McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions

Evidence is presented to suggest that, in three dimensions, spherical 6-designs with N points exist for N = 24, 26, ≥ 28; 7-designs for N = 24, 30, 32, 34, ≥ 36; 8-designs for N = 36, 40, 42, ≥ 44; 9-designs for N = 48, 50, 52, ≥ 54; 10-designs for N = 60, 62, ≥ 64; 11-designs for N = 70, 72, ≥ 74; and 12-designs for 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. 5-designs with 23 and 25 points are presented which, taken together with earlier work of Reznick, show that 5-designs exist for N = 12, 16, 18, 20, ≥ 22. It is conjectured, albeit with decreasing confidence for t ≥ 9, that these lists of t-designs are complete and that no others exist. One of the constructions gives a sequence of putative spherical t-designs with N = 12m points (m ≥ 2) where N = 1 2 t2(1 + o(1)) as t→∞. McLaren’s Improved Snub Cube and Other New Spherical Designs in Three Dimensions R. H. Hardin and N. J. A. Sloane Mathematical Sciences Research Center AT&T Bell Laboratories Murray Hill, NJ 07974 USA