## Elementary derivation of maximum heat in Euclid CAs

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Caenbe
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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

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
``````
0.1485̅