Second noble octagrammic triacontahedron

Second noble octagrammic triacontahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Space	Spherical
Elements
Faces	30 rectangular-symmetric octagrams
Edges	120
Vertices	60
Vertex figure	Butterfly
Measures (edge lengths $\sqrt{5+2\sqrt5}$, $2+\sqrt5$)
Edge length ratio	$\sqrt{\frac{5+2\sqrt5}{5}} \approx 1.37638$
Circumradius	$\sqrt{\frac{11+4\sqrt5}{4}} \approx 2.23295$
Related polytopes
Army	Srid
Dual	Third 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 second noble octagrammic triacontahedron is a noble polyhedron. Its 30 congruent faces are rectangular-symmetric octagrams meeting at congruent order-4 vertices. It is a faceting of a uniform small rhombicosidodecahedron hull.

The ratio between the shortest and longest edges is 1:$\sqrt{\frac{5+2\sqrt5}{5}}$ ≈ 1:1.37638.

Vertex coordinates

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

• $\left(\pm\frac{2+\sqrt5}{2},\,\pm\frac12,\,\pm\frac12\right),$

along with all even permutations of

• $\left(0,\,\pm\frac{3+\sqrt5}{4},\,\pm\frac{5+\sqrt5}{4}\right),$
• $\left(\pm\frac{1+\sqrt5}{4},\,\pm\frac{1+\sqrt5}{2},\,\pm\frac{3+\sqrt5}{4}\right).$

Other noble polyhedra that can have these coordinates are the Crennell number 4 stellation of the icosahedron and the third noble unihexagrammic hexecontahedron.