Medial Triambic Icosahedron
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Medial Triambic Icosahedron
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 (? = -8)
F = 20, E = 60
V = 24 (? = -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
Great triambic icosahedron stellation facets.svg
Stellation diagram
3D model of a medial triambic icosahedron
3D model of a great triambic icosahedron

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.

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 and . The sum of the six angles is , and not as might be expected for a hexagon, because the polygon turns around its center twice. The dihedral angle equals .

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 or triambi. It has 24 vertices: 12 exterior points, and 12 hidden inside. It has 60 edges.

The faces have alternating angles of and . The dihedral angle equals .


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

Ninth stellation of icosahedron.pngStellation icosahedron De2f2.png

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

See also

References

  • 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

External links


  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Medial_triambic_icosahedron
 



 



 
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