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  "body": "\n## Example\n\nA Sturmian word is an infinite word $s = s_0 s_1 s_2 \\cdots$ over $\\set{0, 1}$ that has exactly $n + 1$ distinct factors of length $n$ for every $n \\ge 0$.\nA factor of length $n$ is a block $s_i s_{i+1} \\cdots s_{i+n-1}$.\nEquivalently, there are an [[irrational number|080T2TXVQV]] $\\alpha \\in (0, 1)$ and a real $\\rho$ with the following property.\nFor all $n$, $s_n = \\floor{(n+1)\\alpha + \\rho} - \\floor{n \\alpha + \\rho}$, or the same holds with $\\ceil{\\cdot}$ in place of $\\floor{\\cdot}$.\nThen $s_n = 1$ exactly when the point $n\\alpha + \\rho$ of the circle $\\R/\\Z$ lies in a fixed arc of length $\\alpha$.\nSo $s$ codes the [[irrational rotation|2Q1J7RZ4TK]] by $\\alpha$.\n\nThe Fibonacci word is the main instance.\nDefine $S_0 = 0$, $S_1 = 01$ and $S_{k} = S_{k-1} S_{k-2}$ for $k \\ge 2$.\nEach $S_{k-1}$ is a prefix of $S_k$, and the limit is\n$$f = 0100101001001010010100100101 \\cdots .$$\nCheck for small $n$: the factors of length $1$ are $0, 1$, and those of length $2$ are $00, 01, 10$, since $11$ never occurs.\nSo there are $2$ and $3$ of them, as required.\nFor $k \\ge 1$ the word $S_k$ has $F_k$ zeros and $F_{k-1}$ ones, with $F_k$ the Fibonacci numbers ($F_0 = F_1 = 1$).\nSo the frequency of the letter $1$ is $\\lim F_{k-1}/F_{k+1} = 1/\\varphi^2$, where $\\varphi$ is the [[golden ratio|0SC8T7548D]].\n\n## Remarks\n\nA word with at most $n$ factors of length $n$ for some $n$ is eventually periodic, by a theorem of Morse and Hedlund.\nSo Sturmian words are the aperiodic words of the lowest possible complexity.\nThe closure of the shift orbit of a Sturmian word is a minimal shift space on which the shift has zero entropy.\n\n## Explanations\n\n### L1 human\n\nDraw a straight line of irrational slope across a grid of squares.\nWrite $0$ each time the line crosses a vertical grid line and $1$ each time it crosses a horizontal one.\nThe resulting word is Sturmian: it never repeats periodically, yet it is as simple as a non-repeating word can be.\nThe Fibonacci word, built by gluing the two previous words together, is the best known example.\n",
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