A spherical design, part of
combinatorial design
Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of ''balance'' and/or ''symmetry''. These co ...
theory in
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, is a finite set of ''N'' points on the ''d''-dimensional unit
''d''-sphere ''S
d'' such that the average value of any polynomial ''f'' of degree ''t'' or less on the set equals the average value of ''f'' on the whole sphere (that is, the integral of ''f'' over ''S
d'' divided by the area or
measure of ''S
d''). Such a set is often called a spherical ''t''-design to indicate the value of ''t'', which is a fundamental parameter. The concept of a spherical design is due to Delsarte, Goethals, and Seidel, although these objects were understood as particular examples of
cubature formulas earlier.
Spherical designs can be of value in
approximation theory
In mathematics, approximation theory is concerned with how function (mathematics), functions can best be approximation, approximated with simpler functions, and with quantitative property, quantitatively characterization (mathematics), characteri ...
, in
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
for
experimental design
The design of experiments (DOE), also known as experiment design or experimental design, is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. ...
, in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
, and in
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 ...
. The main problem is to find examples, given ''d'' and ''t'', that are not too large; however, such examples may be hard to come by.
Spherical t-designs have also recently been appropriated in
quantum mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
in the form of
quantum t-designs with various applications to
quantum information theory
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...
and
quantum computing
A quantum computer is a computer that exploits quantum mechanical phenomena. On small scales, physical matter exhibits properties of wave-particle duality, both particles and waves, and quantum computing takes advantage of this behavior using s ...
.
Existence of spherical designs
The existence and structure of spherical designs on the circle were studied in depth by Hong. Shortly thereafter, Seymour and Zaslavsky proved that such designs exist of all sufficiently large sizes; that is, given positive integers ''d'' and ''t'', there is a number ''N''(''d'',''t'') such that for every ''N'' ≥ ''N''(''d'',''t'') there exists a spherical ''t''-design of ''N'' points in dimension ''d''. However, their proof gave no idea of how big ''N''(''d'',''t'') is.
Mimura constructively found conditions in terms of the number of points and the dimension which characterize exactly when spherical 2-designs exist. Maximally sized collections of
equiangular lines In geometry, a set of lines is called equiangular if all the lines intersect at a single point, and every pair of lines makes the same angle.
Equiangular lines in Euclidean space
Computing the maximum number of equiangular lines in ''n''-dimensi ...
(up to identification of lines as antipodal points on the sphere) are examples of minimal sized spherical 5-designs. There are many sporadic small spherical designs; many of them are related to finite
group action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S.
Many sets of transformations form a group under ...
s on the sphere.
In 2013, Bondarenko, Radchenko, and Viazovska obtained the asymptotic upper bound