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The truncated icosahedron is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. It comprises 12 regular pentagonal
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
 faces, 20 regular hexagonal
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
 faces, 60 vertices and 90 edges.
Construction
This polyhedron can be constructed from an icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 with the 12 vertices truncated
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
 (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons.






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Encyclopedia


The truncated icosahedron is an Archimedean solid
Archimedean solid

In geometry an Archimedean solid is a highly symmetric, semi-regular convex set polyhedron composed of two or more types of regular polygons meeting in identical vertex ....
. It comprises 12 regular pentagonal
Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
 faces, 20 regular hexagonal
Hexagon

In geometry, a hexagon is a polygon with six edges and six Vertex . A regular hexagon has Schl?fli symbol ....
 faces, 60 vertices and 90 edges.

Construction


This polyhedron can be constructed from an icosahedron
Icosahedron

In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
 with the 12 vertices truncated
Truncation (geometry)

In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new Facet in place of each vertex....
 (cut off) such that one third of each edge is cut off at each of both ends. This creates 12 new pentagon faces, and leaves the original 20 triangle faces as regular hexagons. Thus the length of the edges is one third of that of the original edges.

Icosahedron

Icosahedron

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated icosahedron centered at the origin are the orthogonal rectangles (0,±1,±3f), (±1,±3f,0), (±3f,0,±1) and the orthogonal cuboid
Cuboid

In geometry, a cuboid is a solid figure bounded by six faces, forming a convex polyhedron. There are two competing and incompatible definitions of a cuboid in the mathematical literature....
s (±2,±(1+2f),±f), (±(1+2f),±f,±2), (±f,±2,±(1+2f)) along with the orthogonal cuboids (±1,±(2+f),±2f), (±(2+f),±2f,±1), (±2f,±1,±(2+f)), where f = (1+v5)/2 is the golden mean
Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller....
. Using f2 = f + 1 one verifies that all vertices are on a sphere, centered at the origin, with the radius squared equal to 9f + 10. The edges have length 2.

Area and volume

The area A and the volume V of the truncated icosahedron of edge length a are:

Geometric relations

The truncated icosahedron easily verifies the Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
:
32 + 60 - 90 = 2.


With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see areas of regular polygons
Regular polygon

A regular polygon is a polygon which is Equiangular polygon and equilateral . Regular polygons may be convex or Star polygon....
).

Applications

Trunc Icosa
C60a
The common soccer
Football (soccer)

Association football, more commonly known as football or soccer, is a team sport played between two teams of eleven players, and is widely considered to be the most popular sport in the world....
 ball
Football (ball)

A football is a ball used to play one of the various sports known as football.In the distant past, crude balls such as inflated pigs' bladders were used, but balls are now designed by teams of engineers to exacting specifications....
 is perhaps the best-known example of a spherical polyhedron
Spherical polyhedron

In mathematics, the surface of a sphere may be divided by line segments into bounded regions, to form a spherical tessellation or spherical polyhedron....
 analog to the truncated icosahedron, found in everyday life. The ball comprises the same pattern of regular pentagons and regular hexagons, but it is more spherical due to the pressure of the air inside and the elasticity of the ball.

This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both the gadget
The gadget

The "gadget" was the code-name given to the first nuclear weapon developed under the Manhattan Project during World War II, which was tested at the Trinity test test site on July 16, 1945....
 and Fat Man
Fat Man

Fat Man is the codename for the atomic bomb that was detonated over Nagasaki, Nagasaki, Japan, by the United States on August 9, 1945, at 11:02 a.m....
 atomic bombs.

The truncated icosahedron can also be described as a model of the Buckminsterfullerene (fullerene) (C60), or "buckyball," molecule, a recently-identified allotrope of elemental carbon. The diameter of the soccer ball and the fullerene molecule are 22 cm and ca. 1 nm
Nanometre

A nanometre is a Units of measurement of length in the metric system, equal to one billionth of a metre .It is one of the more often used units for very small lengths, and equals ten ?ngstr?m, an internationally recognized non-International System of Units of length....
, respectively, hence the size ratio is 220,000,000 : 1.

Truncated icosahedra in the arts

A truncated icosahedron with "solid edges" is a drawing by Lucas Pacioli
Luca Pacioli

Fra Luca Bartolomeo de Pacioli was an Italy mathematician and Franciscan friar, collaborator with Leonardo da Vinci, and seminal contributor to the field now known as accounting....
 illustrating The Divine Proportion.

See also

  • spinning truncated icosahedron
  • dodecahedron
    Dodecahedron

    A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
  • icosidodecahedron
    Icosidodecahedron

    An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 identical edges, each separating a triangle from a pentagon....
  • truncated dodecahedron
    Truncated dodecahedron

    In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangle faces, 60 vertices and 90 edges....
  • truncated rhombic triacontahedron
    Truncated rhombic triacontahedron

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

    The hyperbolic soccerball is a tessellation of a surface frequently used as a manipulative for studying the properties of hyperbolic geometry....
  • fullerene
    Fullerene

    Fullerene are a family of carbon Allotropy, molecules composed entirely of carbon, in the form of a hollow sphere, ellipsoid, cylinder , or plane....


Further reading

(Section 3-9)

External links

  • The Encyclopedia of Polyhedra