[...]And here is the sum of all 31 radii:
(Dialog) In[45]:= RootReduce[ a + 5*(b + c + d + 2*e + f) /. ({%43, %42, %41, %40, %39, %37} /. Equal -> Rule)]
(Dialog) Out[45]= Root[-1420422914555742315579237580166767687895 + 986698570134623752019718445500199526980 #1 + 29500554236063157182483620663202840050 #1^2 - 149707320150923789108393415730476553140 #1^3 + 22402939277976882703556106744155369881 #1^4 + 6381306719751558250584122460830478224 #1^5 - 1693700702675487417920500871791330440 #1^6 - 35309349791131700058455860099349520 #1^7 + 38981029246829481827764941759491826 #1^8 - 2122793423456157043515638726091400 #1^9 - 234604932318709661740722122337716 #1^10 + 17199818186413063901717573711400 #1^11 + 672971253568050367752568404186 #1^12 - 36144272641837305041458024880 #1^13 - 1252583495947208761965493960 #1^14 - 8664849523701754438888016 #1^15 + 38200310480412059474501 #1^16 + 455729543056862323300 #1^17 + 811211149087190290 #1^18 + 376509003678700 #1^19 + 14315068205 #1^20 &, 14]
In[46]:=N[%,33] Out[46]=5.10073616136551344169875454575016
in full agreement with David's approximate value accompanying the diagram http://gosper.org/subopt31D5.png . In this diagram, {31}={a}, {6 mod 6} = {b}, {5 mod 6} = {c}, {4 mod 6} = {d}, {2 mod 6} = {f}, and {1 mod 6} U {3 mod 6} = {e}. They factor over sqrt5 down to 10th, the one for a down to 5th.
Does Lagrange's resolvent sextic apply to extension to sqrt5?
A full RootReduce vetting would probably take many days. This took nearly an hour: (Dialog) In[44]:= RootReduce[ a + 2*(b + c + f) /. ({%43, %42, %41, %40, %39, %37} /. Equal -> Rule)]
(Dialog) Out[44]= 1
And this (stating that five b's surrounf the a) took several hours: (Dialog) In[47]:= RootReduce[ 5 a^4 + 20 a^3 b + 10 a^2 b^2 - 20 a b^3 + b^4 /. ({%37, %40} /. Equal -> Rule)]
(Dialog) Out[47]= 0
This is almost certainly the largest polynomial system I've ever solved. With much help from Neil and Corey.
rwg>Given the consistent superiority of symmetric packings in the smaller cases, I couldn't resist checking David's numbers by exactly solving the D_5 "31" case, which proved challenging. It's easy to write
more than rwg>six trig and pythagorean equations (Neil spotted a "nice" one) rwg> for the six unknown radii, The straightforward method is to use the Law of Cosines on the angles between the centers of circles surrounding a particular circle, summing to 2 pi, generalizing the five-around-the-center equation. But some of these don't yield the nicest polynomials when you equate sin(sum) with 0. What about the e and f circles against the rim? It turns out you can use the same Law of Cosines sum, treating the rim as radius -1 ! You can see this for an e circle (#7) in the diagram. The angle -1,7,21 is the supplement of 21,7,31, and the three sides are e+e, -(e+(-1)), and -(e+(-1)). Etc. rwg> but not so easy (at least for me) to clear the radicals, after which Reduce has been working for days to "univariablize" (triangularize, then back substitute) the six polynomials. In fact, the illustrated case has so far thwarted "polynomialization", but the f equation worked and had a pleasant (and relevant) factor. This reduces the system to an almost embarrassingly modest Timing[Reduce[{b^2 c^4 - 4 b^2 c^3 d - 8 b c^4 d + 6 b^2 c^2 d^2 + 8 b c^3 d^2 + 16 c^4 d^2 - 4 b^2 c d^3 + 8 b c^2 d^3 + 32 c^3 d^3 + b^2 d^4 - 8 b c d^4 + 16 c^2 d^4 - 16 b^2 c^3 e - 34 b c^4 e + 32 b^2 c^2 d e - 40 b c^3 d e - 8 c^4 d e - 16 b^2 c d^2 e - 12 b c^2 d^2 e + 8 c^3 d^2 e - 40 b c d^3 e + 8 c^2 d^3 e - 2 b d^4 e - 8 c d^4 e + 64 b^2 c^2 e^2 - 16 b c^3 e^2 + c^4 e^2 + 32 b c^2 d e^2 - 4 c^3 d e^2 - 16 b c d^2 e^2 + 6 c^2 d^2 e^2 - 4 c d^3 e^2 + d^4 e^2 == 0, b^2 c^4 + 2 b c^5 + c^6 + 2 b c^4 d + 2 c^5 d + c^4 d^2 + 2 b^2 c^3 e + 4 b c^4 e + 2 c^5 e - 2 b^2 c^2 d e + 2 c^4 d e - 2 b c^2 d^2 e + b^2 c^2 e^2 + 2 b c^3 e^2 + c^4 e^2 - 2 b^2 c d e^2 - 2 b c^2 d e^2 + b^2 d^2 e^2 + 2 b^2 c^3 f + 4 b c^4 f + 2 c^5 f + 4 b c^3 d f + 4 c^4 d f + 2 c^3 d^2 f + 2 b^2 c^2 e f + 4 b c^3 e f + 2 c^4 e f - 6 b^2 c d e f - 6 b c^2 d e f - 6 b c d^2 e f - 2 c^2 d^2 e f - 4 b^2 d e^2 f - 6 b c d e^2 f - 2 c^2 d e^2 f - 2 b d^2 e^2 f + b^2 c^2 f^2 + 2 b c^3 f^2 + c^4 f^2 + 2 b c^2 d f^2 + 2 c^3 d f^2 + c^2 d^2 f^2 - 4 b^2 d e f^2 - 6 b c d e f^2 - 2 c^2 d e f^2 - 4 b d^2 e f^2 - 2 c d^2 e f^2 + d^2 e^2 f^2 == 0, a^2 b^3 + 4 a b^4 + 4 b^5 + 2 a^2 b^2 c + 8 a b^3 c + 8 b^4 c + a^2 b c^2 + 4 a b^2 c^2 + 4 b^3 c^2 + 2 a b^3 d + 4 b^4 d - 4 a^2 b c d - 8 a b^2 c d - 4 a^2 c^2 d - 10 a b c^2 d - 4 b^2 c^2 d + b^3 d^2 - 4 a^2 c d^2 - 8 a b c d^2 - 2 b^2 c d^2 + b c^2 d^2 == 0, c e - c f - f^2 + c f^2 + e f^2 + f^3 == 0, 5 a^4 + 20 a^3 b + 10 a^2 b^2 - 20 a b^3 + b^4 == 0, a + 2*b + 2*c + 2*f == 1, .1 < a < .11, .13 < f < .14, .14 < b < .15, .16 < c < .17, .16 < d < .17, .19 < e < .2}, {a, b, c, d, e, f}, Backsubstitution -> True]] however, Reduce does a GroebnerWalk with no sign of returning to the origin. (Does anybody know how to view the Stack of a Level 1 dialog from Level 2?) --rwg