# Fifth noble stellation of rhombic triacontahedron

Fifth noble stellation of rhombic triacontahedron
Rank3
TypeNoble
Elements
Faces30 rectangular-symmetric dodecagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle
Measures (edge lengths ${\displaystyle {\sqrt {15}}-2{\sqrt {3}}}$, ${\displaystyle 2{\sqrt {5}}-3}$, ${\displaystyle {\sqrt {25-10{\sqrt {5}}}}}$)
Edge length ratio${\displaystyle {\sqrt {\frac {25+10{\sqrt {5}}}{3}}}\approx 3.97327}$
Circumradius${\displaystyle {\sqrt {\frac {31-12{\sqrt {5}}}{4}}}\approx 1.02068}$
Number of external pieces300
Level of complexity18
Related polytopes
ArmySemi-uniform Grid, edge lengths ${\displaystyle {\frac {3-{\sqrt {5}}}{2}}}$ (dipentagon-rectangle), ${\displaystyle {\sqrt {5}}-2}$ (dipentagon-ditrigon), ${\displaystyle {\frac {7-3{\sqrt {5}}}{2}}}$ (ditrigon-rectangle)
DualFifth noble faceting of icosidodecahedron
ConjugateFinal stellation of the rhombic triacontahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count720
Euler characteristic–30
OrientableYes
Genus16
Properties
SymmetryH3, order 120
Flag orbits6
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 the same semi-uniform great rhombicosidodecahedron hull as that of the great quasitruncated icosidodecahedron.

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

## Vertex coordinates

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

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

along with all even permutations of:

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

These are the same coordinates as the great quasitruncated icosidodecahedron.