Hofstadter’s INT
It’s a little known fact that before Benoît B. Mandelbrot first pioneered his work on fractal geometry in the 1970s, Douglas Hofstadter stumbled over the phenomenon while working on problems in number theory.
During the 1960s, Hofstadter discovered a family of graphs that exhibited a specific kind of discontinuity with the integers of a particular function. Hofstadter dubbed the main graph INT. The graph of INT(X) contains infinitely many distorted copies of itself. INT(X) is discontinuous at all rational values of X but continuous at all irrational values of X. The function replaces the partial quotient, forming a nearest integer continued fraction, like so [1].
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\[ \displaystyle{\displaylines{a_i+~\leftrightarrow~a_i+1-.}} \] \[0+\cfrac{1}{3-\cfrac{1}{4+\cfrac{1}{5+0}}}\leftrightarrow 1-\cfrac{1}{2+\cfrac{1}{5-\cfrac{1}{6-0}}}. \] \begin{align*} \frac{1}{2}=\frac{1}{2+0}&\leftrightarrow1-\frac{1}{3-0}=\frac{2}{3},&\frac{1}{3}=\frac{1}{3+0}&\leftrightarrow1-\frac{1}{4-0}=\frac{3}{4},\ \end{align*} \begin{align*} \frac{2}{5}=\frac{1}{2+\frac{1}{2+0}}&\leftrightarrow1-\frac{1}{3-\frac{1}{3-0}}=\frac{5}{8},&\frac{3}{5}=1-\frac{1}{2+\frac{1}{2+0}}&\leftrightarrow 0+\frac{1}{3-\frac{1}{3-0}}=\frac{3}{8}. \end{align*}
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