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
(**) Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com
September 11, 1995
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 = t^2/2 (1+o(1)) as t -> infinity.
(*) A different version of this paper appeared in:
Discrete and Computational Geometry, 15 (1996), 429-441.
For the full version see
http://www.research.att.com/~njas/doc/snub.pdf (pdf) or
http://www.research.att.com/~njas/doc/snub.ps (ps)
[note:
Fig. 1a and
Fig. 1b
are in separate files]