Fourth noble kisombreroidal hexecontahedron

Fourth noble kisombreroidal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	60 asymmetric pentagons
Edges	30+60+60
Vertices	60
Vertex figure	Asymmetric pentagon
Measures (edge lengths $4{\sqrt {10-4{\sqrt {5}}}}$, $10-2{\sqrt {5}}$, $4{\sqrt {2}}$)
Edge length ratio	${\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\approx 1.37638$
Circumradius	$2{\sqrt {11-4{\sqrt {5}}}}\approx 2.86756$
Related polytopes
Army	Semi-uniform Ti, edge lengths $4{\sqrt {5}}-8$ (pentagons) and $6-2{\sqrt {5}}$ (between ditrigons)
Dual	Third kisombreroidal hexecontahedron
Conjugate	Third kisombreroidal hexecontahedron
Convex core	Deltoidal hexecontahedron
Abstract & topological properties
Flag count	600
Euler characteristic	–30
Schläfli type	{5,5}
Orientable	Yes
Genus	16
Properties
Symmetry	H3+, order 60
Flag orbits	10
Convex	No
Nature	Tame

The fourth noble kisombreroidal hexecontahedron is a noble polyhedron. Its 60 congruent faces are asymmetric pentagons meeting at congruent order-5 vertices. It is a faceting of the same semi-uniform truncated icosahedron hull as that of the truncated great dodecahedron. The ratio between the shortest and longest edges is 1:${\sqrt {\frac {5+2{\sqrt {5}}}{5}}}$ ≈ 1:1.37638.

Vertex coordinates

The coordinates of a fourth noble kisombreroidal hexecontahedron are all even permutations of:

• $\left(\pm \left(5-{\sqrt {5}}\right),\,\pm \left(3-{\sqrt {5}}\right),\,0\right)$,
• $\left(\pm \left(3-{\sqrt {5}}\right),\,\pm \left(2{\sqrt {5}}-2\right),\,\pm \left({\sqrt {5}}-1\right)\right)$,

plus all permutations of

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

These are the same coordinates as the first noble crossed kignathogrammic hexecontahedron, first kipentagrammic hexecontahedron, first noble ditrapezoidal hexecontahedron, and second kisombreroidal hexecontahedron.