# Fifth noble stellation of rhombic triacontahedron

Fifth noble stellation of rhombic triacontahedron
Rank3
TypeNoble
SpaceSpherical
Elements
Faces30 rectangular-symmetric dodecagrams
Edges180
Vertices120
Vertex figureScalene triangle
Measures (edge lengths ${\displaystyle \sqrt{15}-2\sqrt3}$, ${\displaystyle 2\sqrt5-3}$, ${\displaystyle \sqrt{25-10\sqrt5}}$)
Edge length ratio${\displaystyle \sqrt{\frac{25+10\sqrt5}{3}} \approx 3.97327}$
Circumradius${\displaystyle \sqrt{\frac{31-12\sqrt5}{4}} \approx 1.02068}$
Number of external pieces300
Level of complexity18
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle \frac{3-\sqrt5}{2}}$ (dipentagon-rectangle), ${\displaystyle \sqrt5-2}$ (dipentagon-ditrigon), ${\displaystyle \frac{7-3\sqrt5}{2}}$ (ditrigon-rectangle)
DualFifth noble faceting of icosidodecahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic–30
OrientableYes
Genus16
Properties
SymmetryH3, order 120
ConvexNo
NatureTame
History
Discovered byEdmond Hess
First discovered1876

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:${\displaystyle \sqrt{\frac{25+10\sqrt5}{3}}}$ ≈ 1:3.97327.

## Vertex coordinates

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

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

along with all even permutations of:

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

These are the same coordinates as the great quasitruncated icosidodecahedron.