Noble octagonal triacontahedron

Noble octagonal triacontahedron
Rank3
TypeNoble
Elements
Faces30 rectangular-symmetric octagons
Edges60+60
Vertices60
Vertex figureButterfly
Measures (edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$, ${\displaystyle {\sqrt {\frac {5+{\sqrt {5}}}{2}}}}$)
Edge length ratio${\displaystyle {\sqrt {5+2{\sqrt {5}}}}\approx 3.07768}$
Circumradius${\displaystyle {\sqrt {\frac {17+5{\sqrt {5}}}{8}}}\approx 1.87684}$
Related polytopes
ArmySemi-uniform Ti, edge lengths ${\displaystyle {\frac {{\sqrt {5}}-1}{2}}}$ (pentagons) and 1 (between ditrigons)
DualFirst noble faceting of icosidodecahedron
ConjugateSecond noble octagrammic triacontahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count480
Euler characteristic–30
OrientableNo
Genus32
Properties
SymmetryH3, order 120
Flag orbits4
ConvexNo
NatureTame
History
Discovered byMax Brückner
First discovered1906

The noble octagonal triacontahedron is a noble polyhedron. Its 30 congruent faces are rectangular-symmetric octagons meeting at congruent order-4 vertices. It is a faceting of the same semi-uniform truncated icosahedron hull as that of the truncated great dodecahedron.

The ratio between the shortest and longest edges is 1:${\displaystyle {\sqrt {5+2{\sqrt {5}}}}}$ ≈ 1:3.07768.

Vertex coordinates

A noble octagonal triacontahedron, centered at the origin, has vertex coordinates given by all permutations of:

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

plus all even permutations of:

• ${\displaystyle \left(0,\,\pm {\frac {1}{2}},\,\pm {\frac {5+{\sqrt {5}}}{4}}\right)}$,
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1+{\sqrt {5}}}{2}}\right)}$.

These are the same coordinates as the truncated great dodecahedron.