In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, the truncated
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
is an
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
, one of 13 convex
isogonal nonprismatic solids whose 32
face
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may aff ...
s are two or more types of
regular polygons
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
. It is the only one of these shapes that does not contain triangles or squares. In general usage, the degree of truncation is assumed to be
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
unless specified.
It has 12 regular
pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
al faces, 20 regular
hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ...
al faces, 60 vertices and 90 edges.
It is the
Goldberg polyhedron
In mathematics, and more specifically in polyhedral combinatorics, a Goldberg polyhedron is a convex polyhedron made from hexagons and pentagons. They were first described in 1937 by Michael Goldberg (1902–1990). They are defined by three pro ...
GP
V(1,1) or
1,1, containing pentagonal and hexagonal faces.
This geometry is associated with
footballs
A football is a ball inflated with air that is used to play one of the various sports known as football. In these games, with some exceptions, goals or points are scored only when the ball enters one of two designated goal-scoring areas; football ...
(soccer balls) typically patterned with white hexagons and black pentagons.
Geodesic dome
A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic dom ...
s such as those whose architecture
Buckminster Fuller
Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...
pioneered are often based on this structure. It also corresponds to the geometry of the fullerene
C60 ("buckyball") molecule.
It is used in the
cell-transitive
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its Face (geometry), faces are the same. More specifically, all faces must be not ...
hyperbolic space-filling tessellation, the
bitruncated order-5 dodecahedral honeycomb.
Construction
![Polyhedron 20 max](https://upload.wikimedia.org/wikipedia/commons/2/2e/Polyhedron_20_max.png)
This polyhedron can be constructed from an
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
with the 12 vertices
truncated (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. In addition the shape has 1440 diagonals.
Characteristics
In
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
and
Graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, there are some standard
polyhedron characteristics.
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a ''truncated icosahedron'' centered at the origin are all
even permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...
s of:
:(0, ±1, ±3''φ'')
:(±1, ±(2 + ''φ''), ±2''φ'')
:(±''φ'', ±2, ±(2''φ'' + 1))
where ''φ'' = is the
golden mean. The circumradius is ≈ 4.956 and the edges have length 2.
Orthogonal projections
The ''truncated icosahedron'' has five special
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s, centered, on a vertex, on two types of edges, and two types of faces: hexagonal and pentagonal. The last two correspond to the A
2 and H
2 Coxeter plane
In mathematics, the Coxeter number ''h'' is the order of a Coxeter element of an irreducible Coxeter group. It is named after H.S.M. Coxeter.
Definitions
Note that this article assumes a finite Coxeter group. For infinite Coxeter groups, there a ...
s.
Spherical tiling
The truncated icosahedron can also be represented as a
spherical tiling
In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most c ...
, and projected onto the plane via a
stereographic projection
In mathematics, a stereographic projection is a perspective projection of the sphere, through a specific point on the sphere (the ''pole'' or ''center of projection''), onto a plane (geometry), plane (the ''projection plane'') perpendicular to ...
. This projection is
conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
Dimensions
If the edge length of a truncated icosahedron is ''a'', the
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
of a
circumscribed sphere
In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
(one that touches the truncated icosahedron at all vertices) is:
:
where ''φ'' is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
.
This result is easy to get by using one of the three orthogonal
golden rectangle
In geometry, a golden rectangle is a rectangle whose side lengths are in the golden ratio, 1 : \tfrac, which is 1:\varphi (the Greek letter phi), where \varphi is approximately 1.618.
Golden rectangles exhibit a special form of self-similarity ...
s drawn into the original icosahedron (before cut off) as the starting point for our considerations. The angle between the segments joining the center and the vertices connected by shared edge (calculated on the basis of this construction) is approximately 23.281446°.
Area and volume
The area ''A'' and the volume ''V'' of the truncated icosahedron of edge length ''a'' are:
:
With unit edges, the surface area is (rounded) 21 for the pentagons and 52 for the hexagons, together 73 (see
areas of regular polygons).
The truncated icosahedron easily demonstrates the
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
:
:32 + 60 − 90 = 2.
Applications
The balls used in
association football
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
and
team handball
Handball (also known as team handball, European handball or Olympic handball) is a team sport in which two teams of seven players each (six outcourt players and a goalkeeper) pass a ball using their hands with the aim of throwing it into the g ...
are perhaps the best-known example of a
spherical polyhedron
In geometry, a spherical polyhedron or spherical tiling is a tessellation, tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhe ...
