Home  •  LifeWiki  •  Forums  •  Download Golly

## Elementary derivation of maximum heat in Euclid CAs

A forum where anything goes. Introduce yourselves to other members of the forums, discuss how your name evolves when written out in the Game of Life, or just tell us how you found it. This is the forum for "non-academic" content.

### Elementary derivation of maximum heat in Euclid CAs

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
`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`
0.1485̅
Caenbe

Posts: 51
Joined: September 20th, 2016, 4:24 pm
Location: Nowhere Land, USA

Return to The Sandbox

### Who is online

Users browsing this forum: No registered users and 1 guest 