Great triambic icosahedron

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Great triambic icosahedron Medial triambic icosahedron
DU47 great triambic icosahedron.png DU41 medial triambic icosahedron.png
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 Great triambic icosahedron face.png Medial triambic icosahedron face.png
Duals Great ditrigonal icosidodecahedron.png
Great ditrigonal icosidodecahedron
Ditrigonal dodecadodecahedron.png
Ditrigonal dodecadodecahedron
Stellation
Icosahedron: W34
Ninth stellation of icosahedron facets.png
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[edit]

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[edit]

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:

Uniform tiling 65-t0.png

As a stellation[edit]

Ninth stellation of icosahedron.pngStellation icosahedron De2f2.png

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

See also[edit]

References[edit]

External links[edit]

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
Zeroth stellation of icosahedron.png First stellation of icosahedron.png Ninth stellation of icosahedron.png First compound stellation of icosahedron.png Second compound stellation of icosahedron.png Third compound stellation of icosahedron.png Sixteenth stellation of icosahedron.png Third stellation of icosahedron.png Seventeenth stellation of icosahedron.png
Zeroth stellation of icosahedron facets.png First stellation of icosahedron facets.png Ninth stellation of icosahedron facets.png First compound stellation of icosahedron facets.png Second compound stellation of icosahedron facets.png Third compound stellation of icosahedron facets.png Sixteenth stellation of icosahedron facets.png Third stellation of icosahedron facets.png Seventeenth stellation of icosahedron facets.png
The stellation process on the icosahedron creates a number of related polyhedra and compounds with icosahedral symmetry.