# Sixth noble stellation of rhombic triacontahedron

Sixth noble stellation of rhombic triacontahedron
Rank3
TypeNoble
Elements
Faces30 rectangular-symmetric dodecagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle
Measures (edge lengths ${\displaystyle {\sqrt {5-2{\sqrt {5}}}}}$, ${\displaystyle 3}$, ${\displaystyle {\sqrt {5+2{\sqrt {5}}}}}$)
Edge length ratio${\displaystyle 2+{\sqrt {5}}\approx 4.23607}$
Circumradius${\displaystyle {\sqrt {\frac {11}{4}}}\approx 1.65831}$
Number of external pieces1020
Level of complexity54
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (decagons), ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (ditrigon-rectangle)
DualSixth noble faceting of icosidodecahedron
ConjugateSixth noble stellation of rhombic triacontahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic–30
OrientableYes
Genus16
Properties
SymmetryH3, order 120
Flag orbits6
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 the same semi-uniform great rhombicosidodecahedron hull as that of the quasitruncated dodecadodecahedron.

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

## Vertex coordinates

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

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

along with all even permutations of:

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

These are the same coordinates as the quasitruncated dodecadodecahedron.