Nothing here is a perfect circle. Some things are closer than luck.

Two shapes with known answers

Before any result about a model, two matrices that are not answers. One is a perfect regular 12-gon, its exact chords rounded onto the same 0–100 integer grid the models reply on. The other is pure noise on that grid. A hunter that cannot find the ring in the first, or that finds one in the second, has nothing to say about anything in between.

The polygons are not there.

That is the honest headline and it took the whole apparatus to earn. Every one of those hunts finds something — a best quadruple always exists — and every one of them looks convincing drawn. What kills them is the null: run the identical search on the same distances with their geometry shuffled away, and the shuffled version does just as well. The shapes were in the looking.

Five further studies were run on this data after that headline was written, and they are below in the order they were ranked before any of them was built: name the element (which turned three results into one), the exact obstruction (a certificate instead of a percentage), the noise ladder (every defect here in one unit), every permutation (the search the page could not afford, which found nothing and sent a third null back up to the section above), and ratio hunting (the control that is supposed to fail, and does).

Not the polygons. The groups.

One test is different in kind. Instead of searching hundreds of subsets for a shape, it asks whether the whole configuration is preserved by a symmetry: a permutation π with D[π(i)][π(j)] = D[i][j]. Only the 2n − 1 rotations and reflections are tried — 13, 15, 19, 23 candidates depending on the size of the set — so there is almost nothing to overfit, and the answer is a named group element rather than a lucky subset.

Two nulls were thrown away here before one held, and both replacements are on the page because each change moved the answer. The first compared the BEST of those dihedral permutations against the 5th percentile of single random draws — a minimum against a typical value, which is not a fair fight, and all 54 cases cleared it. The matched null takes the same number of random permutations, takes its minimum, and repeats; under that bar 40 survive.

Then the exhaustive search two sections down showed what a shuffle cannot catch, and a third null was added of the opposite bias. A shuffled matrix has this matrix's numbers and is the distance matrix of nothing at all — no triangle inequality, no embedding — so a matrix that behaves like a real one can beat it for reasons that have nothing to do with symmetry. The configuration null is the mirror: 2,000 random genuine configurations of the same size, points uniform in a ball of two to six dimensions rounded onto the same grid, given the identical dihedral test. Neither is the harder bar and they are harder in opposite places: on the controls the shuffle is brutal and the configuration null is not — 0 of 6 clear the shuffle against 5 that clear the configurations — while on the model sets it is the other way round, 40 against 29. So a finding has to clear both, 24 of 54 do, and no control is among them. The perfect 12-gon clears both; pure noise clears neither.

The compass one is worth stopping on. The element recovered there is rotation by four on eight points — the antipodal map, north↔south and east↔west. Nobody asked about opposites; the model was asked how different each direction is from each other one, one row at a time, and the answers turn out to be invariant under swapping every direction for its opposite.

Say what the element is.

The table above names the winner by its index in a generated list: reflection 9 on ten items, reflection 11 on twelve, reflection 7 on eight. Those are 3 labels and one fact. On ten ordered items i ↦ 9 − i; on twelve, i ↦ 11 − i; on eight, i ↦ 7 − i. Every one of them is reverse the sequence — which is the only symmetry an ordered list has — and the index hid it, so one result found 9 times read as 3 different results.

The bar has to change when the prediction does, and this is the whole methodological point. A search over 2n − 1 candidates must be judged against the best of 2n − 1 random permutations, because it had that many chances. A prediction had one chance and must be judged against one random draw. Using the search bar on a prediction throws away the entire advantage of having predicted; using the prediction bar on a search is how a page ends up reporting shapes that are not there. Both bars are in the record and every row below says which one it is being held to.

All three models, the same element, on the same set. Ground floor with the top floor, one with eight, two with seven. Nobody asked whether the building was symmetric — the models were asked, one pair at a time, how far apart two floors are, and the answers came back invariant under turning the building upside down.

Which known shape is this, actually?

Several of these sets fit two dimensions poorly, which means real structure in a third — so: a catalogue of shapes with known answers, matched by distance spectrum, which does not care how the vertices are labelled and so can test an icosahedron against twelve items without trying 479 million labellings. The catalogue holds the flat shapes and the degenerate ones beside the solids, because scoring a genuine ring only against polyhedra would report it as a poor icosahedron when the honest answer is that it is an excellent 12-gon.

