# Noble octagonal triacontahedron

Noble octagonal triacontahedron Rank3
TypeNoble
SpaceSpherical
Elements
Faces30 rectangular-symmetric octagons
Edges120
Vertices60
Vertex figureButterfly
Measures (edge lengths $\frac{\sqrt5-1}{2}$ , $\sqrt{\frac{5+\sqrt5}{2}}$ )
Edge length ratio$\sqrt{5+2\sqrt5} \approx 3.07768$ Circumradius$\sqrt{\frac{17+5\sqrt5}{8}} \approx 1.87684$ Related polytopes
ArmySemi-uniform Ti, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons) and 1 (between ditrigons)
DualFirst noble faceting of icosidodecahedron
Convex coreRhombic triacontahedron
Abstract & topological properties
Flag count480
Euler characteristic–30
OrientableNo
Genus32
Properties
SymmetryH3, order 120
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 a semi-uniform truncated icosahedron hull.

The ratio between the shortest and longest edges is 1:$\sqrt{5+2\sqrt5}$ ≈ 1:3.07768.

## Vertex coordinates

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

• $\left(\pm\frac{\sqrt5-1}{4},\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{3+\sqrt5}{4}\right),$ plus all even permutations of:

• $\left(0,\,\pm\frac12,\,\pm\frac{5+\sqrt5}{4}\right),$ • $\left(\pm\frac12,\,\pm\frac{1+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{2}\right).$ These are the same coordinates as the truncated great dodecahedron.