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Home/Thermodynamics/Wolfram Elementary Cellular Automata

Wolfram Elementary Cellular Automata

1D binary CA with Wolfram code R∈[0,255]: space–time raster, periodic or null boundaries, single-cell or random IC, rule-bit strip, and informal class hints for textbook rules (e.g. 30, 90, 110, 184).

Simulation

Single-cell IC is the classic textbook picture; random IC stresses the same rule on disordered seeds.

Parameters

30
257
384

Presets

Neighborhood index (L,C,R) maps to bit position in R: output = (R >> (4L+2C+R)) & 1. Wolfram’s four classes are qualitative; the sidebar lists a short table for common rules.

Shortcuts

  • •Space / Enter — toggle random vs single-cell IC
  • •P — new random top row (random IC mode)
  • •R — reset to Rule 30 defaults

Measured values

8-bit pattern (111→000)00011110
Class (informal)III
Mean density ⟨s⟩0.0000
W × T257 × 384

About this model

Elementary cellular automata in Wolfram’s 1D, two-state setup update each cell from its left–center–right neighborhood through an 8-bit rule code R ∈ [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.

Key equations

s_i^{t+1} = (R >> (4 s_{i-1}^t + 2 s_i^t + s_{i+1}^t)) & 1 — R = Σ_{abc} b_{abc} 2^{4a+2b+c} with b_{abc} ∈ {0,1}.

Frequently asked questions

Which neighborhood bit is “most significant”?
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.