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 ball type was introduced to the
World Cup in 1970 (starting in
2006
File:2006 Events Collage V1.png, From top left, clockwise: The 2006 Winter Olympics open in Turin; Twitter is founded and launched by Jack Dorsey; The Nintendo Wii is released; Montenegro 2006 Montenegrin independence referendum, votes to declare ...
, this iconic design has been superseded by
alternative patterns).
British traffic signs indicating football grounds use a uniformly-colored
hexagonal tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling).
English mathemat ...
section to represent a football, rather than a truncated icosahedron. This angered mathematician and comedian
Matt Parker
Matthew Thomas Parker (born 22 December 1980) is an Australian recreational mathematician, author, comedian, YouTube personality and science communicator based in the United Kingdom. His book ''Humble Pi'' was the first maths book in the UK to ...
, who started a petition to the UK government to have these signs changed to be geometrically accurate. The petition was ultimately declined.
Geodesic dome
A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The triangular elements of the dome are structurally rigid and distribute the structural stress throughout the structure, making geodesic dom ...
s are typically based on triangular facetings of this geometry with example structures found across the world, popularized by
Buckminster Fuller
Richard Buckminster Fuller (; July 12, 1895 – July 1, 1983) was an American architect, systems theorist, writer, designer, inventor, philosopher, and futurist. He styled his name as R. Buckminster Fuller in his writings, publishing more t ...
.
A variation of the icosahedron was used as the basis of the honeycomb wheels (made from a polycast material) used by the
Pontiac Motor Division between 1971 and 1976 on its
Trans Am
Trans Am may refer to:
* Pontiac Firebird Trans Am, an automobile model
* Trans Am (band), an American post-rock band
** ''Trans Am'' (album), their 1996 debut album
*** ''Trans Am'' (1996 song), their eponymous song from the eponymous album, see ...
and
Grand Prix
Grand Prix ( , meaning ''Grand Prize''; plural Grands Prix), is a name sometimes used for competitions or sport events, alluding to the winner receiving a prize, trophy or honour
Grand Prix or grand prix may refer to:
Arts and entertainment ...
.
This shape was also the configuration of the lenses used for focusing the explosive shock waves of the detonators in both
the gadget
Trinity was the code name of the first detonation of a nuclear weapon. It was conducted by the United States Army at 5:29 a.m. on July 16, 1945, as part of the Manhattan Project. The test was conducted in the Jornada del Muerto desert abo ...
and
Fat Man
"Fat Man" (also known as Mark III) is the codename for the type of nuclear bomb the United States detonated over the Japanese city of Nagasaki on 9 August 1945. It was the second of the only two nuclear weapons ever used in warfare, the fir ...
atomic bomb
A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions (thermonuclear bomb), producing a nuclear explosion. Both bomb ...
s.
The truncated icosahedron can also be described as a model of the
Buckminsterfullerene
Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a soccer ball. Each of its 60 carbon atoms is bonded ...
(fullerene) (C
60), or "buckyball", molecule – an
allotrope
Allotropy or allotropism () is the property of some chemical elements to exist in two or more different forms, in the same physical state, known as allotropes of the elements. Allotropes are different structural modifications of an element: the ...
of elemental carbon, discovered in 1985. The diameter of the football and the fullerene molecule are 22 cm and about 0.71
nm, respectively, hence the size ratio is ≈31,000,000:1.
In popular craft culture, large
sparkleballs can be made using
icosahedron patternand plastic, styrofoam or paper cups.
In the arts
File:Comparison of truncated icosahedron and soccer ball.png, The truncated icosahedron (left) compared with an association football
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...
.
File:Buckminsterfullerene Model in Red Beads.jpg, Fullerene
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
C60 molecule
File:Peter Stehlik 2010.08.03 003.jpg, Truncated icosahedral radome
A radome (a portmanteau of radar and dome) is a structural, weatherproof enclosure that protects a radar antenna (radio), antenna. The radome is constructed of material transparent to radio waves. Radomes protect the antenna from weather and ...
on a weather station
A weather station is a facility, either on land or sea, with instruments and equipment for measuring atmospheric conditions to provide information for weather forecasts and to study the weather and climate. The measurements taken include tempera ...
File:Truncated icosahedron by Sean Journot.jpg, Truncated icosahedron machined out of 6061-T6 aluminum
6061 ( Unified Numbering System (UNS) designation A96061) is a precipitation-hardened aluminium alloy, containing magnesium and silicon as its major alloying elements. Originally called "Alloy 61S", it was developed in 1935. It has good mechani ...
File:Truncated icosahedron cherry model by George W. Hart.jpg, A wooden truncated icosahedron artwork by George W. Hart
George William Hart (born 1955) is an American sculptor and geometer. Before retiring, he was an associate professor of Electrical Engineering at Columbia University in New York City and then an interdepartmental research professor at Stony B ...
