Fifth noble stellation of rhombic triacontahedron

Fifth noble stellation of rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Space	Spherical
Elements
Faces	30 rectangular-symmetric dodecagrams
Edges	180
Vertices	120
Vertex figure	Scalene triangle
Measures (edge lengths $\sqrt{15}-2\sqrt3$, $2\sqrt5-3$, $\sqrt{25-10\sqrt5}$)
Edge length ratio	$\sqrt{\frac{25+10\sqrt5}{3}} \approx 3.97327$
Circumradius	$\sqrt{\frac{31-12\sqrt5}{4}} \approx 1.02068$
Number of external pieces	300
Level of complexity	18
Related polytopes
Army	Semi-uniform Grid, edge lengths $\frac{3-\sqrt5}{2}$ (dipentagon-rectangle), $\sqrt5-2$ (dipentagon-ditrigon), $\frac{7-3\sqrt5}{2}$ (ditrigon-rectangle)
Dual	Fifth noble faceting of icosidodecahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–30
Orientable	Yes
Genus	16
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame
History
Discovered by	Edmond Hess
First discovered	1876

The fifth noble stellation of rhombic triacontahedron is a noble polyhedron. Its 30 congruent faces are rectangular-symmetric dodecagrams meeting at congruent order-3 vertices. It is a faceting of a semi-uniform great rhombicosidodecahedron hull.

The ratio between the shortest and longest edges is 1:$\sqrt{\frac{25+10\sqrt5}{3}}$ ≈ 1:3.97327.

Vertex coordinates

A fifth noble stellation of rhombic triacontahedron, centered at the origin, has vertex coordinates given by all permutations of

• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac{2\sqrt5-3}{2}\right),$

along with all even permutations of:

• $\left(\pm\frac12,\,\pm\frac{\sqrt5-2}{2},\,\pm\frac{4-\sqrt5}{2}\right),$
• $\left(\pm1,\,\pm\frac{3-\sqrt5}{4},\,\pm\frac{7-3\sqrt5}{4}\right),$
• $\left(\pm\frac{3-\sqrt5}{4},\,\pm3\frac{\sqrt5-1}{4},\,\pm\frac{3-\sqrt5}{2}\right),$
• $\left(\pm\frac{\sqrt5-1}{2},\,\pm\frac{3\sqrt5-5}{4},\,\pm\frac{5-\sqrt5}{4}\right).$

These are the same coordinates as the great quasitruncated icosidodecahedron.