Small triambic icosahedron

Small triambic icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Stai
Coxeter diagram	m5/2o3o3*a ()
Elements
Faces	20 triambi
Edges	60
Vertices	12+20
Vertex figure	20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius	${\frac {3{\sqrt {30}}+5{\sqrt {6}}}{24}}\approx 1.19496$
Volume	${\frac {175{\sqrt {2}}+75{\sqrt {10}}}{24}}\approx 20.19409$
Surface area	${\frac {15{\sqrt {15}}+25{\sqrt {3}}}{2}}\approx 50.69801$
Dihedral angle	$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$
Central density	2
Number of external pieces	60
Related polytopes
Dual	Small ditrigonary icosidodecahedron
Conjugate	Great triambic icosahedron
Convex hull	Non-Catalan Pentakis dodecahedron
Convex core	Icosahedron
Abstract & topological properties
Flag count	240
Euler characteristic	–8
Orientable	Yes
Genus	5
Properties
Symmetry	H3, order 120
Convex	No
Nature	Tame

The small triambic icosahedron is a uniform dual polyhedron. It consists of 20 irregular hexagons, more specifically equilateral triambuses.

If its dual, the small ditrigonary icosidodecahedron, has an edge length of 1, then the edges of the hexagons will measure ${\frac {3{\sqrt {10}}-5{\sqrt {2}}}{5}}\approx 0.48315$.

If its convex core, the icosahedron, has an edge length of 1, then the edges of the hexagons will measure ${\frac {\sqrt {10}}{5}}\approx 0.63246$.

The hexagons have alternating interior angles of $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$, and $\arccos \left({\frac {1}{4}}\right)+60^{\circ }\approx 135.52249^{\circ }$.

It is a rare example of a polyhedron with icosahedral symmetry that can be built out of green Zome tools. This is because the compound of five octahedra is its edge stellation.

Vertex coordinates

A small triambic icosahedron with dual edge length 1 has vertex coordinates given by all even permutations of:

• $\left(\pm {\frac {5-{\sqrt {5}}}{10}},\,\pm {\frac {5+{\sqrt {5}}}{10}},\,0\right),$
• $\left(\pm {\frac {{\sqrt {5}}-1}{2}},\,\pm {\frac {3-{\sqrt {5}}}{2}},\,0\right),$
• $\left(\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {\sqrt {5}}{5}},\,\pm {\frac {\sqrt {5}}{5}}\right).$

External links

• Klitzing, Richard. "stai".
• Wikipedia contributors. "Small triambic icosahedron".
• McCooey, David. "Small Triambic Icosahedron"