ConwayLife.com

...x * y, yet entierely explored as start figures ?

5 * 5, 10 * 10, 20 * 20,...?

???
What are we talking about?

All start's combinaisons possible for a surface x,y (a bounding box)
example (symmetry or rotation are not inspected)
for a surface 1 * 1 there are 2 possibilities
for a surface 1 * 2 there are 4 possibilities
for a surface 1 * 3 there are 5 possibilities
for a surface 2 * 2 there are 14 possibilities
for a surface 2 * 3 there are 1 248
for a surface 3 * 3 there are 643 531
etc...

I think he means number of possible patterns in x*y bounding box, excluding rotations/reflections. There is function to compute this, but I don't know it exactly

number of possible patterns in x*y bounding box, excluding rotations/reflections.

Yes that is exactly that

If the surface 5*5 is fully explored, no need to search inside for find something new!

So at this date any idea of the biggest size of the bounding box fully explored ?

I think he means number of possible patterns in x*y bounding box, excluding rotations/reflections. There is function to compute this, but I don't know it exactly

It's possible using Burnside's Lemma, although we have to consider five separate cases:

Rectangular:
2m by 2n: (2^(4mn) + 2^(2mn) + 2^(2mn) + 2^(2mn))/4
2m+1 by 2n: (2^(4mn + 2n) + 2^(2mn + n) + 2^(2mn + 2n) + 2^(2mn + n))/4
2m+1 by 2n+1: (2^(4mn + 2n + 2m + 1) + 2^(2mn + n + 2m + 1) + 2^(2mn + 2n + m + 1) + 2^(2mn + n + m + 1))/4

Square:
2k by 2k: (2^(4k²) + 2^(2k²) + 2^(2k²) + 2^(2k²) + 2^(k²) + 2^(k²) + 2^(2k² + k) + 2^(2k² + k))/8
2k+1 by 2k+1: (2^(4k² + 4k + 1) + 2^(2k² + 3k + 1) + 2^(2k² + 3k + 1) + 2^(2k² + 2k + 1) + 2^(k² + 2k + 1) + 2^(k² + 2k + 1) + 2^(2k² + 3k + 1) + 2^(2k² + 3k + 1))/8

Also, you can replace any '2^' with 'n^' for the equivalent formula for n-state cellular automata.

calcyman wrote:

I think he means number of possible patterns in x*y bounding box, excluding rotations/reflections. There is function to compute this, but I don't know it exactly

It's possible using Burnside's Lemma, although we have to consider five separate cases:
(…)

Doesn't suffice for excluding translations, tho. This still counts a pattern such as "O.." distinct from a pattern such as ".O." (An extra constraint needed seems to be having cells on all edges of the bounding box in consideration.)

Infinite growth studies cover at least the entirety of the 5x5, 2x12 and 1x39 areas (possibly a little higher?), but I don't think that necessarily counts as having "entirely explored" the start figures in question. Eg. there might be a 5x5 pattern that evolves into some (larger) novel oscillator, but this result wasn't recorded?

All oscillators up to at least 6x7 are apparently known (according to the LifeWiki article on Mold). I'm not sure where the limit is on spaceships.

—For exhausting "small" patterns, I'm tempted to agree that the bounding box is a better constraint than the amount of live cells; the combinatorics of glider/still life/methuselah collisions start piling up ridiculously fast.