Truncated rhombic triacontahedron

Truncated rhombic triacontahedron

The truncated rhombic triacontahedron is a convex polyhedron constructed from the rhombic triacontahedron by truncating the twelve vertices where five faces meet at their acute corners.

The 12 vertices can be truncated such that all edges are equal length. The original 30 rhombic faces become non-regular hexagons, and the truncated vertices become regular pentagons.

The hexagon faces can be equilateral but not regular with D2 symmetry. The angles at the two vertices with vertex configuration "6.6.6" are arccos(-1/sqrt(5)) = 116.565 degrees, and at the remaining four vertices with "5.6.6", they are 121.717 degrees each.

Related polyhedra

This polyhedron looks very similar to the uniform truncated icosahedron which has 12 pentagons, but only 20 hexagons.

Note that this name is ambiguous since only 12 vertices were truncated, and different polyhedra can be generated by truncating the other 20 vertices, or all 32 vertices of the original rhombic triacontahedron.

Chemistry

This is the shape of the fullerene C80; sometimes this shape is denoted C80(Ih) to describe its icosahedral symmetry and distinguish it from other less-symmetric 80-vertex fullerenes. It is one of only four fullerenes found by harvtxt|Deza|Deza|Grishukhin|1998 to have a skeleton that can be isometrically embeddable into an L1 space.

References

*citation
last1 = Deza | first1 = A. | last2 = Deza |first2 = M. | last3 = Grishukhin | first3 = V.
title = Fullerenes and coordination polyhedra versus half-cube embeddings
journal = Discrete Mathematics | volume = 192 | issue = 1 | year = 1998 | pages = 41–80
url = http://www.ehess.fr/centres/cams/papers/144.ps.gz | doi = 10.1016/S0012-365X(98)00065-X.

See also

* Truncated rhombic dodecahedron

External links

* [http://www.georgehart.com/virtual-polyhedra/conway_notation.html VTML polyhedral generator] Try "t5daD" (Conway polyhedron notation)
* [http://wackel.home.comcast.net/Zometool/ITruncRTiacontahedron.html Zometool model]
* Fullerene C80
*# [http://www.cochem2.tutkie.tut.ac.jp/Fuller/fsl/c80.html] (Number 7 -Ih)
*# [http://www.chem.qmul.ac.uk/iupac/fullerene2/327.html]


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Rhombic triacontahedron — In geometry, the rhombic triacontahedron is a convex polyhedron with 30 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. It is the polyhedral dual of the icosidodecahedron, and it is a zonohedron.The ratio of the long diagonal… …   Wikipedia

  • Truncated rhombic dodecahedron — The truncated rhombic dodecahedron is a convex polyhedron constructed from the rhombic dodecahedron by truncating the 6 (order 4) vertices. The 6 vertices are truncated such that all edges are equal length. The original 12 rhombic faces become… …   Wikipedia

  • Medial rhombic triacontahedron — Type Star polyhedron Elements F = 30, E = 60 V = 24 (χ = −6) Symmetry group Ih, [5,3], *532 …   Wikipedia

  • Truncated icosahedron — The truncated icosahedron is an Archimedean solid. It comprises 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges. Construction This polyhedron can be constructed from an icosahedron with the 12 vertices truncated… …   Wikipedia

  • Truncated icosidodecahedron — (Click here for rotating model) Type Archimedean solid Uniform polyhedron Elements F = 62, E = 180, V = 120 (χ = 2) Faces by sides …   Wikipedia

  • Truncated great dodecahedron — Type Uniform star polyhedron Elements F = 24, E = 90 V = 60 (χ = −6) Faces by sides 12{5/2}+12{10} Wythof …   Wikipedia

  • Truncated great icosahedron — Type Uniform star polyhedron Elements F = 32, E = 90 V = 60 (χ = 2) Faces by sides 12{5/2}+20{6} Wythoff symb …   Wikipedia

  • Truncated cuboctahedron — (Click here for rotating model) Type Archimedean solid Uniform polyhedron Elements F = 26, E = 72, V = 48 (χ = 2) Faces by sides …   Wikipedia

  • Truncated octahedron — (Click here for rotating model) Type Archimedean solid Uniform polyhedron Elements F = 14, E = 36, V = 24 (χ = 2) Faces by sides 6 …   Wikipedia

  • Rhombic dodecahedron — The rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron. Properties It is the polyhedral dual of the cuboctahedron, and a zonohedron. The long… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”