# Small triambic icosahedron

Small triambic icosahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymStai
Coxeter diagramm5/2o3o3*a ()
Elements
Faces20 triambuses
Edges60
Vertices20+12
Vertex figure20 triangles, 12 pentagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {3{\sqrt {30}}+5{\sqrt {6}}}{24}}\approx 1.19496}$
Volume${\displaystyle {\frac {175{\sqrt {2}}+75{\sqrt {10}}}{24}}\approx 20.19409}$
Surface area${\displaystyle {\frac {15{\sqrt {15}}+25{\sqrt {3}}}{2}}\approx 50.69801}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density2
Number of external pieces60
Related polytopes
DualSmall ditrigonary icosidodecahedron
ConjugateGreat triambic icosahedron
Convex hullNon-Catalan Pentakis dodecahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Euler characteristic–8
OrientableYes
Genus5
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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 ${\displaystyle {\frac {3{\sqrt {10}}-5{\sqrt {2}}}{5}}\approx 0.48315}$.

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

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

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:

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