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 are visually identical dual uniform polyhedra. The exterior surface also represents the De₂f₂ stellation of the icosahedron. The only way to differentiate these two polyhedra is to mark which intersections between edges are true vertices and which are not. In the above images, true vertices are marked by gold spheres.

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

Great triambic icosahedron

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

Medial triambic icosahedron

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

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:

As a stellation

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

See also

• Triakis icosahedron
• Small triambic icosahedron

References

1. The Regular Polyhedra (of index two), David A. Richter

• 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 730208.
• 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

External links

• 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.