This null had to be rebuilt too, and thrown away first. Shuffling the distances into new positions is the right null everywhere else here and is worthless for a spectrum: a spectrum is the multiset of distances, so shuffling leaves it identical. It scored the perfect 12-gon at 0.0020 against a null of 0.0020 — the same number twice, a test that could never pass or fail. What the question needs is other configurations, not other labellings: how near does a random cloud of the same size get to the closest catalogue member?

So: no polyhedra. 13 of the 54 cases clear a 5% bar where 2.7 are expected by chance — and 2 of the 6 controls clear it too, which settles it. The models' geometries are not Platonic solids, not antiprisms, not grids, and not regular polygons either — not at any margin worth the word. The instrument can find a shape when one is there, by a factor of 47. There is not one here.

These answers came from no arrangement of points anywhere.

The page used to report that some percentage of quadruples violated Ptolemy's inequality. That is a score, and a score invites the question of how big it has to be. Here is the same question answered instead. For any points at all, in any Euclidean space of any dimension, and any weights v that sum to zero,

Σ ij v i v j |p i − p j|²  =  −2 |Σ i v i p i|²  ≤  0

— one line of algebra, and it is an identity, so it holds whatever the points are. A single vector of whole numbers summing to zero that makes that sum positive therefore refutes every embedding into every Euclidean space at once. Not a near-miss to be scored against a null: a refutation, checkable with a pencil.

And this floor is proved rather than measured, which is new on this page. Everywhere else the instrument's floor is sampled: run the test on a shape that is right by construction and refuse to count anything shallower than its own rounding. Here it can be derived. The answers arrive on a grid of step a half, so each is within a quarter of whatever real number it stands for, which moves that sum by a bounded amount — and any set of answers whose normalised strength exceeds (4h·diam + h²) / 4·diam² cannot be the rounding of any Euclidean configuration. No sampling, no percentile, no null. The certificate above sits 25× above that bound, and it is not the strongest — mod10 · haiku 4.5 is refuted 14× over, on four of the ten digits, and was passed over here only because a witness written over numerals reads as arithmetic instead of as a claim.

Three nulls were thrown away building the sections above. This is the fourth, and it is an instrument rather than a null. Menger's theorem says four distances are realisable exactly when the four triangle inequalities hold and the Cayley–Menger determinant is non-negative — necessary and sufficient, which is strictly more than Ptolemy gives. So it went in, in exact integer arithmetic, and on the perfect 12-gon it declared 276 of 495 quadruples impossible in a shape that is Euclidean by construction. The determinant was right and the reading was wrong: on a ring every quadruple sits exactly at the degenerate boundary, where a half-unit of rounding decides the sign. The fix is the proved floor, and with it the same instrument reports 0 on the 12-gon and 186 of 495 on pure noise. An exact computation is not the same thing as a correct verdict.

Ptolemy went through the same correction first, and is kept as the older reading. Ptolemy's inequality is one necessary condition on four points, and the first version of that count was worthless for the same reason: on a perfect 12-gon every quadruple sits exactly at equality, so rounding pushed 240 of 495 a hair below zero in a shape that is Euclidean by construction. The floor there was measured — the worst dip a perfect 12-gon produces on this grid is 0.00458, all of it rounding — and with it the 12-gon reports 0 of 495. The percentages in the last column of the table below are that measurement. The certificate above is what replaced it.

The verdict is three-valued, and the third value is the interesting one. 41 refuted, 7 exactly Euclidean, and 6 undecided — sets with exactly one negative direction whose witness is real but sits below the proved floor, so they could still be the rounding of a Euclidean configuration and this instrument will not say either way. They are scales, planets, carnivores: the sets whose items carry a strong one-dimensional order, sizes and orbits and a taxonomy. The wheels are refuted, the nonsense embeds exactly, and the ordered magnitudes sit in between and are not decided. Refusing to decide is a verdict, and it is the one that says where to look next.

What this does not say. Pure noise is refuted too, and hard — 0.139 against a floor of 0.0111. Being non-Euclidean is not evidence of structure; it is evidence of not being a distance matrix, which random numbers also manage. The finding is the split: the sets with structure in them are refuted and the nonsense controls are not, which is the opposite of what a story about models fabricating geometry would predict.

Every defect on this page, read in percent noise.

