A one-dimensional cellular automaton. Two states. Three neighbours. Eight instructions.
01 / THE MODELOne bit, no dice.
At generation zero, only x = 0 is on. All other cells are zero. Every row updates simultaneously using L XOR (C OR R). Reading the eight outputs as binary gives 00011110₂ = 30.
The display computes through generation 1,024. Its zero padding extends beyond every displayed causal cone; there is no wraparound.
Wolfram’s Atlas of Simple Programs
02 / A PROVEN STRUCTUREMove with the edge.
Set d = t − x. Keeping d fixed follows a right diagonal. Rowland proved that these diagonals are periodic, with periods that are powers of two, for the single-cell initial condition.
Hidden order aligns the moving right boundary and enlarges its first 33 diagonals. The period label reports a match across this finite sample, requiring at least four repeats. It is not a new proof.
Eric Rowland · Local Nested Structure in Rule 30 (2006)
03 / AN OPEN QUESTIONThe centre keeps its secret.
The official Rule 30 prize asks whether the single-seed centre column is nonperiodic, whether its limiting fraction of ones is ½, and whether its nth bit can be computed in sublinear time.
A finite sample, a balanced count, or an irregular melody settles none of these questions. This exhibit makes no claim to prove randomness or computational irreducibility.
The Wolfram Rule 30 Prize Problems
04 / THE COUNTERFACTUALChange one fact.
One-bit echo copies the original evolution, flips exactly one cell at the chosen generation, then applies the same rule. Gold shows unchanged on-cells. Cyan marks cells where the two futures disagree: A XOR B.
The initial intervention is at (t, x) = (32, 0). Click the tapestry to move it, or scrub time and choose Flip the selected centre cell. An influence can travel at most one cell per generation.
05 / THE COMPOSITIONThe rule becomes a score.
Each one-bit triggers a quiet pluck. A three-bit window selects a pitch from a D-minor pentatonic palette: centre in the upper register, edge in the lower register. Zero is silence. Sound begins only when you choose Listen.
The tuning and light are artistic choices. The cell values, coordinates, comparisons and measurements are computed data. No random generator supplies the artwork.
06 / USE THE INSTRUMENTLook. Unfold. Intervene.
Space pauses time. 1, 2 and 3 switch views. M toggles sound. F opens fullscreen. Escape closes these notes or fullscreen. Scrub any generation; choose a diagonal to hear its rhythm.
The tour spends 20 seconds in each view. It respects reduced motion and starts with your current sound setting. For a lab demonstration, unfold the edge and select diagonal 4.
Inspiration · Eric Rowland’s The Hidden Structure of Rule 30
Built as an artistic mathematical instrument by Hasan Tayyar Besik. Mathematical results belong to their cited authors. All visuals are generated in your browser.