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Concept

Sturmian word

A two-letter word that never repeats yet has only n + 1 different blocks of each length n, the fewest possible. Every one is the record of a straight line of irrational slope crossing a square grid, and each is balanced: equal-length stretches differ by at most one letter.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A straight line across a grid, and the word of walls it crosses. A line of slope 0.6180 crossing a unit grid, with each crossing marked V or H, beside the same path folded into a square as a billiard trajectory; the word begins VHVHVVHVHVVHVVHVHVVH.

The word a straight line spells

A ball rolling across a square table, forever, hits walls in some order: V for a side wall, H for the end wall. Unfold the table and the ball becomes a straight line across a grid, and the order of walls becomes a word — VHVHVVHVHVVHV… for the golden slope. That word has exactly n + 1 different blocks of every length n, the fewest any word that never repeats can have; every stretch of it holds its fair share of H's to within one; and its blocks occur with at most three different frequencies.

dynamics · Billiards A ball in a cube and the word of walls it hits, direction (1, √2, √3). A cube drawn in perspective with a billiard path of 30 bounces inside it, each bounce point coloured by the pair of walls it is on, and the word ZYXZYZXYZZXYZYXZYZXYZZYXZYZXYZ below.

The word a line spells in a cube

A ball bouncing in a cube hits three kinds of wall, and the order spells a word in three letters. On a square table the word had n + 1 different blocks of length n; in the cube it has $n^2 + n + 1$ — three, seven, thirteen, twenty-one — for a direction in general position. What general position has to exclude turns out to be more than any rational relation among the three speeds.

dynamics · Billiards

Named alongside it

The objects these essays reach for when they reach for this one.

BilliardsSymbolic dynamicsUnfoldingComplexityFibonacciGolden ratioIrrational rotationSubstitutionTorusWord complexity

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