# Great triambic icosahedron

Great triambic icosahedron Rank3
TypeUniform dual
Notation
Bowers style acronymGatai
Coxeter diagramm3/2o3o5*a (      )
Elements
Faces20 triambuses
Edges60
Vertices20+12
Vertex figure20 triangles, 12 pentagons
Measures (edge length 1)
Inradius${\frac {3{\sqrt {30}}-5{\sqrt {6}}}{24}}\approx 0.17434$ Volume${\frac {175{\sqrt {2}}-75{\sqrt {10}}}{24}}\approx 0.42986$ Surface area${\frac {15{\sqrt {15}}-25{\sqrt {3}}}{2}}\approx 7.39674$ Dihedral angle$\arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }$ Central density6
Number of external pieces60
Related polytopes
DualGreat ditrigonary icosidodecahedron
ConjugateSmall triambic icosahedron
Abstract & topological properties
Flag count240
Euler characteristic–8
OrientableYes
Genus5
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

It appears the same as the medial triambic icosahedron.

If its dual, the great ditrigonary icosidodecahedron, has an edge length of 1, then the edges of the hexagons will measure ${\frac {5{\sqrt {2}}+3{\sqrt {10}}}{5}}\approx 3.31158$ . ​The hexagons have alternating interior angles of $\arccos \left({\frac {1}{4}}\right)-60^{\circ }\approx 15.52249^{\circ }$ , and $\arccos \left(-{\frac {1}{4}}\right)\approx 104.47751^{\circ }$ .

## Vertex coordinates

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

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