A set of N points is called a spherical t-design if the integral of any polynomial of degree at most t over the sphere is equal to the average value of the polynomial over the set of N points.
Given a value of N, one wishes to choose the N points so as to maximize t.
Evidence is presented to suggest that, in three dimensions, spherical
1-designs with N points exist iff N >= 2 (this is known);
2-designs with N points exist iff N = 4 or >= 6 (this is known);
3-designs with N points exist iff N = 6, 8, >= 10;
4-designs with N points exist iff N = 12, 14 >= 20;
5-designs with N points exist iff N = 12, 16, 18, 20 >= 22 ;
6-designs with N points exist iff N = 24 , 26, >= 28 ;
7-designs with N points exist iff N = 24 , 30, 32, 34, >= 36 ;
8-designs with N points exist iff N = 36 , 40, 42, >= 44 ;
9-designs with N points exist iff N = 48 , 50, 52, >= 54 ;
10-designs with N points exist iff N = 60 , 62, >= 64 ;
11-designs with N points exist iff N = 70 , 72, >= 74 ;
12-designs with N points exist iff N = 84 , >= 86 .
The existence of some of these designs is established analytically, while
others are given by very accurate numerical coordinates.
The 24-point 7-design was first found by
McLaren in 1963, and
--- although not identified as such by McLaren --- consists of the vertices of an ``improved'' snub cube,
obtained from Archimedes' regular snub cube (which is only a 3-design)
by slightly shrinking each square face and expanding each triangular face.
It is worth remarking that one of the constructions in the paper gives a sequence of putative spherical t-designs with N= 12m points (m >= 2) where N= (1/2) t^2 (1+o(1)) as t to infinity .
Let tau (N) denote the largest value of t for which an N-point 3-dimensional spherical t-design exists.
Since a t-design is also a t'-design for all t' <= t ,
an N-point spherical t-design exists if and only if tau (N) > t .
Our main results in 3 dimensions
are summarized in the following table,
which gives what we believe are the values of tau (N) for N <= 100 .
The assertions made in the begiining of the abstract can then be simply read off the table.
The table also gives, in columns 4 and 5, the largest symmetry group we have found for such a design (using the notation of Coxeter and Moser (1984)), and in some cases a list of the sizes of the orbits under this group and a description of the polyhedron formed by the points. In most cases the designs found were not unique.
For every value of N in the table we have found very accurate numerical coordinates for a putative spherical t-design with t equal to the value given in column 2. Furthermore, after a considerable amount of searching, we have been unable to find a (t+1)-design, and so we conjecture that the entries in column 2 do indeed give the exact values of tau(N) .
In a number of cases we have proved that there is a spherical t-design that is very close to our numerical approximation.
A symbol V1 in the third column of the table indicates that we have an algebraic proof of the existence of the design, V2 that we have a proof by interval methods, and V3 that we have a numerical solution with discrepancy (defined in the paper) at most 10^-26 .
