Map of fractions

These are based on Farey sequences, (well, more properly, the left half (0\le\frac{y}{x}\le1) of the Stern-Brocot tree) drawn in the positive x-y plane between (1,0) and (1,1), and expanded (i.e. by 8) anti-clockwise around the origin so that the fraction value \quad[\frac{0}{1}\ldots\frac{y}{x}\ldots\frac{1}{1}]\quad is represented as \quad[0\ldots\theta\ldots2\pi]\quad radians, i.e. in \quad[0^{\circ}\ldots360^{\circ}] (from 3 o’clock anticlockwise around to 3, in clock-speak).

Each new row of the Stern-Brocot tree is made from the previous one by putting a new fraction between each pair, adding together the numerators and denominators (i.e. top and bottom) to get the new values, i.e.

    \[\{\frac{0}{1},\frac{1}{1}\}\quad \{\frac{0}{1},\frac{1}{2},\frac{1}{1}\}\quad \{\frac{0}{1},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{1}{1}\}\quad \{\frac{0}{1},\frac{1}{4},\frac{1}{3},\frac{2}{5},\frac{1}{2},\frac{3}{5},\frac{2}{3},\frac{3}{4},\frac{1}{1}\}\]

Continuing on in this way, the set will contain every fraction between 0 and 1, and all in lowest terms! (i.e. never \frac{4}{8} or \frac{33}{99} etc)
It has many amazing properties and applications, but strangely was not described until the 19th C.

The big gap on the middle left hand side is the region of \frac{1}{2}, and then gaps clockwise around from there, each one smaller, are \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \frac{1}{6}, \frac{1}{7}, etc. The ones on the underside are, from left, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7} etc.

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