Hofstadterās INT
Itās a little known fact that before BenoĆ®t 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*}















