## Spherical Designs

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

A library of putatively optimal spherical t-designs.

• How should one place N points on a sphere for use in numerical integration with equal weights?

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.

• Reference:
McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions, R. H. Hardin and N. J. A. Sloane, Discrete and Computational Geometry, 15 (1996), pp. 429-441. [note: Fig. 1a and Fig. 1b are in separate files]

• From the abstract of that paper (somewhat expanded):

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 .

• Go to library of 3-d designs | library of 4-d designs (not yet installed)

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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