Spherical code
Encyclopedia
In geometry
and coding theory
, a spherical code with parameters (n,N,t) is a set of N points on the unit hypersphere
in n dimensions for which the dot product
of any two points is less than or equal to t. The kissing number problem
may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters (n,N,1/2) exists. The Tammes problem
may be stated as the problem of finding a spherical code with minimal t for given n and N.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and coding theory
Coding theory
Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error-correction and more recently also for network coding...
, a spherical code with parameters (n,N,t) is a set of N points on the unit hypersphere
Hypersphere
In mathematics, an n-sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension. For any natural number n, an n-sphere of radius r is defined as the set of points in -dimensional Euclidean space which are at distance r from a central point, where the radius r may be any...
in n dimensions for which the dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
of any two points is less than or equal to t. The kissing number problem
Kissing number problem
In geometry, a kissing number is defined as the number of non-overlapping unit spheres that touch another given unit sphere. For a lattice packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another...
may be stated as the problem of finding the maximal N for a given n for which a spherical code with parameters (n,N,1/2) exists. The Tammes problem
Tammes problem
In geometry, Tammes problem is a problem in packing a given number of circles on the surface of a sphere such that the minimum distance between circles is maximized. It is named after a Dutch botanist who posed the problem in 1930 while studying the distribution of pores on pollen grains...
may be stated as the problem of finding a spherical code with minimal t for given n and N.