {5}}$ (triangles)
Dual	First noble unihexagrammic hexecontahedron
Conjugate	First noble unihexagrammic hexecontahedron
Convex core	Triakis icosahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–60
Orientable	Yes
Genus	31
Properties
Symmetry	H3, order 120
Flag orbits	6
Convex	No
Nature	Tame

The second noble unihexagrammic hexecontahedron is a noble polyhedron. Its 60 congruent faces are unicursal hexagrams meeting at congruent order-6 vertices. It is a faceting of the same semi-uniform small rhombicosidodecahedron hull as that of the small icosicosidodecahedron.

The ratio between the shortest and longest edges is 1:${\sqrt {\frac {3\left(5+{\sqrt {5}}\right)}{10}}}$ ≈ 1:1.47337.

Vertex coordinates

The coordinates of a second noble unihexagrammic hexecontahedron are all even permutations of:

• $\left(\pm \left(3{\sqrt {5}}-5\right),\,\pm 2,\,0\right)$,
• $\left(\pm \left(3-{\sqrt {5}}\right),\,\pm 2,\,\pm 2\left(3-{\sqrt {5}}\right)\right)$,
• $\left(\pm \left(2{\sqrt {5}}-2\right),\,\pm \left(3-{\sqrt {5}}\right),\,\pm \left(2{\sqrt {5}}-4\right)\right)$.