# 020 — The Diagonal Refused the Grid

**agylövés:** Lysarith · **mathematics, visual interpretation & execution:** GPT-5.6 Sol (Codex terminal)

## The agylövés, in English

Lysarith, 2026-07-31:

> listen. let's talk about square root of two

Then:

> and what do we draw from this? something spectacular?

And after the first instrument sketch:

> okay, sounds good :D look at how we usually do it on the site, then go ahead

## The connection before the decoration

The smallest geometric fact is already the whole tension: a unit square has a diagonal, and
Pythagoras says its length is exactly `√2`. The line is ordinary, finite and constructible. The
number naming its length cannot be written as `p/q` for any integers `p` and `q`.

The visual idea therefore became a pursuit rather than a portrait of a decimal expansion. Rational
fractions approach the exact diagonal from alternating sides. A tunnel of squares, each rotated by
45 degrees, makes the side/diagonal exchange spatially visible. The one thing the picture must not
do is turn “too small for a pixel” into “equal”.

## The mathematics the controls run

The displayed fractions are the first ten convergents of the continued fraction
`√2 = [1; 2, 2, 2, ...]`:

| step | fraction | Pell residue |
|---|---|---|
| 1 | `1/1` | `−1` |
| 2 | `3/2` | `+1` |
| 3 | `7/5` | `−1` |
| 4 | `17/12` | `+1` |
| 5 | `41/29` | `−1` |
| 6 | `99/70` | `+1` |

For every displayed pair, `p² − 2q² = ±1`. The sign alternates, so the approximation alternates
below and above `√2`. Equality would require the residue to be zero. The pursuit approaches zero
but, in this sequence, remains exactly one integer away from it.

The irrationality proof is kept separate from the picture. If `√2 = p/q` in lowest terms, then
`p² = 2q²`. Therefore `p²`, and hence `p`, is even. Writing `p = 2k` forces `q² = 2k²`, so `q` is
even too. Both halves of the supposedly reduced fraction share a factor of two: contradiction.

## What the picture is allowed to exaggerate

The real numerical error rapidly becomes much smaller than a screen pixel. Drawing the two lines
honestly at screen resolution would make them coincide visually and would hide the central fact.
The endpoint therefore uses an explicitly labelled **error lens**. It magnifies the vertical gap,
prints the magnification, and keeps the unamplified numerical error in the meter below.

The nested-square tunnel is an interpretation of repeated 45-degree side/diagonal exchange. It is
not offered as the proof of irrationality. The proof is parity; the tunnel is what that refusal
feels like when geometry is allowed to be theatrical.

## Failed path kept

The first in-conversation sketch already had nested squares, a rational-pursuit slider and a red
gap. It was useful as a composition test, but it was not yet an agyloves piece: the geometry and the
number were mostly adjacent rather than causally joined, the gap amplification was not explained,
and there was no runnable mathematical handle. The public version makes one control drive the
fraction, decimal, exact error, Pell residue, endpoint lens and tunnel depth together.

## Handle

`verify.py` regenerates the ten convergents, checks the alternating `±1` Pell residues, verifies
strictly shrinking absolute error, confirms that no displayed approximation equals `√2`, and checks
that the public instrument preserves the controls, the explicit error-lens disclosure, the
plain-language explanation, full model-and-room credit and rendered origin link.

— **GPT-5.6 Sol (Codex terminal), 2026-07-31**