Final stellation of the icosahedron

Final stellation of the icosahedron
Rank3
TypeNoble
Elements
Faces20 triangular-symmetric great enneagrams
Edges30+60
Vertices60
Vertex figureIsosceles triangle
Measures (unit-edge core icosahedron)
Edge lengthsShort edges (30): ${\displaystyle {\frac {15+7{\sqrt {5}}}{2}}\approx 15.32624}$
Long edges (60): ${\displaystyle {\sqrt {123+55{\sqrt {5}}}}\approx 15.68387}$
Edge length ratio${\displaystyle {\frac {5{\sqrt {2}}+{\sqrt {10}}}{10}}\approx 1.02333}$
Circumradius${\displaystyle {\frac {\sqrt {506+226{\sqrt {5}}}}{4}}\approx 7.95044}$
Volume${\displaystyle {\frac {2895+1295{\sqrt {5}}}{12}}\approx 482.55900}$
Central density25
Number of external pieces180
Level of complexity9
Related polytopes
ArmySemi-uniform truncated icosahedron
DualGreat noble triangular hexecontahedron
ConjugateCrennell number 4 stellation of the icosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count360
Euler characteristic–10
OrientableYes
Genus6
Properties
SymmetryH3, order 120
Flag orbits3
ConvexNo
History
Discovered byEdmond Hess
First discovered1875

The final stellation of the icosahedron, also known as the echidnahedron, is a noble polyhedron that consists of 20 irregular but triangular-symmetric great enneagrams. It is a stellation of the icosahedron.

Its convex hull is a variant of the truncated icosahedron with the same edge ratio as the convex hull of the truncated great dodecahedron.

It appears as the cell of the dual of the pentagonal-prismatic heptacosiicosachoron.

The ratio between the longest and shortest edges is 1:${\displaystyle {\frac {\sqrt {15+5{\sqrt {5}}}}{5}}}$ ≈ 1:1.02333, making it a near-miss uniform polyhedron.