Elementary derivation of maximum heat in Euclid CAs

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Elementary derivation of maximum heat in Euclid CAs

Most of the work was already done in this video:
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
`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:
`Max. heat   inf  -----                  -----  \           1          \=  >      --------- * 4   >    X(k)  /       sqrt(N)^4      /  -----                  -----   N=1                    k|N   inf  -----    -----  \        \      4*X(k)=  >        >     ------  /        /       N^2  -----    -----   N=1      k|NSince (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`
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