First noble pterogrammic hexecontahedron

The  is a noble polyhedron. Its 60 congruent faces are mirror-symmetric hexagons meeting at congruent order-6 vertices. It is a faceting of a semi-uniform truncated icosahedron hull.

The ratio between the shortest and longest edges is 1:$$\sqrt{\frac{5+\sqrt5+\sqrt{22+10\sqrt5}}{2}}$$ ≈ 1:2.63595.

Vertex coordinates
The vertex coordinates of a are given by all even permutations of:
 * $$\left(0,\,\pm\left(\sqrt{3+\sqrt5} + \sqrt{4+2\sqrt5}\right),\,\pm\left(\sqrt{1+\sqrt5} - \sqrt2\right)\right),$$
 * $$\left(\pm\sqrt2,\,\pm\sqrt{2\left(2+\sqrt5\right)},\,\pm\sqrt{2\left(2+\sqrt5+2\sqrt{2+\sqrt5}\right)}\right),$$
 * $$\left(\pm\sqrt{1+\sqrt5},\,\pm\sqrt{3+\sqrt5},\,\pm\left(\sqrt2 + \sqrt{4+2\sqrt5}\right)\right).$$

Related polyhedra
It shares its vertex coordinates with the fifth noble unihexagrammic hexecontahedron.