Great triambic icosahedron

Great triambic icosahedron
(3D model) (OFF file)
Rank	3
Type	Uniform dual
Notation
Bowers style acronym	Gatai
Coxeter diagram	m3/2o3o5*a ()
Elements
Faces	20 nonconvex triambi
Edges	60
Vertices	12+20
Vertex figure	20 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 density	6
Number of external pieces	60
Related polytopes
Dual	Great ditrigonary icosidodecahedron
Conjugate	Small triambic icosahedron
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 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:

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

External links

• Klitzing, Richard. "gidtid".
• Wikipedia contributors. "Great triambic icosahedron".
• McCooey, David. "Great Triambic Icosahedron"