Curve: prediction that ignores spatial correlations. Dots: measured density pairs from this run.
The science
Rule. Each cell is alive (1) or dead (0). In B/S notation a dead cell is Born if its number n of live neighbours (of 8) is in the B set, and a live cell Survives if n is in the S set. Conway's Life is B3/S23. The update is synchronous, so the whole grid is computed from the previous generation.
Mean-field map. If live cells were placed independently with probability p, a cell would see n live neighbours with probability C(8,n) p^n (1-p)^(8-n) and the next density would be f(p) = Σ C(8,n) p^n (1-p)^(8-n) [ p S(n) + (1-p) B(n) ]. After the first step of a random start the measured density follows this curve; later, spatial structures (blocks, gliders) break the independence assumption and the dots leave the curve. That gap is the signal that the pattern has organised itself.
Langton lambda. The fraction of the 512 possible neighbourhood configurations that map to "alive", λ = Σ C(8,n)(B(n)+S(n)) / 512. Conway's Life has λ = 0.273.
Patterns. A block is a still life, a blinker has period 2, a glider moves one cell diagonally every 4 generations, and the Gosper gun (period 30) adds one 5-cell glider per period, so the population grows by 5 every 30 generations while the gliders are still on the board.
Simplifications. The board is finite (160 x 100); with the torus switch on, gliders wrap around and may collide with the gun after a long time, with it off the edges are dead. Other rules are shown as an exploration tool; their qualitative behaviour (chaos, replication, mazes) is observed, not proven.