Great triambic icosahedron

The great triambic icosahedron is the dual of the great ditrigonal icosidodecahedron, U47. It has 20 inverted-hexagonal (triambus) faces, shaped like a three-bladed propeller. It has 32 vertices: 12 exterior points, and 20 hidden inside. It has 60 edges.

The faces have alternating angles of ${\displaystyle \arccos({\frac {1}{4}})-60^{\circ }\approx 15.522\,487\,814\,07^{\circ }}$ and ${\displaystyle \arccos(-{\frac {1}{4}})\approx 104.477\,512\,185\,93^{\circ }}$. The sum of the six angles is ${\displaystyle 360^{\circ }}$, and not ${\displaystyle 720^{\circ }}$ as might be expected for a hexagon, because the polygon turns around its center twice. The dihedral angle equals ${\displaystyle \arccos(-{\frac {1}{3}})\approx 109.471\,220\,634\,49}$.

The medial triambic icosahedron is the dual of the ditrigonal dodecadodecahedron, U41. It has 20 faces, each being simple concave isogonal hexagons or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.

The faces have alternating angles of ${\displaystyle \arccos({\frac {1}{4}})-60^{\circ }\approx 15.522\,487\,814\,07^{\circ }}$ and ${\displaystyle \arccos(-{\frac {1}{4}})+120^{\circ }\approx 224.477\,512\,185\,93^{\circ }}$. The dihedral angle equals ${\displaystyle \arccos(-{\frac {1}{3}})\approx 109.471\,220\,634\,49}$.

Unlike the great triambic icosahedron, the medial triambic icosahedron is topologically a regular polyhedron of index two.[1] By distorting the triambi into regular hexagons, one obtains a quotient space of the hyperbolic order-5 hexagonal tiling:

It is Wenninger's 34th model as his 9th stellation of the icosahedron

• Triakis icosahedron
• Small triambic icosahedron
• Medial rhombic triacontahedron

• Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
• Wenninger, Magnus (1983). Dual Models. Cambridge University Press. ISBN 978-0-521-54325-5. MR 0730208.
• H.S.M. Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, 3.6 6.2 Stellating the Platonic solids, pp.96-104

• Weisstein, Eric W. "Great triambic icosahedron". MathWorld.
• Weisstein, Eric W. "Medial triambic icosahedron". MathWorld.
• gratrix.net Uniform polyhedra and duals
• bulatov.org Medial triambic icosahedron Great triambic icosahedron

Notable stellations of the icosahedron
Regular	Uniform duals			Regular compounds			Regular star	Others
(Convex) icosahedron	Small triambic icosahedron	Medial triambic icosahedron	Great triambic icosahedron	Compound of five octahedra	Compound of five tetrahedra	Compound of ten tetrahedra	Great icosahedron	Excavated dodecahedron	Final stellation
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.