# Second noble octagrammic triacontahedron

Second noble octagrammic triacontahedron
Rank3
TypeNoble
SpaceSpherical
Elements
Faces30 rectangular-symmetric octagrams
Edges120
Vertices60
Vertex figureButterfly
Measures (edge lengths ${\displaystyle \sqrt{5+2\sqrt5}}$, ${\displaystyle 2+\sqrt5}$)
Edge length ratio${\displaystyle \sqrt{\frac{5+2\sqrt5}{5}} \approx 1.37638}$
Circumradius${\displaystyle \sqrt{\frac{11+4\sqrt5}{4}} \approx 2.23295}$
Related polytopes
ArmySrid
DualThird 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 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:${\displaystyle \sqrt{\frac{5+2\sqrt5}{5}}}$ ≈ 1:1.37638.

## Vertex coordinates

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

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

along with all even permutations of

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

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