First noble ditrapezoidal hexecontahedron

First noble ditrapezoidal hexecontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	60 mirror-symmetric hexagons
Edges	60+120
Vertices	60
Vertex figure	Asymmetric hexagon
Measures (edge lengths $4{\sqrt {10-4{\sqrt {5}}}}$, $10-2{\sqrt {5}}$)
Edge length ratio	${\frac {\sqrt {5+{\sqrt {5}}}}{2}}\approx 1.34500$
Circumradius	$2{\sqrt {11-4{\sqrt {5}}}}\approx 2.86756$
Number of external pieces	2160
Related polytopes
Army	Semi-uniform Ti, edge lengths $4{\sqrt {5}}-8$ (pentagons) and $6-2{\sqrt {5}}$ (between ditrigons)
Dual	Third noble unihexagrammic hexecontahedron
Conjugate	Third noble unihexagrammic hexecontahedron
Convex core	Deltoidal hexecontahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–60
Schläfli type	{6,6}
Orientable	Yes
Genus	31
Properties
Symmetry	H3, order 120
Flag orbits	6
Convex	No
Nature	Tame

The first noble ditrapezoidal hexecontahedron is a noble polyhedron. Its 60 congruent faces are mirror-symmetric hexagons meeting at congruent order-6 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:${\frac {\sqrt {5+{\sqrt {5}}}}{2}}$ ≈ 1:1.34500.

Vertex coordinates

The coordinates of a first noble ditrapezoidal 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, fourth kisombreroidal hexecontahedron, and second kisombreroidal hexecontahedron.