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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 ...
, a spherical polyhedron or spherical tiling is a tiling 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 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 a ...
s into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra 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 squa ...
. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
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
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party in Germany to descr ...
. 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 inScotland
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 th ...
, 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 part ...
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 used them to enumerate all but one of the uniform polyhedra, 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, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
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 toprojective 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) – 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 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
See also
*Spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
*Spherical trigonometry
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 grea ...
*Polyhedron
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 ...
* Projective polyhedron
* Toroidal polyhedron
* Conway polyhedron notation
References
Further reading
* * * {{Tessellation Polyhedra Tessellation Spheres