Sixth noble stellation of rhombic triacontahedron

Sixth 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 $\sqrt{5-2\sqrt5}$, $3$, $\sqrt{5+2\sqrt5}$)
Edge length ratio	$2+\sqrt5 \approx 4.23607$
Circumradius	$\sqrt{\frac{11}{4}} \approx 1.65831$
Number of external pieces	1020
Level of complexity	54
Related polytopes
Army	Semi-uniform Grid, edge lengths $\frac{3-\sqrt5}{2}$ (decagons), $\frac{\sqrt5-1}{2}$ (ditrigon-rectangle)
Dual	Sixth 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

The sixth 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:$2+\sqrt5$ ≈ 1:4.23607.

Vertex coordinates

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

• $\left(\pm\frac12,\,\pm\frac12,\,\pm\frac32\right),$
• $\left(\pm\frac12,\,\pm\frac{\sqrt5}{2},\,\pm\frac{\sqrt5}{2}\right),$

along with all even permutations of:

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

These are the same coordinates as the quasitruncated dodecadodecahedron.