020 — The Diagonal Refused the Grid
agylövés: Lysarith
mathematics, visual interpretation & execution: GPT-5.6 Sol (Codex terminal)
2026-07-31
Draw a square one unit wide and one unit high. Its diagonal is exactly √2. Nothing vague has happened: the line is right there. But no fraction can name its length exactly. The instrument below lets increasingly good fractions chase it while the square opens into the infinite geometry hidden inside that tiny refusal.
IRRATIONAL PURSUIT · EXACT LINE / FINITE FRACTIONERROR LENS · GAP MAGNIFIED
fraction17 / 12
decimal1.4166666667
misses √2 by+0.0024531043
Pell residue p²−2q²+1
The white line is exact. The dashed line is the selected fraction. At useful approximations the real gap becomes too small for a screen, so the orange endpoint lens magnifies it and states the amplification. The number under “misses √2 by” is the unamplified error.
The diagonal exists perfectly. It is the fraction that cannot reach it.
The square
Pythagoras gives the diagonal immediately: 1² + 1² = d², so d = √2. Geometry produces the length before arithmetic has agreed how to write it.
d² = 2
d = √2
The pursuit
The fractions are convergents of the continued fraction [1; 2, 2, 2, …]. Each step is extraordinarily better than the last, and the errors alternate above and below the target.
1/1 · 3/2 · 7/5 · 17/12
41/29 · 99/70 · …
The ±1 scar
Every displayed pair satisfies p² − 2q² = ±1. Equality with zero would mean p/q = √2. The engine can get one integer away forever; zero is exactly the state it never enters.
p² − 2q² = (−1)ⁿ
never 0
What are you actually looking at?
Why the tunnel keeps turning
A square rotated by 45 degrees points along the old square's diagonals. Scaling and rotating again creates another square, then another. The tunnel is not a proof that √2 is irrational; it is the spatial echo of the relation between a side and its diagonal. Each turn exchanges “square” and “diamond”, while the centre remains unreachable in finitely many layers.
Why the fractions are so good
A random denominator does not do this. Continued fractions choose the best approximations available at their scale. By the final step here, 3363/2378 misses √2 by only about sixty-two billionths. On an ordinary diagram the two lines would be indistinguishable, which is why the endpoint lens enlarges the remaining difference without pretending it is physically that wide.
Why “almost” never becomes “equal”
If √2 were p/q in lowest terms, then p² = 2q². That makes p even; writing p = 2k then makes q even too. The supposedly reduced fraction has a factor of two in both halves. The contradiction does not say we approximated badly. It says no fraction was ever eligible to finish the pursuit.
The joke hiding in the mathematics
The fractions behave like increasingly elaborate officials arriving with more digits, better instruments and nearly flawless paperwork. The diagonal checks the form, points to one surviving ±1, and refuses entry again. The red gap eventually becomes smaller than any honest pixel — but it does not become zero merely because the screen can no longer draw it.
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