Final stellation of the icosahedron

Final stellation of the icosahedron
(3D model) (OFF file)
Rank	3
Type	Noble
Elements
Faces	20 triangular-symmetric great enneagrams
Edges	30+60
Vertices	60
Vertex figure	Isosceles triangle
Measures (unit-edge core icosahedron)
Edge lengths	Short edges (30): ${\frac {15+7{\sqrt {5}}}{2}}\approx 15.32624$
	Long edges (60): ${\sqrt {123+55{\sqrt {5}}}}\approx 15.68387$
Edge length ratio	${\frac {5{\sqrt {2}}+{\sqrt {10}}}{10}}\approx 1.02333$
Circumradius	${\frac {\sqrt {506+226{\sqrt {5}}}}{4}}\approx 7.95044$
Volume	${\frac {2895+1295{\sqrt {5}}}{12}}\approx 482.55900$
Central density	25
Number of external pieces	180
Level of complexity	9
Related polytopes
Army	Semi-uniform truncated icosahedron
Dual	Great noble triangular hexecontahedron
Conjugate	Crennell number 4 stellation of the icosahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	360
Euler characteristic	–10
Orientable	Yes
Genus	6
Properties
Symmetry	H3, order 120
Flag orbits	3
Convex	No
History
Discovered by	Edmond Hess
First discovered	1875

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:${\frac {\sqrt {15+5{\sqrt {5}}}}{5}}$ ≈ 1:1.02333, making it a near-miss uniform polyhedron.

Gallery

External links

• Wikipedia contributors. "Final stellation of the icosahedron".
• Verse and Dimensions Wiki Contributors. "Echidnahedron".