Everything about Spherical Design totally explained
A
spherical design, part of
combinatorial design theory in
mathematics, is a finite set of points on the
d-dimensional unit
hypersphere Sd 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
Sd divided by the area or
measure of
Sd). Such a set is often called a
spherical t-
design to indicate the value of
t, which is a fundamental parameter.
Spherical designs can be of value in
approximation theory, in
statistics for
experimental design (being usable to construct
rotatable designs), in
combinatorics, and in
geometry. 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.
The concept of a spherical design is due to Delsarte, Goethals, and Seidel (1977). The existence and structure of spherical designs with
d = 1 (that is, in a circle) was studied in depth by Hong (1982). Shortly thereafter, Seymour and Zaslavsky (1984) proved that such designs exist of all sufficiently large sizes; that is, there's 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. Good estimates for that were found later on. Besides these "large" sizes, there are many sporadic small spherical designs; many of them are related to finite
group actions on the sphere and are of great interest in themselves.
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