Spherical Designs

R. H. Hardin and N. J. A. Sloane

A library of putatively optimal spherical t-designs.

Summary of results

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



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