# Fibonacci Word curve

See Wikipedia on the Fibonacci Word and the Fibonacci word fractal. Extract below.

A Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name "Fibonacci word" has also been used to refer to the members of a formal language L consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs to L, but so do many other strings. L has a Fibonacci number of members of each possible length.

Definition Let ${S}_{0}$ be "0" and ${S}_{1}$ be "01". Now ${S}_{\mathrm{n}}$ (the concatenation of the previous sequence and the one before that). The infinite Fibonacci word is the limit ${S}_{\mathrm{\infty}}$ , that is, the (unique) infinite sequence that contains each ${S}_{\mathrm{n}}$ , for finite $n$ , as a prefix. Enumerating items from the above definition produces:

- ${S}_{0}$ 0
- ${S}_{1}$ 01
- ${S}_{2}$ 010
- ${S}_{3}$ 01001
- ${S}_{4}$ 01001010
- ${S}_{5}$ 0100101001001

= bits / segments