# 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 $\sqrt{5-2\sqrt5}$ , $3$ , $\sqrt{5+2\sqrt5}$ )
Edge length ratio$2+\sqrt5 \approx 4.23607$ Circumradius$\sqrt{\frac{11}{4}} \approx 1.65831$ Number of external pieces1020
Level of complexity54
Related polytopes
ArmySemi-uniform Grid, edge lengths $\frac{3-\sqrt5}{2}$ (decagons), $\frac{\sqrt5-1}{2}$ (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:$2+\sqrt5$ ≈ 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.