Spark & Anvil Privacy

The Generative Bench

A generative rule isn’t magic or randomness — it’s a tiny piece of code you can predict. Read the rule, predict the next row, and the bench runs the code and renders the pattern it grows. Same rule, same result, every time.

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Where this idea comes from

The triangle that came before Pascal

Write 1 at the top, then make each number the sum of the two above it, and you get the triangle now called Pascal’s. In China, Jia Xian described it around 1050 and Yang Hui printed it in 1261; mathematicians in Persia and India also knew it. Blaise Pascal wrote his treatise on it in 1654.

One thing to hold: A triangle of numbers, each one the sum of the two above it.

Why it matters: Colour in only the odd numbers and the triangle becomes the Sierpinski triangle. Rule 90 draws exactly that from a single cell, because adding and keeping only “odd or even” is the same as its left-XOR-right rule.

What we don’t know: Jia Xian’s own book is lost; we know his triangle because Yang Hui copied it and named him as the source.

Dates:

  1. 1050 Jia Xian describes the triangle (about this year).
  2. 1261 Yang Hui prints it, crediting Jia Xian.
  3. 1654 Pascal writes his treatise on the arithmetical triangle.

Sources: A. W. F. Edwards, Pascal’s Arithmetical Triangle (Griffin, 1987) · J. Needham, Science and Civilisation in China, vol. 3 (Cambridge University Press, 1959) · S. Wolfram, A New Kind of Science (Wolfram Media, 2002), ch. 2

A shell that looks like a rule

In 1983 Stephen Wolfram studied Rule 30, a one-line rule whose output from a single cell looks random. He later pointed out that the cone snail Conus textile grows a shell covered in a very similar pattern of triangles.

One thing to hold: A cone-snail shell patterned with brown and white triangles.

Why it matters: A shell grows only at its edge, and the cells along that edge lay down pigment or don’t, partly depending on their neighbours: one row at a time, like a cellular automaton. Simple local rules can make patterns that look designed or random.

What we don’t know: Nobody claims the snail “runs” Rule 30. Its pigment chemistry is a different, more complicated rule that happens to draw similar shapes, and biologists model it in other ways.

Dates:

  1. 1983 Wolfram publishes his study of the 256 simple rules, Rule 30 among them.
  2. 1995 Meinhardt models how sea shells lay down their patterns.
  3. 2002 Wolfram sets the Conus shell beside Rule 30.

Sources: S. Wolfram, “Statistical mechanics of cellular automata”, Reviews of Modern Physics 55(3), 1983 · S. Wolfram, A New Kind of Science (Wolfram Media, 2002), ch. 8 · H. Meinhardt, The Algorithmic Beauty of Sea Shells (Springer, 1995)