.
Related polyhedra
These
uniform star-polyhedra, and one icosahedral stellation have nonuniform truncated icosahedra
convex hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
s:
This polyhedron looks similar to the uniform
chamfered dodecahedron
In geometry, the chamfered dodecahedron is a convex polyhedron with 80 vertices, 120 edges, and 42 faces: 30 hexagons and 12 pentagons. It is constructed as a chamfer (edge-truncation) of a regular dodecahedron. The pentagons are reduced in s ...
which has 12 pentagons, but 30 hexagons.
Truncated icosahedral graph
In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
graph theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...
, a truncated icosahedral graph is the
graph of vertices and edges of the ''truncated icosahedron'', one of the
Archimedean solid
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...
s. It has 60
vertices and 90 edges, and is a
cubic Archimedean graph.
History
![Piero della Francesca - Libellus de quinque corporibus regularibus - p52b (cropped)](https://upload.wikimedia.org/wikipedia/commons/0/04/Piero_della_Francesca_-_Libellus_de_quinque_corporibus_regularibus_-_p52b_%28cropped%29.jpg)
The truncated icosahedron was known to
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
, who classified the 13 Archimedean solids in a lost work. All we know of his work on these shapes comes from
Pappus of Alexandria
Pappus of Alexandria (; grc-gre, Πάππος ὁ Ἀλεξανδρεύς; AD) was one of the last great Greek mathematicians of antiquity known for his ''Synagoge'' (Συναγωγή) or ''Collection'' (), and for Pappus's hexagon theorem i ...
, who merely lists the numbers of faces for each: 12 pentagons and 20 hexagons, in the case of the truncated icosahedron. The first known image and complete description of a truncated icosahedron is from a rediscovery by
Piero della Francesca
Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...
, in his 15th-century book ''
De quinque corporibus regularibus
''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is ...
'', which included five of the Archimedean solids (the five truncations of the regular polyhedra). The same shape was depicted by
Leonardo da Vinci
Leonardo di ser Piero da Vinci (15 April 14522 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, Drawing, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially res ...
, in his illustrations for
Luca Pacioli
Fra Luca Bartolomeo de Pacioli (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as accounting ...
's plagiarism of della Francesca's book in 1509. Although
Albrecht Dürer
Albrecht Dürer (; ; hu, Ajtósi Adalbert; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer (without an umlaut) or Due ...
omitted this shape from the other Archimedean solids listed in his 1525 book on polyhedra, ''Underweysung der Messung'', a description of it was found in his posthumous papers, published in 1538.
Johannes Kepler
Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...
later rediscovered the complete list of the 13 Archimedean solids, including the truncated icosahedron, and included them in his 1609 book, ''
Harmonices Mundi
''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...
''.
See also
*
Fullerene
A fullerene is an allotrope of carbon whose molecule consists of carbon atoms connected by single and double bonds so as to form a closed or partially closed mesh, with fused rings of five to seven atoms. The molecule may be a hollow sphere, ...
**
Buckminsterfullerene
Buckminsterfullerene is a type of fullerene with the formula C60. It has a cage-like fused-ring structure (truncated icosahedron) made of twenty hexagons and twelve pentagons, and resembles a soccer ball. Each of its 60 carbon atoms is bonded ...
(C
60)
*
Hyperbolic soccerball
In geometry, the order-7 truncated triangular tiling, sometimes called the hyperbolic soccerball, is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional ...
*
Snyder equal-area projection Snyder equal-area projection is a polyhedral map projection used in the '' ISEA (Icosahedral Snyder Equal Area) discrete global grids''. It is named for John P. Snyder, who developed the projection in the 1990s.
Snyder, J. P. (1992), “An Equa ...
*
Soccer ball
A football (also known as football ball, soccer ball, or association football ball specifically in the United Kingdom) is the ball used in the sport of association football. The name of the ball varies according to whether the sport is called " ...
**
Adidas Telstar
Telstar is a football made by Adidas. The iconic 32-panel alternating black-and-white design of the ball, based on the work of Eigil Nielsen, has since become a global standard design used to portray a football in different media.
Ball
The ba ...
Notes
References
* (Section 3-9)
*
External links
*
**
*
Editable printable net of a truncated icosahedron with interactive 3D viewThe Uniform Polyhedra''The Encyclopedia of Polyhedra''
3D paper data visualization World Cup ball
{{Polyhedron navigator
Archimedean solids
Goldberg polyhedra
Individual graphs
Truncated tilings
Uniform polyhedra