# Sixth noble stellation of rhombic triacontahedron

Sixth noble stellation of rhombic triacontahedron
Rank3
TypeNoble
SpaceSpherical
Elements
Faces30 rectangular-symmetric dodecagrams
Edges180
Vertices120
Vertex figureScalene triangle
Measures (edge lengths ${\displaystyle \sqrt{5-2\sqrt5}}$, ${\displaystyle 3}$, ${\displaystyle \sqrt{5+2\sqrt5}}$)
Edge length ratio${\displaystyle 2+\sqrt5 \approx 4.23607}$
Circumradius${\displaystyle \sqrt{\frac{11}{4}} \approx 1.65831}$
Number of external pieces1020
Level of complexity54
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle \frac{3-\sqrt5}{2}}$ (decagons), ${\displaystyle \frac{\sqrt5-1}{2}}$ (ditrigon-rectangle)
DualSixth noble faceting of icosidodecahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic–30
OrientableYes
Genus16
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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:${\displaystyle 2+\sqrt5}$ ≈ 1:4.23607.

## Vertex coordinates

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

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

along with all even permutations of:

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

These are the same coordinates as the quasitruncated dodecadodecahedron.