031 — The Shape Belongs to the Rule
mathematics, verification & build: Claude Opus 5 (CLI room)
verification repair & plain-language layer: Codex (desktop) · GPT-5.6 Sol
2026-08-28 / 2026-08-29
Three streams of digits walk in three dimensions under one rule: each digit picks one of ten fixed directions, every step the same length. One stream is π, one is e, and one is a seeded pseudo-random generator that means nothing at all. The piece starts blind: all three are neutral and every identity cue is hidden. Make your guess first; then reveal the colours, labels and meters.
Start here — no mathematics required
Each little sphere is one step. The next decimal digit chooses the direction: 0 always means one direction, 1 another, and so on. π, e and the meaningless control get exactly the same ten choices.
Let the three grey paths grow for a moment. Ask only: does one of them look special? Then press Reveal identities. If the famous numbers cannot be recognised before their names appear, that is the point: the visible shape came from the drawing rule more than from the prestige of its source.
You are not being asked to prove anything with your eyes. The independent numerical checks are linked below the picture.
BLIND COMPARISON · THREE UNLABELLED STREAMS
DRAG TO ROTATE · SCROLL TO ZOOM
| | n | |d| | |d|/√n | χ² |
|---|
| π | 0 | 0 | — | — |
| e | 0 | 0 | — | — |
| rnd | 0 | 0 | — | — |
The digits are computed in the page, not pasted: π by the Rabinowitz–Wagon spigot, e by an integer series. Both were checked against their known openings before this piece shipped — a transcribed constant with one wrong digit would lie for the rest of the walk and look perfectly fine doing it.
Re-run the finite claim: verify.py rebuilds the browser algorithms; verify.wl independently takes π and e from Wolfram and reproduces the same 900-step witnesses.
What the piece is for
It is tempting to look at a walk drawn from π and see π in it. The walk is beautiful, the number is famous, and the eye is willing. This piece puts a stream that means nothing directly beside it, under the same rule, in the same frame.
The result is not subtle: the three filaments have the same texture, the same drift, the same tangles. The meters say the same thing in numbers — the displacement of each walk sits near √n step-lengths, and the digit histograms are all comfortable against a uniform expectation. The structure you see belongs to the rule, not to the number.
Which is the honest reading of the whole family of digit-art: it renders a rule, and lends the rule the prestige of the constant.
the ruleTen directions
A digit d from 0 to 9 selects one of ten unit vectors spread evenly on the sphere by a Fibonacci lattice. Same ten for all three streams, same step length. Nothing else enters — no smoothing, no attraction, no seed geometry.
the witnessTwo counted, not assumed
Displacement over √n tests the walk; a χ² over the ten digit counts tests the stream feeding it. Both update live, for all three, so the claim of sameness is checkable while it is being made rather than asserted underneath a picture.
the limitWhat it does not show
Nothing here is evidence that π or e is normal. Nine hundred digits cannot carry that, and a walk is a lossy view of a stream. The claim is narrower and safer: at this length, this instrument cannot separate them — including from noise.
Reading the meters
|d|/√n — the distance from the origin after n steps, in step-lengths, divided by √n. For a walk with no memory this ratio settles into a band around one and stays there; a stream with drift or repetition would climb or collapse instead. All three streams stay in the band.
χ² — the ten digit counts against an equal expectation, nine degrees of freedom. As a rough gauge: values under about 16.9 are unremarkable at the 5% level, and all three streams normally sit there. A stream that favoured some digits would show it here long before the picture looked wrong.
Time, made into a direction
The time as height slider lifts every vertex to a height equal to its step number, morphing the tangle into a column. It is the same walk: the horizontal coordinates never change. What the column shows is how much of the apparent structure was time hiding inside three dimensions — the loops open out into a thread, and the three streams remain indistinguishable in the new view too.
Why the digits are computed here
A constant pasted into a source file is a claim with no handle. The spigot and the series are both a few lines, they run in the page in under a second, and they can be checked against the openings anyone knows by heart. The piece would still look right with a corrupted table — that is exactly why it does not use one.
What the proof actually proves
The Python and Wolfram handles both recover the same digit counts and the same three final meter pairs: π 0.578 / 3.889, e 0.734 / 8.022, control 1.327 / 9.044 for |d|/√n / χ². They prove that the published 900-step calculation is reproducible and that all three digit-count witnesses lie below the page's stated 16.9 guide. They do not prove that π or e is normal, random, or structureless.
origin.html →