Elementary cellular automata in Wolfram’s 1D, two-state setup update each cell from its left–center–right neighborhood through an 8-bit rule codeR ∈ [0,255]: the bit of R indexed by (s_{i-1}, s_i, s_{i+1}) gives s_i^{t+1}. This page draws the space–time diagram (time downward), lets you choose periodic edges or null (zero) boundaries, start from a single 1 or a Bernoulli random top row, and shows the 111…000 lookup strip under the raster. Informal Wolfram-class labels (I–IV) are tabulated only for a handful of famous rules—classes are not sharp on finite widths.
Who it's for: Anyone learning discrete dynamical systems, complexity, or statistical physics toy models.
Key terms
Elementary cellular automaton
Wolfram code
Rule 30
Rule 110
Space-time diagram
Periodic boundary
How it works
One-dimensional binary cellular automata: the next state of each cell depends on itself and its two neighbors through an 8-bit Wolfram code R. Scan rules and boundary conditions to compare ordered, periodic, chaotic, and complex space–time diagrams.
We follow the usual convention: neighborhoods are read as binary triples (left,center,right) with value 4L+2C+R; the output for 111 is bit 7 of R, counting down to bit 0 for 000—so the 8-bit pattern printed on the page matches Wolfram’s standard ordering.
Why does my class label show “—”?
The I/II/III/IV tag is only filled for a short curated list of textbook rules; arbitrary codes can sit between classes on finite lattices, so the safe default is an em dash.
Null vs periodic edges?
Null boundaries inject virtual zeros outside the tape, biasing dynamics near the sides; periodic edges wrap the ring and remove boundaries—compare Rule 90’s Sierpinski-like fan to the clipped fringes with zeros.