McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions
| dc.creator | Hardin, R. H. | |
| dc.creator | Sloane, N. J. A. | |
| dc.date | 2002-07-23 | |
| dc.date.accessioned | 2026-07-07T04:49:48Z | |
| dc.date.available | 2026-07-07T04:49:48Z | |
| dc.description | 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 = t^2/2 (1+o(1)) as t -> infinity. | |
| dc.description | 16 pages, 1 figure | |
| dc.identifier | https://arxiv.org/abs/math/0207211 | |
| dc.identifier | http://arxiv.org/abs/math/0207211 | |
| dc.identifier | Discrete Computational Geometry, 15 (1996), 429-441 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/64566 | |
| dc.subject | Combinatorics | |
| dc.subject | 52B11 (05B30) | |
| dc.title | McLaren's Improved Snub Cube and Other New Spherical Designs in Three Dimensions | |
| dc.type | text |