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 of the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

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 g ...

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 ...

is most conveniently derived in this way. The most familiar spherical polyhedron is the soccer ball, 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, 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 ha ...

. Some "improper" polyhedra, such as hosohedra 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, P ...

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

, 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 pa ...

period (the New Stone Age). During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (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 ...

, through the construction of kaleidoscopes ( Wythoff construction).

Examples

All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings: Sphere5tesselation

Improper cases

Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as , and dihedra: 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 (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 b ...

) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin. 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 inv ...

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 e ...

s, as well as two infinite classes of even dihedra and hosohedra: * Hemi-cube, /2 * Hemi-octahedron, /2 * Hemi-dodecahedron, /2 * Hemi-icosahedron, /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