# Medial triambic icosahedron

Medial triambic icosahedron
Rank3
TypeUniform dual
Notation
Bowers style acronymMatai
Coxeter diagramm5/3o3o5*a ()
Elements
Faces20 nonconvex triambi
Edges60
Vertices12+12
Vertex figure12 pentagons, 12 pentagrams
Measures (edge length 1)
Inradius${\displaystyle {\frac {\sqrt {6}}{12}}\approx 0.20412}$
Dihedral angle${\displaystyle \arccos \left(-{\frac {1}{3}}\right)\approx 109.47122^{\circ }}$
Central density4
Number of external pieces60
Related polytopes
ConjugateMedial triambic icosahedron
Convex coreIcosahedron
Abstract & topological properties
Flag count240
Euler characteristic–16
OrientableYes
Genus9
Properties
SymmetryH3, order 120
ConvexNo
NatureTame

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

It appears the same as the great triambic icosahedron.

If its dual, the ditrigonary dodecadodecahedron, has an edge length of 1, then the edges of the hexagons will measure ${\displaystyle 2{\sqrt {2}}\approx 2.82843}$.

If its convex core, the icosahedron, has an edge length of 1, then the edges of the hexagons will measure ${\displaystyle {\frac {3{\sqrt {2}}+{\sqrt {10}}}{2}}\approx 3.70246}$.

The hexagons have alternating interior angles of ${\displaystyle \arccos \left({\frac {1}{4}}\right)-60^{\circ }\approx 15.52249^{\circ }}$, and ${\displaystyle \arccos \left(-{\frac {1}{4}}\right)+120^{\circ }\approx 224.47751^{\circ }}$.

This is an abstractly regular polyhedron.

## Vertex coordinates

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

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