Ratios and Geometry

Rational Sky

One Map for Euclid's Orchard, Farey Rows, Ford Circles, and Stern-Brocot Paths

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Orchard

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Continued fractions are a valid way to design good rational gear ratios; the resulting gear train then realizes the chosen fixed ratio.
Visible lattice points from the origin correspond to coprime integer pairs.

Fun Fact

Brocot

Achille Brocot was a clockmaker. The Stern-Brocot tree has real gear-ratio history, not just abstract number theory.

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