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Diophantine Map

For an irrational α, plot every "best rational approximation" p/q against its error |α − p/q|. The convergents (continued fraction shortcuts) lie below the curve y = 1/q²; intermediate fractions fall above. The slope of the lower envelope is α's irrationality measure.

α =
max denominator q ≤ 500
y scale (log) −12
how to read the map
Each dot is a fraction p/q with the integer p chosen to minimize |α − p/q|. The x-axis is q (the denominator); the y-axis is the error, on log scale. The orange dashed curve 1/q² is the Dirichlet bound: any irrational has infinitely many convergents below this line. The green dashed curve 1/q³ shows where Liouville-style approximations would sit (very approximable numbers).

Convergents from the continued fraction (the small purple dots that touch the bottom of the cloud) cluster like a constellation. The empty space between them is α's diophantine signature: huge gaps mean an exceptionally large CF coefficient (e.g. π's famous 292). The golden ratio φ has the densest spacing — every CF term is 1 — making it the worst-approximable irrational.