Great triambic icosahedron

	Great triambic icosahedron	Medial triambic icosahedron
Types	Dual uniform polyhedra
Symmetry group	Ih
Name	Great triambic icosahedron	Medial triambic icosahedron
Index references	DU47, W34, 30/59	DU41, W34, 30/59
Elements	F = 20, E = 60 V = 32 (χ = -8)	F = 20, E = 60 V = 24 (χ = -16)
Isohedral faces
Duals	Great ditrigonal icosidodecahedron	Ditrigonal dodecadodecahedron
Stellation
Icosahedron: W34
Stellation diagram

In geometry, the great triambic icosahedron and medial triambic icosahedron (or midly triambic icosahedron) are visually identical dual uniform polyhedra. The exterior surface also represents the De₂f₂ stellation of the icosahedron. These figures can be differentiated by marking which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres, which can be seen in the concave Y-shaped areas. Alternatively, if the faces are filled with the even–odd rule, the internal structure of both shapes will differ.

The 12 vertices of the convex hull matches the vertex arrangement of an 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 isotoxal 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.