Fifth noble stellation of rhombic triacontahedron

Fifth 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 ${\sqrt {15}}-2{\sqrt {3}}$, $2{\sqrt {5}}-3$, ${\sqrt {25-10{\sqrt {5}}}}$)
Edge length ratio	${\sqrt {\frac {25+10{\sqrt {5}}}{3}}}\approx 3.97327$
Circumradius	${\sqrt {\frac {31-12{\sqrt {5}}}{4}}}\approx 1.02068$
Number of external pieces	300
Level of complexity	18
Related polytopes
Army	Semi-uniform Grid, edge lengths ${\frac {3-{\sqrt {5}}}{2}}$ (dipentagon-rectangle), ${\sqrt {5}}-2$ (dipentagon-ditrigon), ${\frac {7-3{\sqrt {5}}}{2}}$ (ditrigon-rectangle)
Dual	Fifth noble faceting of icosidodecahedron
Conjugate	Final stellation of the 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
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 the same semi-uniform great rhombicosidodecahedron hull as that of the great quasitruncated icosidodecahedron.

The ratio between the shortest and longest edges is 1:${\sqrt {\frac {25+10{\sqrt {5}}}{3}}}$ ≈ 1:3.97327.

Vertex coordinates

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

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

along with all even permutations of:

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

These are the same coordinates as the great quasitruncated icosidodecahedron.