What does a small disturbance in Rule 110's periodic background turn into? Time runs down the board: the top row is the seed, and each tick adds the next generation below it.
Try this Choose the Random Density preset (seed random-bits, density 45) and press Play. Within a few dozen rows, stretches of the ring settle into the same ether, and the boundaries between them move at fixed slopes as gliders. seed and density apply only at setup, so press Reset after changing either.
About this model Where it ends up
Rule 110 is one of the 256 elementary cellular automata, in which each cell of a row is updated from itself and its two neighbors. Stephen Wolfram's classification puts it in class 4: localized structures (gliders) moving through a periodic background (the ether). Matthew Cook proved it Turing complete (Complex Systems, 2004) by arranging gliders to emulate a cyclic tag system.
Setup writes the top row from the seed param. ether-glider, the default, lays the ether, the 14-cell pattern 11111000100110 repeated six times around the ring, then clears the eight cells from px = −4 to 3 except px = −2. single-point lights one cell, and random-bits lights each cell with probability density percent (the Random Density preset uses 45). Each tick fills the next row: a cell is live when the three cells above it read 110, 101, 011, 010 or 001. The outputs for the eight neighborhoods 111 down to 000 are 01101110, which is 110 in binary. The ring is 84 cells, a multiple of 14, so the ether closes on itself without a seam; the run stops after 80 ticks, at the bottom of the board.
Cook's construction needs an unbounded row built from carefully spaced blocks, so nothing on an 84-cell ring computes. What fits is the raw material: the ether, which repeats every 14 cells and every 7 rows, and a glider moving through it.
Idea Stephen Wolfram · Matthew Cook
MathPets MathPets · org