References to the literature indicate who first proved the existence of some spherical t-design with this number of points (not necessarily the particular design described in the table).
Conjectured values of tau (N) , the largest t for which an N-point configuration on the sphere in 3 dimensions forms a spherical t-design.
N tau(N) Proof Group Order Orbits (Description) 1 0 V1 infinity infinity 1 (single point) 2 1 V1 infinity infinity 2 (2 antipodal points) 3 1 V1 [2,3] 12 3 (equilateral triangle) 4 2 V1 [3,3] 24 4 (regular tetrahedron) 5 1 V1 [2,3] 12 3+2 (triangular bipyramid) 6 3 V1 [3,4] 48 6 (regular octahedron) 7 2 [Mimura 1990] [3] 6 3^2+1 8 3 V1 [3,4] 48 8 (cube) 9 2 [Mimura 1990] [2,3] 12 6+3 (triangular biprism) 10 3 [Bajnok 1993] [2^+,10] 20 10 (pentagonal prism) 11 3 [Bajnok 1993] [2,3]^+ 6 6+3+2 12 5 V1 [3,5] 120 12 (regular icosahedron) 13 3 [Bajnok 1993] [4] 8 4^3+1 14 4 [Hardin-Sloane 1992] [2,3]^+ 6 6^2+2 15 3 [Bajnok 1993] [2,5] 20 10+5 16 5 [Hardin-Sloane 1992] [3,3]^+ 12 12+4 (hexakis truncated tetrahedron) 17 4 [Hardin-Sloane 1992] [2,3]^+ 6 6^2+3+2 18 5 [Reznick 1995] [2^+,6] 12 12+6 19 4 [Hardin-Sloane 1992] [3] 6 6^2+3^2+1 20 5 V1 [3,5] 120 20 (regular dodecahedron) 21 4 [Hardin-Sloane 1992] [2,3] 12 12+6+3 22 5 [Reznick 1995] [2^+,10] 20 10^2+2 23 5 V2 [2,3]^+ 6 6^3+3+2 24 7 McL63 [3,4]^+ 24 24 (improved snub cube) 25 5 V1 [2,5]^+ 10 10^2+5 26 6 V3 [2,3]^+ 6 6^4+2 27 5 [Reznick 1995] [2,3] 12 12^2+3 28 6 V3 [2^+,4] 8 8^3+4 29 6 V3 [2]^+ 2 2^ 14 +1 30 7 V1 [3,4]^+ 24 24+6 (tetrakis snub cube) 31 6 V3 [5]^+ 5 5^6+1 32 7 V1 [3,4]^+ 24 24+8 (snub cube + cube) 33 6 V3 [2,3]^+ 6 34 7 V3 [2,4]^+ 8 35 6 V3 [2,5]^+ 10 10^3+5 36 8 V3 [3,3]^+ 12 12^3 (3 snub tetrahedra) 37 7 V3 [3]^+ 3 38 7 V3 [3,4]^+ 24 24+8+6 39 7 V3 [2,3]^+ 6 40 8 V3 [3,3]^+ 12 12^3+4 41 7 V3 [2,3]^+ 6 42 8 V3 [2,4]^+ 8 43 7 V3 [6]^+ 6 44 8 V3 [3,3]^+ 12 12^3+4^2 45 8 V3 [2]^+ 2 46 8 V3 [2,4]^+ 8 47 8 V3 [2,3]^+ 6 48 9 V1 [3,4]^+ 24 24^2 (two snub cubes) 49 8 V3 [4]^+ 4 50 9 V3 [2,6]^+ 12 12^4+2 51 8 V3 [2,3]^+ 6 52 9 V3 [3,3]^+ 12 12^4+4 53 8 V3 [2,3]^+ 6 54 9 V3 [3,4]^+ 24 24^2+6 55 9 V3 [2]^+ 2 56 9 V3 [3^+,4] 24 24^2+8 57 9 V3 [2,3]^+ 6 58 9 V3 [2,4]^+ 8 59 9 V3 [2,3]^+ 6 60 10 V3 [3,3]^+ 12 12^5 (5 snub tetrahedra) 61 9 V3 [6]^+ 6 62 10 V3 [2,3]^+ 6 63 9 V3 [2,7]^+ 14 14^4+7 64 10 V3 [3,3]^+ 12 12^5+4 65 10 V3 [2]^+ 2 66 10 V3 [2,4]^+ 8 67 10 V3 [2]^+ 2 68 10 V3 [2^+,4] 8 69 10 V3 [4]^+ 4 70 11 V3 [2,5]^+ 10 10^7 71 10 V3 [2,3^+] 6 72 11 V3 [3,5]^+ 60 60+12 (pentakis truncated icosahedron) 73 10 V3 [4]^+ 4 74 11 V3 [2,6]^+ 12 12^6+2 75 11 V3 [2]^+ 2 76 11 V3 [3,3]^+ 12 12^6+4 77 11 V3 [4]^+ 4 78 11 V3 [3,4]^+ 24 24^3+6 79 11 V3 [2]^+ 2 80 11 V3 [3,5]^+ 60 60+20 (hexakis truncated icosahedron) 81 11 V3 [4]^+ 4 82 11 V3 [2^+,10^+] 10 10^8+2 83 11 V3 [2,3]^+ 6 84 12 V3 [3,3]^+ 12 12^7 (7 snub tetrahedra) 85 11 V3 [2,5]^+ 10 86 12 V3 [2,2]^+ 4 87 12 V3 [1]^+ 1 88 12 V3 [3,3]^+ 12 12^7+4 89 12 V3 [2]^+ 2 90 12 V3 [2,4]^+ 8 91 12 V3 [2]^+ 2 92 12 V3 [3,3]^+ 12 12^7+4^2 93 12 V3 [4]^+ 4 94 13 V3 [2^+,2^+] 2 95 12 V3 [2]^+ 2 96 13 V3 [3,3]^+ 12 12^8 (8 snub tetrahedra) 97 12 V3 [4]^+ 4 98 13 V3 [2,4]^+ 8 99 12 V3 [2] 4 100 13 V3 [3,3]^+ 12 12^8+4
A similar table for 4-dimensions is in preparation