McLaren's improved snub cube and other new spherical designs in three dimensions

Abstract

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 = 1/1t2(1 + o(1)) as t → ∞.