Alexey Tarasov
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 May 21 comment Packing twelve spherical caps to maximize tangencies I was fast because I already have working approach. Graphs are generated by program plantri and filtered by my program. Total times about 15 minutes. Program for solving Tammes problem for N=13 is quite complicated. It is written on the perl it's size 1800 lines. Part used here more simple. Program will available soon on my webpage with dcs.isa.ru/taras/tammes13 May 19 comment Packing twelve spherical caps to maximize tangencies I recalculated for all good graphs. Total amount of graphs with 24 edges and more is 221501. By elimination program only two graph were survived. These graphs are described in the question. Does anybody know about existing non-computer proof of this fact? May 18 comment Packing twelve spherical caps to maximize tangencies I was wrong then exclude septagons. I did it for Tammes problem because where is lemma that septagons are impossible for irreducible graphs. But here they are not irreducible. I will recalculate with all faces tomorrow (as well question for 24 edges). About rhombus, from spherical Pythagorean theorem where is constrain : $cot(a/2)*cot(b/2) = cos d$, where d - length of the side, a,b - angles. Maximum sum of angles applies for regular quadrilateral, so yes maximal summ is $2 pi - 2 \arccos(1/3)$ May 18 awarded Supporter May 18 awarded Editor May 18 awarded Teacher May 18 revised Packing twelve spherical caps to maximize tangencies added 711 characters in body May 18 answered Packing twelve spherical caps to maximize tangencies