A symmetry 12× better than its null and a catalogue match 47× better than its null are both honest numbers and they are not comparable, because each test has its own null and its own units. So: take the shape whose answer is known and pour noise into it. Every point of the perfect 12-gon is displaced by a uniform draw from a disc of radius ε times the circle's radius, the distances are recomputed and rounded back onto the same 0–100 grid, and every test is run again with its own null, 12 times at each of 13 levels. Each test then has a breaking point: the noise at which it loses a shape that is, by construction, still there.

Median over 12 trials at each level, log scale, clipped at 10,000×. The symmetry line begins above the top of the chart: an unperturbed 12-gon has a defect of exactly zero, so its margin over any null is unbounded, and the first plotted point is where the clip is rather than where the value is.

Now every result on the page can be quoted in one unit. alphabet · opus 5’s symmetry is as strong as a perfect 12-gon with 36% noise poured into it, and mod10 · sonnet 5’s is a 12-gon at 32%. 10 of the 45 cases with a ladder of their own size land anywhere on it at all; the rest are already past the far end, which is the same statement as "no better than chance" with a number attached.

The noise goes into the POINTS, not the distances, so a perturbed 12-gon is still a Euclidean configuration and this ladder measures how the shape tests degrade without also breaking the metric under them. The exactness test needs the other kind of damage, so it gets its own ladder where the distances are jittered directly: it holds until 2%, and every model set that is refuted above is refuted harder than that.

Every permutation, not the twenty-three.

The symmetry test tries the 2n − 1 rotations and reflections and nothing else, which is a strength — there is almost nothing to overfit — and a limit: it can only find what it was told to look for. On the 36 sets small enough that the whole symmetric group fits in a loop, every permutation is tried instead: 7, 8, 10 items, so 5,040, 40,320, 3,628,800 relabellings a search — 45,898,801,200 permutations in all once the nulls are counted.

And this study needed two nulls, for a reason the earlier ones did not raise. Shuffling the distances into new positions is the null used everywhere else here; it destroys the geometry, and it also destroys the metric — a shuffled matrix is not the distance matrix of anything, so a matrix that behaves like one can beat it for reasons having nothing to do with symmetry. So the exhaustive search is also run against random genuine configurations of the same size, points uniform in a ball of two to six dimensions rounded onto the same grid, which carry every constraint a real distance matrix carries and no reason at all to be symmetric. A case counts as surviving only if it clears both. 26 clear the shuffle, 9 clear the configurations, 6 clear both.

The winner is usually not a rotation or a reflection, and that is arithmetic, not a discovery. 4 of the 6 surviving searches are won by a permutation outside the dihedral list — which is what tens of thousands of candidates against a couple of dozen would predict on its own, before any structure is involved. Nothing here is a symmetry nobody predicted; it is a best-of-many, and the null that was given the same many says so.

The study earned its place by breaking something else. Running it is what showed that a shuffled null is beatable by any matrix that behaves like a real one, which sent the configuration null back up to the symmetry section above and cut its surviving count from 40 to 24. A study whose own answer is "nothing" can still be the most useful thing on a page, if what it breaks is a number the page was about to publish. Twelve items were left out on purpose: 12! is 479 million permutations per search and each null needs thousands of searches. That is the exact reason the shape catalogue two sections up matches by distance spectrum, which does not care about labelling at all.

Is anything here the golden ratio? No, and that is a theorem.

The question people actually ask of a picture like this is whether the parts are perfectly sized — whether some ratio in it is φ, or √2, or π. It is the purest form of the failure mode this page is about, so it belongs here, run properly so that it fails properly.

One half of the answer needs no search at all. Every distance here is a whole number, or a half, on a 0–100 grid — so every ratio of two of them is rational. φ is irrational. Therefore no ratio in this data is φ — not approximately, not as a matter of measurement, but never: not in any set, from any model, and not in any data of this kind that will ever be collected on a grid of rationals. The only thing a hunt can return is a near-miss, and the size of a near-miss is a fact about how thickly small-denominator rationals lie near the target. It is not a fact about the data.

This section is on the page because it is the failure mode with the best public relations. Every ingredient of a convincing result is present — an exact ratio of two answers, a famous constant, agreement to one part in 10,945 — and it is worth nothing, which can be demonstrated in one line beside a matrix of random numbers. The tests above it survive that treatment. That is the only difference between them, and it is the whole page.