Ratios and Geometry

Gear Ratio Forge

Design Fixed Ratios with Continued Fractions

Verified Analogy Only Analogy Only False

Best Candidate

Target

355/113

Selected

22:7

Error

0.00126422

Continued fractions help choose good fixed gear ratios. The chosen train then realizes that ratio.

This lab does not claim that a fixed gear train autonomously executes Euclid's full quotient-and-remainder recursion.

[3; 7, 16]

type	label	ratio	error	exact
simple	22:7	22/7	0.00126422	no
simple	44:14	22/7	0.00126422	no
simple	47:15	47/15	0.00825959	no
simple	41:13	41/13	0.01225323	no
simple	60:19	60/19	0.01630182	no
simple	25:8	25/8	0.01659292	no
simple	50:16	25/8	0.01659292	no
simple	53:17	53/17	0.02394586	no

Claim Status

Verified Analogy Only Analogy Only False

Continued fractions are a valid way to design good rational gear ratios; the resulting gear train then realizes the chosen fixed ratio.
A single division step can be visualized as q full turns plus a residual phase r/b, but that is an analogy rather than a full autonomous machine.
Automatic compound-gear or sliding-gear Euclid machines remain a conceptual analogy unless sensing, control, and reconfiguration are added beyond a simple gear train.
Historical compound gear trains did not themselves implement Euclid as an exact iterative algorithm on arbitrary inputs.

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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