Fifth noble stellation of rhombic triacontahedron

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Fifth noble stellation of rhombic triacontahedron
Rank3
TypeNoble
Elements
Faces30 rectangular-symmetric dodecagrams
Edges60+60+60
Vertices120
Vertex figureScalene triangle
Measures (edge lengths , , )
Edge length ratio
Circumradius
Number of external pieces300
Level of complexity18
Related polytopes
ArmySemi-uniform Grid, edge lengths (dipentagon-rectangle), (dipentagon-ditrigon), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{7-3\sqrt5}{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:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\frac{25+10\sqrt5}{3}}} ≈ 1:3.97327.

Vertex coordinates[edit | edit source]

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

  • Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\pm\frac12,\,\pm\frac12,\,\pm\frac{2\sqrt5-3}{2}\right),}

along with all even permutations of:

These are the same coordinates as the great quasitruncated icosidodecahedron.