Noble octagonal triacontahedron

Noble octagonal triacontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Space	Spherical
Elements
Faces	30 rectangular-symmetric octagons
Edges	120
Vertices	60
Vertex figure	Butterfly
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
Army	Semi-uniform Ti, edge lengths $\frac{\sqrt5-1}{2}$ (pentagons) and 1 (between ditrigons)
Dual	First noble faceting of icosidodecahedron
Convex core	Rhombic triacontahedron
Abstract & topological properties
Flag count	480
Euler characteristic	–30
Orientable	No
Genus	32
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame
History
Discovered by	Max Brückner
First discovered	1906

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.