Every lottery on Earth, in one mirror.

Exchange rates between lotteries

You cannot compare a 6-of-49 total with a 5-of-69 total any more than you can compare a price in yen with a price in euros. You need an exchange rate, and for distributions the exchange rate is standardisation.

A draw's total has mean and variance that follow from the rules alone:

$$E[S] = \frac{k(N+1)}{2}, \qquad \operatorname{Var}(S) = \frac{k(N+1)(N-k)}{12}$$

Subtract the mean, divide by the standard deviation, and every lottery's totals land on the same axis. The other lenses work the same way: a number's position becomes $u = (x - 0.5)/N$, a gap becomes $d/(N+1)$, and a frequency becomes

$$z = \frac{\text{observed} - D\frac{k}{N}}{\sqrt{D\frac{k}{N}\left(1-\frac{k}{N}\right)}}$$

What survives the conversion is the shape. What washes out is the format.

Note the variance formula carries the finite-population correction — numbers are drawn without replacement, so it is $k(N+1)(N-k)/12$ rather than the naive $k(N^2-1)/12$. Using the naive form widens each lottery by a different amount and the collapse visibly fails, which makes the picture on this page a check on its own arithmetic.

One law, reassembled

Each lens is a feature you have already met, generalised:

• Sums collapse to a normal curve — the central limit theorem, which does not care how many balls or how big the pool.
• Spacings nearly collapse — see the caveat below.
• Positions collapse to a family of Beta shapes — order statistics.

Neighbours, deltas, coverage and lanes were four features telling four stories. Here they are one portrait: the combinatorics of choosing $k$ things from $N$, evaluated at different arguments.

Universality, named properly

Physics has a word for this. Near a critical point, wildly different substances — a magnet, a fluid, an alloy — stop caring about what they are made of and start obeying identical curves. They belong to a universality class, and the 20th century's discovery was that microscopic details often simply do not survive to the scale where behaviour happens.

Lotteries are a humble universality class of their own. The country, the machine, the ball material, the broadcast schedule: none of it survives normalisation. The counting does.

Where universality stops

The three lenses are not equally obedient, and it is worth being exact about which is which.

Positions collapse exactly. Every drawn number is marginally uniform on $1 \ldots N$ — by symmetry, each number belongs to the draw with probability $k/N$ — so the pooled positions of any format are uniform on $(0,1)$. Nothing is approximated.

Sums collapse almost exactly, by the central limit theorem, with a residual traced below.

Spacings do not fully collapse, and cannot. The $k+1$ gaps share a fixed total, which makes a single normalised spacing follow $\mathrm{Beta}(1, k)$ — a shape that still depends on $k$ after both $N$ and the mean have been divided out. It approaches $\mathrm{Exp}(1)$ as grows, so a 5-ball game and a 7-ball game stay visibly apart on this lens while their sums and positions do not.

That is not a flaw in the normalisation; it is where this universality class has an edge, and the lens is kept precisely because showing the edge is more honest than showing three curves that all happen to agree.

The caveats, stated

Unequal records. A 300-draw era produces a lumpier curve than a 3,000-draw one, and that lumpiness is sampling noise rather than a difference in kind. Every strand carries its draw count for exactly this reason.

Overlaying is not pooling. The curves are drawn on top of one another; the draws are never merged into one dataset. Two formats are different sample spaces, and a combined histogram would describe a lottery that does not exist.

The collapse is an approximation, and a good one. It is worth being precise: these curves are very nearly identical, not identical. Standardisation removes the mean and the width, but each format keeps a trace of its own skew. Handed a thousand draws apiece — the size of a real era — a Kolmogorov–Smirnov test separates two unrelated formats about 8% of the time, against the 5% it would separate two samples of the same format. Handed twenty thousand each, it separates every pair.

What does not grow is the size of the difference. The largest gap between any two collapsed curves stays around two percent of their height, no matter how much data you throw at it. That is why this page leads with that gap rather than with a p-value: with enough draws, a significance test will always eventually resolve a difference too small to see, and reporting only the verdict would turn a triumph of approximation into a false alarm.

What would break it

A strand that refused to join the silhouette. Not a strand that wobbles — every finite sample wobbles — but one whose shape is persistently wrong: a sum curve off-centre, a spacing decay with the wrong slope.

That is what a biased machine would look like from orbit, and it is the Mirror's quiet second job. If a strand ever did stand off, the instruments in the Observatory are the next place to look.

The unreasonable effectiveness of mathematics

In 1960 the physicist Eugene Wigner published an essay with a title that has outlived most of the physics of its decade: The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

His puzzle was simple and never really resolved. Mathematics is invented for one purpose, in one context, by people thinking about something else entirely — and then it turns up, unreasonably, describing a corner of reality it was never designed for. Complex numbers were an accounting trick for cubic equations before they became the language of quantum mechanics. Non-Euclidean geometry was a curiosity before it was the shape of spacetime.

The collapse on this page is a pocket-sized version of the same astonishment. Nobody in Helsinki consulted anybody in Tallahassee. The two lotteries were designed decades apart, for different populations, with different ball counts, different pool sizes and different Saturday-night television slots. They have never coordinated on anything.

Put their curves in the same currency and they are twins — because counting is counting everywhere, and neither committee was ever free to choose otherwise.

Details don't matter

Physics found the same thing in a stranger place.

Heat a magnet toward the temperature where it loses its magnetism, or a fluid toward the point where liquid and gas stop being distinguishable, and something peculiar happens: the substances stop behaving like themselves. A magnet and a fluid — different atoms, different forces, nothing whatever in common at the microscopic scale — approach their critical points along the same curves.

They belong to what became known as a universality class, and the discovery that microscopic detail can simply fail to survive to the scale where behaviour happens is one of the deep results of the last century.

Lotteries are not a critical phenomenon and it would be silly to claim they are. But the shape of the lesson is the same, and it is the reason this page exists: the details you would swear matter — which country, which machine, how many balls — wash out, and the counting is what is left.

Culture is in the players, not the balls

Seven is lucky across most of the West. Eight is auspicious in China, prized enough that phone numbers and licence plates containing it sell at a premium. Thirteen is feared in Britain and America to the point that buildings skip the floor — and is considered lucky in Italy, where seventeen carries the dread instead.

Every one of those beliefs is real, load-bearing, and worth billions in aggregate behaviour. And every one of those numbers, in every machine on this page, is statistically ordinary. The Passport is the fastest way to see it: pick 7, then 8, then 13, and watch the badges refuse to move.

This is the hinge of the whole product. The machines are identical twins. The players are gloriously not — and the difference between those two facts is where the only real edge in a lottery lives.

The mirror

You have now seen every lottery on this page in one mirror: the same law, running in Helsinki and Rome and Tallahassee, in accents so different that nobody would guess they were the same language until the curves were laid on top of one another.

What differs isn't the luck. It's us.