Comparison of truncated icosahedron and soccer ball

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 ...

, a spherical polyhedron or spherical tiling is a

tiling Tiling may refer to: *The physical act of laying tiles *Tessellations Computing *The compiler optimization of loop tiling *Tiled rendering, the process of subdividing an image by regular grid *Tiling window manager People *Heinrich Sylvester The ...

of the

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

in which the surface is divided or partitioned by great arcs into bounded regions called

spherical polygon Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are gr ...

s. Much of the theory of symmetrical

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

is most conveniently derived in this way. The most familiar spherical polyhedron is the

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 " ...

, thought of as a spherical

truncated icosahedron In geometry, the truncated icosahedron is an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares. ...

. The next most popular spherical polyhedron is the

beach ball ''Beach Ball'' is a 1965 American beach party movie starring Edd Byrnes and partly financed by Roger Corman.Ed. J. Philip di Franco, ''The Movie World of Roger Corman'', Chelsea House Publishers, 1979, p. 145 It features appearances by The Supre ...

, thought of as a

hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havin ...

. Some "improper" polyhedra, such as

hosohedra In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune havi ...

and their

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, Pas ...

dihedra A dihedron is a type of polyhedron, made of two polygon faces which share the same set of ''n'' edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat ...

, exist as spherical polyhedra, but their flat-faced analogs are

degenerate Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...

. The example hexagonal beach ball, is a hosohedron, and is its dual dihedron.

History

The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in

Scotland Scotland (, ) is a country that is part of the United Kingdom. Covering the northern third of the island of Great Britain, mainland Scotland has a border with England to the southeast and is otherwise surrounded by the Atlantic Ocean to the ...

, and appear to date from the

neolithic The Neolithic period, or New Stone Age, is an Old World archaeological period and the final division of the Stone Age. It saw the Neolithic Revolution, a wide-ranging set of developments that appear to have arisen independently in several parts ...

period (the New Stone Age). During the 10th Century, the Islamic scholar

Abū al-Wafā' Būzjānī Abū al-Wafāʾ, Muḥammad ibn Muḥammad ibn Yaḥyā ibn Ismāʿīl ibn al-ʿAbbās al-Būzjānī or Abū al-Wafā Būzhjānī ( fa, ابوالوفا بوزجانی or بوژگانی) (10 June 940 – 15 July 998) was a Persian mathematician ...

(Abu'l Wafa) wrote the first serious study of spherical polyhedra. Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra. In the middle of the 20th Century,

Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington to ...

used them to enumerate all but one of the

uniform polyhedra In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fa ...

, through the construction of kaleidoscopes (

Wythoff construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

Examples

All

regular polyhedra A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...

semiregular polyhedra In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

, and their duals can be projected onto the sphere as tilings: Sphere5tesselation

Improper cases

Spherical tilings allow cases that polyhedra do not, namely

: figures as , and

: figures as . Generally, regular hosohedra and regular dihedra are used.

Relation to tilings of the projective plane

Spherical polyhedra having at least one inversive symmetry are related to

projective polyhedra In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projec ...

(tessellations of the

real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...

) – just as the sphere has a 2-to-1

covering map A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...

of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under

reflection through the origin In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

. The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the

centrally symmetric In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s, as well as two infinite classes of even

and

: * Hemi-cube, /2 *

Hemi-octahedron A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron. It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the hemicube. It can be realized as a projective polyhedron (a tes ...

, /2 *

Hemi-dodecahedron A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 6 pentagons), which can be visualized by const ...

, /2 *

Hemi-icosahedron A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron (a tessellation of the real projective plane by 10 triangles), which can be visualized by constr ...

, /2 * Hemi-dihedron, /2, p>=1 * Hemi-hosohedron, /2, p>=1

History

Examples

Improper cases

Relation to tilings of the projective plane

See also

References

Further reading