Systems based on numbers reveal that even familiar arithmetic processes can behave like complex programs. Explore five laboratories that turn digit sequences into living patterns, from binary counting tapes to chaotic logistic maps.
Stack the digit expansions of counting numbers and exponential sequences to expose nested regularities or apparent randomness. Change the base, watch the tape extend, and quantify how structured the columns remain.
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Transition rate measures how often digits flip between adjacent rows, while entropy captures the diversity of digits across the tape. Powers with large exponents typically resemble random sources, unlike the nested structure visible when counting in base 2.
Iterate classical and fast-growing recurrences, then view their digit tapes in any base. Compare linear growth to explosive Ackermann-style behavior and nonlinear modular dynamics.
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The Sylvester recurrence produces astronomically large values after only a few steps, while the nonlinear modular system settles into complex but bounded digit patterns. Entropy approximates how evenly digits are distributed in the rendered tape.
Watch a sieve-like automaton discover the primes. Each iteration marks multiples of the current base prime, revealing the seemingly erratic spacing of primes within a regular scanning procedure.
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The automaton follows Eratosthenes' logic: sweep left to right, elevate a new prime, then eliminate its multiples. The prime density converges slowly toward 0 as the limit increases, illustrating the erratic yet structured spacing of primes.
Sample digits from π, e, or √2 and run quick uniformity checks. Slide across the stream, measure entropy, and inspect frequency histograms that hint at built-in randomness.
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For a perfectly random source each digit would appear 10% of the time. π, e, and √2 pass basic frequency tests across modest windows, though deeper randomness questions remain unsolved.
Iterate the logistic map and follow both its real-valued trajectory and the evolving digit stream. Estimate the Lyapunov exponent and see how sensitive parameters feed apparent randomness in the digits.
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Positive Lyapunov exponents indicate sensitive dependence on initial conditions. The digit plot shows why chaotic maps look random even though the underlying iteration is entirely deterministic.