Wolfram Rule 30 Cellular Automaton - 3D Tower

An elementary cellular automaton updates a row of binary cells: each new cell depends on itself and its two neighbours, so there are 2³ = 8 neighbourhoods and 2⁸ = 256 rules. The rule number is the binary word of the eight outputs. In this 3D view the row is closed into a ring and each new generation is stacked as a new layer, so time becomes the height of the tower - the newest generation is at the top.

• Rule 30: new = left XOR (centre OR right). From one live cell it grows an irregular pattern whose centre column passes many randomness tests; its left edge is perfectly regular.
• Rule 90: left XOR right - Sierpinski triangle with exactly 2^popcount(t) live cells in generation t (on a ring that is wide enough that the pattern does not meet itself).
• Rule 110 is computationally universal; Rule 184 models single-lane traffic and conserves the number of cars.
• The red damage curve compares the pattern with a copy whose initial centre cell was flipped; in chaotic rules the difference spreads at about the light-cone speed until it fills the ring.

Checks. The same four numerical checks as the 2D page run on an independent flat grid: table vs boolean formula for rule 30, the exact Rule 90 cell counts, the Rule 30 left edge, and the fair-coin z-score of the 2000-bit centre column.

Simplifications. A ring of N cells (32-96) is a finite periodic system: once the pattern wraps around it interacts with itself, unlike the infinite line used in Wolfram's pictures, so the 2D page is the better place to study the single-seed limit. Randomness claims are empirical statistics of finite samples. The Blender-built plinth, core and pillars are decoration; the tower itself is drawn as instanced cell blocks.