DOI: 10.5281/zenodo.5595365
title: 198 Spherical Codes in S2 for Smale's Problem 7 - 0 to 65 points - global minimums

description: Spherical Codes for 0 to 65 points on the S2 sphere<br> (1/log, 1/r, and 1/r^2 potentials)<br> ---------------------------------------------------<br> <br> Introduction &gt;<br> ------------<br> <br> Smale's Problem #7 asks for an arrangement of n-points on a S2 sphere, such that the pairwise sum of all the reciprocals of the logarithm of the distance of any 2 points is minimized. This is equivalent to the J. J. Thomson Problem in physics, or minimizing the potential energy for all points. See https://en.wikipedia.org/wiki/Smale%27s_problems<br> <br> Smale asks for the global minimal solution, not just a local minimum solution which can be infinite in number.<br> <br> In the spherical code datasets given, not only was the log force law answers found, but also the Coulomb attraction (1/r) and the inverse square law (1/r^2) force laws solutions for the points N=1..65 also found as either 40 digit or 80 digit floats.<br> <br> 40 digits of accuracy are given for the 3d configuration of the points for n-points under the 1/r (Coulomb) and 1/r^2 (inverse-square) force law while 80 digits of accuracy are given for the points under the 1/log law (JJ Thomson problem)<br> <br> NOTE: The spherical code is the floating point approximation of the answer which has been determined to be the root of an algebraic number (root of a monomial polynomial equation) 49 algebraic numbers have been successfully recovered but 109 remain to be found, however the spherical codes in the database ARE the optimal global solution (and proved mathematically numerically)<br> <br> Description &gt;<br> -----------<br> <br> 1. There are 198 files of spherical codes (floats)<br> <br> The files are arranged by the force law and then the dimension, then by number of points and finally by the count of float digits:<br> <br> log.3.[0...65].80 for 66 files<br> <br> r1.3.[0..65].40 for 66 files<br> <br> r2.3. [0..65].40 for 66 files<br> ---<br> 198 files total<br> <br> NOTE: Only 3d points on the S2 sphere were considered<br> <br> 2. The float precision is given by the last number of the file name, thus all the log configurations contain 80 digits of precision and both the 1/r and 1/r^2 configurations have 40 digits of precision.<br> <br> 3. The solutions sets by force law are available in zip file format:<br> <br> log.0-65.zip<br> r1.0-65.zip<br> r2.0-65.zip<br> <br> ---------------<br> - FILE FORMAT -<br> ---------------<br> <br> Example:<br> <br> r1.3.4.40 (tetrahedron)<br> <br> p=[<br> [-0.3365532784838319691031161303207511198229, -0.6059398890165475593599566114651164660267, 0.7208111691978685424318098323811596649287],<br> [-0.5092599564667882530696631587397749138869, -0.1705056542849786943591080778850668158280, -0.8435532695664786882743495411824978650915],<br> [0.9720056552661925898658589741289275620993, -0.1755432104873921130532233410620585904927, -0.1561716599845142109837275941326816962649],<br> [-0.1261924203155723676930796850684015283895, 0.9519887537889183667722880304122418723474, 0.2789137603531243568262673029340198964277]<br> ];<br> ener=3.674234614174767147295926112058837087949<br> <br> p are the n 3d points given as a vector set of n - [x,y,z] vectors<br> 3d Cartesian coordinates of an isomorphism are given<br> <br> ener is the global minimal energy of this n-point set configuration<br> <br> All files are GP-Pari format, but easily converted to ASCII format<br> <br> <br> Citation &gt;<br> --------<br> <br> Please see https://arxiv.org/abs/2008.04880<br> <br> Some spherical codes in S2 and their algebraic numbers<br> Randall L Rathbun, Wesley JM Ridgway<br> <br> Correspondence Author &gt;<br> ---------------------<br> Randall L Rathbun (randallrathbun@protonmail.com) Special Note &gt; ------------ The 1/r (Coulomb) values are initially taken from Neil J.A. Sloane's "Spherical codes for minimal energy" database, and improved from 12 digits accuracy to 38 digits accuracy, thus extending the precision of his points. Special care has been taken to keep the order of the points, exactly as in his tables. See http://neilsloane.com/electrons/dim3/
description: 3 additional files are added: index, metadata, and the README file which explains the contents.