Comparison of truncated icosahedron and soccer ball

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a spherical polyhedron or spherical tiling is a tiling of the

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, 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 arc In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

s into bounded regions called '' spherical polygons''. A

polyhedron In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...

whose vertices are equidistant from its center can be conveniently studied by projecting its edges onto the sphere to obtain a corresponding spherical polyhedron. The most familiar spherical polyhedron is the soccer ball, thought of as a spherical

truncated icosahedron In geometry, the truncated icosahedron is a polyhedron that can be constructed by Truncation (geometry), truncating all of the regular icosahedron's vertices. Intuitively, it may be regarded as Ball (association football), footballs (or soccer ...

. 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, ...

, 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

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 (, ; 10 June 940 – 15 July 998) was a Persian mathematician and astronomer who worked in Baghdad. He made import ...

(Abu'l Wafa) studied spherical polyhedra as part of a work on the geometry needed by craftspeople and architects. The work of

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 ...

geodesic dome A geodesic dome is a hemispherical thin-shell structure (lattice-shell) based on a geodesic polyhedron. The rigid triangular elements of the dome distribute stress throughout the structure, making geodesic domes able to withstand very heavy ...

s in the mid 20th century triggered a boom in the study of spherical polyhedra. At roughly the same time, Coxeter 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—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...

, 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 eq ...

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 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 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. Pr ...

(tessellations of the

real projective plane In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...

) – just as the sphere has a 2-to-1

covering map In topology, a covering or covering projection is a map between topological spaces that, intuitively, locally acts like a projection of multiple copies of a space onto itself. In particular, coverings are special types of local homeomorphisms ...

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

Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

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

Hemi-octahedron In geometry, a hemi-octahedron is an abstract polytope, 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 (geometry), hemicube ...

, /2 *

Hemi-dodecahedron In geometry, a hemi-dodecahedron is an abstract polytope, abstract, regular polyhedron, containing half the Face (geometry), faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective pla ...

, /2 *

Hemi-icosahedron In geometry, a hemi-icosahedron is an abstract polytope, 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), ...

, /2 * Hemi-dihedron, /2, p≥1 * Hemi-hosohedron, /2, p≥1

References

{{Tessellation Polyhedra Tessellation Spheres

History

Examples

Improper cases

Relation to tilings of the projective plane

See also

References