# Second noble octagrammic triacontahedron

Second noble octagrammic triacontahedron
Rank3
TypeNoble
Elements
Faces30 rectangular-symmetric octagrams
Edges60+60
Vertices60
Vertex figureButterfly
Measures (edge lengths ${\displaystyle {\sqrt {5+2{\sqrt {5}}}}}$, ${\displaystyle 2+{\sqrt {5}}}$)
Edge length ratio${\displaystyle {\sqrt {\frac {5+2{\sqrt {5}}}{5}}}\approx 1.37638}$
Circumradius${\displaystyle {\sqrt {\frac {11+4{\sqrt {5}}}{4}}}\approx 2.23295}$
Related polytopes
ArmySrid
DualThird noble faceting of icosidodecahedron
ConjugateNoble octagonal 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 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:${\displaystyle {\sqrt {\frac {5+2{\sqrt {5}}}{5}}}}$ ≈ 1:1.37638.

## Vertex coordinates

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

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

along with all even permutations of

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

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