Elementary derivation of maximum heat in Euclid CAs

[Caenbe]

Relevant thread: viewtopic.php?f=11&t=3599

Most of the work was already done in this video:
https://www.youtube.com/watch?v=NaL_Cb42WyY
The whole video is actually a pretty neat proof of pi/4 = 1 - 1/3 + 1/5 - ... and I recommend watching it. But the specific result I'll use is that the number of lattice points a distance sqrt(N) from the origin is 4 times sum of X(k) over all k|N, where

Code: Select all

X(k) = 0 if k mod 4 = 0, 2
       1 if k mod 4 = 1
      -1 if k mod 4 = 3

So, adding up the contributions from all the cells, with multiplicity:

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Max. heat

   inf
  -----                  -----
  \           1          \
=  >      --------- * 4   >    X(k)
  /       sqrt(N)^4      /
  -----                  -----
   N=1                    k|N


   inf
  -----    -----
  \        \      4*X(k)
=  >        >     ------
  /        /       N^2
  -----    -----
   N=1      k|N

Since (N,k) run over all solutions of N=jk over the natural numbers, rewrite as

   inf      inf
  -----    -----
  \        \      4*X(k)
=  >        >     ------
  /        /      (jk)^2
  -----    -----
   j=1      k=1


      inf         inf
     -----       -----
     \       1   \      X(k)
=  4  >     ---   >     ----
     /      j^2  /      k^2
     -----       -----
      j=1         k=1


= 4 * (pi^2 / 6) * G