Chapter 12: The Principle of Computational Equivalence

Section 6: Computational Irreducibility

[Examples of] short computations

Some properties include:

(a) The regions are bounded by the hyperbolas x y == Exp[n/2] for successive integers n.

(d) There is approximate repetition associated with rational approximations to π (for example with period 22), but never precise repetition.

(e) The pattern essentially shows which x are divisors of y, just as on pages 132 and 909.

(h) Mod[Quotient[s, 2^n], 2] extracts the digit associated with 2^n in the base 2 digit sequence of s.

(i) Like (e), except that colors at neighboring positions alternate.

(l) See page 613.

(m) The pattern can be generated by a 2D substitution system with rule {1 -> {{0, 0}, {0, 1}}, 0 -> {{1, 1}, {1, 0}}} (see page 583).

(See also page 870.)

Even though standard mathematical functions are used, few of the pictures can readily be generalized to continuous values of x and y.

From Stephen Wolfram: A New Kind of Science [citation]