Fibonacci numbers
Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.
Counting in a base that is not a whole number
Multiply by β and keep the fractional part, over and over, and the whole parts you throw away are the digits of the starting number in base β — even when β is the golden ratio. Which digit strings can ever appear is decided by one string alone: the way the number 1 is written in that base. In base φ it is .11, so 11 is the only thing forbidden; in base 1.8 it never ends, and no finite list of rules describes what is allowed.
A growth rate no step contains
Start with 1, 1 and at every step either add the last two numbers or subtract them, choosing by a coin. The sequence wanders, shrinks, even returns near zero — and grows, almost surely, like 1.13198824 to the power of the step. That number is a Lyapunov exponent of a product of random matrices, and it cannot be found by averaging anything about a single step: the two steps do not commute, and the rate depends on which way the vector is pointing when each one is applied.
Fractions repeat and roots look random
Almost every number between 0 and 1 has base-φ digits with the frequencies Parry's measure predicts. Which particular numbers do? Every fraction, provably, does not: its expansion repeats from the first digit, with a period equal to the period of the Fibonacci numbers modulo its denominator. And √2 − 1 and 1/π, computed exactly to twelve thousand digits, match every predicted frequency to within sampling error — which proves nothing about them at all.
How often a number can appear in Pascal's triangle
Every number bigger than one appears in Pascal's triangle at least twice, next to the 1 at each end of its own row. A few appear more often: 120 appears six times, and 3003 eight — as C(3003, 1), C(78, 2), C(15, 5) and C(14, 6), each twice. Counting every entry up to a million million finds seven numbers that appear more than four times and none that appear five, seven or more than eight. Whether any number appears ten times, or whether there is any limit at all, is unknown.
A chaotic map that always comes back
Arnold's cat map stretches a square by the golden ratio squared and wraps it round, and any two nearby points fly apart: a difference of a million-millionth fills the square in twenty-seven steps. Apply the same map to a picture made of pixels and it dissolves into noise within a few steps — and then, after exactly thirty steps on a 61 × 61 grid, every pixel is back where it started. Chaos and exact return are both true, and the Fibonacci numbers decide how long the return takes.
Named alongside it
The objects these essays reach for when they reach for this one.
Golden ratioLyapunov exponentSensitive dependenceBeta expansionBinomial coefficientConjectureDiophantine equationErgodic theoremExhaustive searchItineraryNormal numberPascals triangle