Solutions to such a problem can be expressed in genetic form, suitable for being acted upon by evolutionary forces. The position of each point is expressed in altitude and azimuth angles, sufficient to cover the surface of the unit sphere. (Note to self: explore using quaternions.) The simulation is then run, with the function that determines evolutionary fitness being the minimum distance between any two points. Solutions can then be visualized as polyhedra, where the unit sphere is cut by planes normal to the vector from the origin to each point. This results in an N-sided polyhedron — although sometimes a rather weird, lumpy one at first.<\/p>\n\n\n\n For “easy” problems such as N=6, this approach\nquickly produces a nearly-optimal solution. Six equally-spaced points on the\nunit sphere will be farthest apart when they are in the form of a cube. For\nexample, points at (1,0,0) and (0,1,0) could be two such points, 90 degrees\napart. The optimal Tammes distance for N=6, then, is sqrt(2).<\/p>\n\n\n\n For N=6, the genetic approach described above very quickly converges to the expected cube solution, with point positions very nearly the expected distance apart. (A typical result differs from sqrt(2) by well under one part per million, if left to run for a while.)<\/p>\n\n\n\n Some values of N produce interesting results. For N=8, the most typical result is an anti-prism, similar to the platonic octohedron, but with four of the points rotated by 45 degrees. (When you think about it, it makes sense; the points are farther apart that way.)<\/p>\n\n\n\n Larger values seem to pose greater challenges to this\napproach; N=12 sometimes produces the expected Platonic regular dodecahedron,\nbut more often will get stuck in a local minimum, producing a somewhat\nsymmetrical shape. (Like in nature, Darwinian evolution doesn’t always produce\na *perfect* solution — just whatever seems to get the job done best. )<\/p>\n\n\n\n The next goal: try to grow a regular icosahedron. <\/p>\n","protected":false},"excerpt":{"rendered":" Genetic algorithms can be used to evolve regular and nearly-regular polyhedra. Here’s how. Continue reading ![\"\"](\"https:\/\/www.paleotechnologist.net\/wp-content\/uploads\/2019\/12\/cube_193200_ppm_error.jpg\")
Not very impressive at some 19.3% error — but that quickly improves!<\/figcaption><\/figure>\n\n\n\n![\"\"](\"https:\/\/www.paleotechnologist.net\/wp-content\/uploads\/2019\/12\/cube_12ppm_error.jpg\")
a cube with a measurement error of ~12.293 parts per million.)<\/figcaption><\/figure>\n\n\n\n![\"\"](\"https:\/\/www.paleotechnologist.net\/wp-content\/uploads\/2019\/12\/2019-12-30-18.11.53-1024x576.jpg\")
![\"\"](\"https:\/\/www.paleotechnologist.net\/wp-content\/uploads\/2019\/12\/2019-12-30-18.13.21-1024x576.jpg\")
All except for the 8-gon are irregular, but with striking symmetries.
From left to right: Irregular 10-sider; antiprism 8-sider; 7-hedron; 9-hedron.<\/figcaption><\/figure>\n\n\n\n