I have a more easily-computable measure, which avoids those axis issues. -------- Let P be a pattern, which is the disjoint union of cells (x1, y1), (x2, y2), ..., (xn, yn). Then, we define a constant: S(P) = x1^2 + y1^2 + x2^2 + y2^2 + ... + xn^2 + yn^2. This is simply the sum of squared distances from the origin. Clearly, we have S(P disjointunion Q) = S(P) + S(Q), which is rather cute. Next, we want to handle translations. If our pattern P has centroid (x,y), and we translate it to form a pattern P' with centroid (x',y'), then the following identity holds thanks to the Huygens-Steiner theorem: S(P) - n(x^2 + y^2) = S(P') - n(x'^2 + y'^2) This gives us S(P') = S(P) + n(x'^2 + y'^2 - x^2 - y^2). -------- We now define R(P) = exp(-S(P)), so that summing over all translates of a pattern (and, indeed, all patterns) yields a finite measure. Disjoint union is now multiplicative: R(P disjointunion Q) = R(P) R(Q). And the Huygens-Steiner translation formula becomes: R(P') = R(P) exp(x^2 + y^2 - x'^2 - y'^2)^n. -------- Suppose P has a centroid (x,y). Then, we can compute the sum over all translations of P: Sigma(R(Translate[P, (i,j)])) = R(P) (LM) where L = Sum[integer i] [exp(n x^2 - n (x + i)^2)] and M = Sum[integer j] [exp(n y^2 - n (y + j)^2)] [Aside: The maths is much nicer if we integrated over all *real* translations of P instead of summing over all integer translations. Then, L = Sqrt[Pi/n] exp(n x^2) and M = Sqrt[Pi/n] exp(n y^2).] -------- Let's suppose we want to find the total value of R, summing over all Life patterns. We use the disjoint union formula: R(P disjointunion Q) = R(P) R(Q) where Q is just a single cell. Then, we have: R(P) + R(P disjointunion Q) = R(P) (1 + R(Q)) So, the total value of R is equal to the infinite product: Product[integer i, integer j] [1 + exp(-i^2-j^2)] According to Mathematica, this evaluates to: 13.229455772329303730238455403464234174634112652323001 034270847525567752900584753556103622375911163436936902 250061828936681214420033729814493443269508795000058792 023403572348671341891245501751290476764 -------- Sincerely, Adam P. Goucher