Sixth noble stellation of rhombic triacontahedron

Sixth noble stellation of rhombic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	30 rectangular-symmetric dodecagrams
Edges	60+60+60
Vertices	120
Vertex figure	Scalene triangle
Measures (edge lengths ${\sqrt {5-2{\sqrt {5}}}}$, $3$, ${\sqrt {5+2{\sqrt {5}}}}$)
Edge length ratio	$2+{\sqrt {5}}\approx 4.23607$
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 ${\frac {3-{\sqrt {5}}}{2}}$ (decagons), ${\frac {{\sqrt {5}}-1}{2}}$ (ditrigon-rectangle)
Dual	Sixth noble faceting of icosidodecahedron
Conjugate	Sixth noble stellation of rhombic triacontahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	720
Euler characteristic	–30
Orientable	Yes
Genus	16
Properties
Symmetry	H3, order 120
Flag orbits	6
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 the same semi-uniform great rhombicosidodecahedron hull as that of the quasitruncated dodecadodecahedron.

The ratio between the shortest and longest edges is 1:$2+{\sqrt {5}}$ ≈ 1:4.23607.

Vertex coordinates

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

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

along with all even permutations of:

• $\left(\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {{\sqrt {5}}-1}{2}}\right)$,
• $\left(\pm {\frac {3+{\sqrt {5}}}{4}},\,\pm {\frac {3-{\sqrt {5}}}{4}},\,\pm 1\right)$.

These are the same coordinates as the quasitruncated dodecadodecahedron.