Truncated rhombic triacontahedron

Truncated rhombic triacontahedron
Type	Conway polyhedron
Faces	12 pentagons 30 hexagons
Edges	120 (2 types)
Vertices	80 (2 types)
Vertex configuration	(60) 5.6.6 (20) 6.6.6
Symmetry group	Icosahedral (Ih)
Dual polyhedron	Pentakis icosidodecahedron
Properties	convex, equilateral-faced

A net for truncated rhombic triacontahedron

The truncated rhombic triacontahedron is a convex polyhedron constructed as a truncation of the rhombic triacontahedron. It can more accurately be called a pentatruncated rhombic triacontahedron because only the order-5 vertices are truncated.

These 12 order-5 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.

Full truncation

If all 32 vertices of a rhombic triacontahedron are truncated, the resulting solid has 12 regular pentagons, 20 equilateral triangles and 30 irregular octagons, with 180 edges in all, and 120 vertices. Its dual is a triangular hecatonicosahedron known as the tripentakis icosidodecahedron, a solid formed by adding a low pyramid to each face of a uniform icosidodecahedron. This figure is to the dodecahedron as the truncated cube is to the tetrahedron.

• 

  The Truncated Rhombic Triacontahedron as shown

• 

  The full truncation, shown with orange octagons, purple pentagons, and cyan triangles.

Related polyhedra and polytopes

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

• 

  Truncated rhombic triacontahedron

• 

  Truncated icosahedron

• 

  cell-centered orthogonal projection of the 120-cell

It also represents the exterior envelope of a cell-centered orthogonal projection of the 120-cell, one of six (convex regular 4-polytopes).

This polyhedron is also a part of a sequence of truncated rhombic polyhedra and tilings with [n,3] Coxeter group symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces. The sequence has two vertex figures (n.6.6) and (6,6,6).

Polyhedra			Euclidean tiling	Hyperbolic tiling
[3,3]	[4,3]	[5,3]	[6,3]	[7,3]	[8,3]
Cube	Rhombic dodecahedron	Rhombic triacontahedron	Rhombille
Alternate truncated cube	Truncated rhombic dodecahedron	Truncated rhombic triacontahedron	Hexagonal tiling

Chemistry

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

References

• Deza, A.; Deza, M.; Grishukhin, V. (1998), "Fullerenes and coordination polyhedra versus half-cube embeddings", Discrete Mathematics 192 (1): 41–80, doi:10.1016/S0012-365X(98)00065-X, http://www.ehess.fr/centres/cams/papers/144.ps.gz .

See also

• Truncated rhombic dodecahedron
• Near-miss Johnson solid

External links

• VTML polyhedral generator Try "t5daD" (Conway polyhedron notation)
• Zometool model
• Fullerene C80
  1. [1] (Number 7 -Ih)
  2. [2]