How complex can the simplest possible computation get? A 1-D row of cells, each updated from only itself and its two neighbours — can such a trivial rule make order, fractals, randomness, or universal computation?
▶ Launch the interactive simulationA cell's next state is a function of (left, self, right) — 8 inputs, so the rule is one of just 256 bytes (Wolfram). Stack the rows into a spacetime picture and measure what three rules do: rule 90's fractal dimension by mass-scaling (live cells in L rows ∝ L^D), rule 30's centre-column density and block-entropy (is it random?), and rule 110's behaviour.
the whole Wolfram zoo from 8 bits: rule 90 paints the SIERPIŃSKI triangle with fractal dimension D=log₂3≈1.585 (measured to the digit, since the live-cell count in 2^k rows is exactly 3^k); rule 30 manufactures RANDOMNESS — its centre column has density ≈½ and per-bit entropy ≈1.0, indistinguishable from a coin (Wolfram used it as a PRNG); rule 110 is COMPLEX — gliders on a textured background, and provably Turing-complete (Cook 2004). Determinism alone yields order, fractals, chaos, and computation