I’ve been trying to get a better solution to the heptagon packing in P2 than previously( https://milotorda.net/index.php/packing-of-regular-heptagons-in-the-p2-group-using-the-hypertorus-igo/ ) and I did. It just take long. The decay of the neighbourhoods has to be very slow ie. 1/1.25. The density of the structure is 0.892690680150324.
Left: Found configuration; Right: Zoom of the configuration
An experiment if I could get the a tiling with a random triangle with the hIGO. The triangle is constructed via 3 randomly picked points on an unit circle. The density of the packing is 0.999999961042656.
Left: Found configuration; Right: Zoom of the configuration
An experiment if I could get the Kisrhombille tiling with the hIGO. The density of the packing is 0.999999998714671.
Left: Found configuration; Right: Zoom of the configuration
Best density found by the hIGO is 0.892690301112988. The densest packing of heptagons in P2 (proved by Greg Kuperberg and Włodzimierz Kuperberg) is ~0.892690686126508.
Left: Found configuration; Right: Zoom of the configuration
Best density found is 0.906163111422202. Optimal P1 density is ~ 0.906163678643946. Looks like optimal configurations in P1 and P2 coincide. This could be true for all centrally symmetric polygons.
Left: Found configuration; Right: Zoom of the configuration
I’ve searched the P1 group to find the densest packing of regular octagons since since the densest packing of convex set with central symmetry in the plane is a lattice packing ie a member of P1 plane group. The density I found is 0.906163653094707. The analytical result for the optimal density is ~ 0.906163678643946. ( Packing Regular Octagons ). The minimum distance between the octagons in the packing is 1.950763550695456e-09.
Left: Found configuration; Right: Zoom of the configuration
Finally some good results. I’ve used the hypertorus IGO to search the P2 plane group for the optimal packing of the of regular pentagons. The density found by the first run of the algorithm was 0.921310671743827. The optimal packing of regular pentagons proved by Hales and Kusner is ~0.921310674166737 ( https://milotorda.net/index.php/plane-group-packings-with-regular-pentagons/ ). The minimum distance between the pentagons in the packing is 3.937183912228193e-11. The algorithm can refine this just needs to run longer.
Left: Found configuration; Right: Zoom of the configuration
Setting: initial learning rates = [1,1]; runs=1; shrinkage covariance as previously; and resetting the learning rates to initial values left after 100 iterations right after 200 iterations.
Setting: adaptive learning rates with 2*delta and delta set to eta_{*}/3. 100 samples per run. Starting distribution: uniform. Shrinkage covariance computed with this: http://ledoit.net/shrinkDiag.m
Starting learning rates: eta_d=10; eta_D=10; iterations=1000; runs=1;
Starting learning rates: eta_d=10; eta_D=10; iterations=100; runs=10;
Starting learning rates: eta_d=10; eta_D=1; iterations=100; runs=10;
Starting learning rates: eta_d=10; eta_D=10; iterations=100; runs=10;
Starting learning rates: eta_d=1; eta_D=1; iterations=100; runs=10;
Learning rates: d update step size=1; D update step size =1
Number of samples per iteration = 500;
