Map of Math · PG-01 · explainer · all explainers
Seven pictures that build the carpet experiments from nothing. Every number on this page is worked out in your browser while you watch. None of it is copied from the lab.
Take a 9 × 9 grid and remove some cells. The outer ring always stays. That is the pattern. Now treat every cell that is left as a small copy of the whole grid and remove the same cells inside it. Then do it again.
Two patterns run through the whole programme. They are called T and X.
Dark is kept, blank is removed. Three rounds give a grid 729 cells across. The lab goes one round further, to 6,561.
Neighbouring cells are joined by links. Give every link a random number between 0 and 1, and open the links whose number is below the dial. Turn the dial up and islands of joined cells grow.
At some setting the left side first reaches the right. One link completes it. That setting is this draw's crossing point.
Each fresh set of random numbers crosses at a different setting. Draw many and they pile up. Three things describe the pile: where its middle is, how wide it is, and which way it leans.
On a bigger carpet the pile is narrower. Each time the side grows nine times, the width shrinks by a steady factor, and the factor is different for T and X. The lab calls it the sharpening rate.
| size | draws | middle | width | lean |
|---|
Width is the span holding the middle 60% of draws. Lean is the skewness: above zero means a longer tail toward late crossings. Its ± is large at a few hundred draws, which is why the lab draws 8,000 per carpet before reading it.
For a single 9 × 9 block there is one curve: the chance it is crossed, at each dial setting. A cheap way to imitate a bigger carpet is to pretend every link is itself a whole block. That means feeding the curve into itself.
Each feeding makes the curve steeper, and all of them pass through one fixed setting. Now overlay the real carpet at the matching size and compare where each one steepens.
| size | shortcut middle | real middle | shortcut width | real width |
|---|
At 9 × 9 the two agree because they are the same thing. After that the shortcut's middle climbs toward its fixed setting while the real carpet's stays lower. The real columns use the draws from Fig 3 and sharpen as that figure fills.
The shortcut treats a block as a switch: crossed from side to side, or not. A real route does not respect that. It enters a block, turns, and leaves by the top. Or it dips in and backs out the way it came.
Here is one route at the moment of crossing, with each 9 × 9 block marked by how the route uses it.
Between the shortcut and the real carpet there is a middle option. Keep the real blocks, but let neighbours touch at three chosen rows per side instead of all nine. Then everything a block needs to tell its neighbour passes through three points, and the bookkeeping becomes exact again.
Which three rows? It matters where a contact lands. On T the middle row opens into the block. On X the middle row meets a wall and has to walk along the edge first.
Shortest route, in links, from each left contact to the nearest right contact inside one block.
Two neighbouring blocks and the seam between them. With the same random numbers, three contacts can only cross later than all nine: every route that exists with three also exists with nine.
Two links in a row need both open: the chance is p × p. Two links side by side need either: 1 − (1 − p)². Swap open for closed and the two curves trade places.
Patterns with a two-cell bottleneck turn out to be exactly two mirror-image channels side by side, so their curve is the second formula applied to one channel's curve.
The lab goes one size further, to 6,561 cells across and 16.8 million cells, with 8,000 draws per carpet. At that size the lean of T's pile turns back down, which the shortcut never does at any size. That turn is still unexplained.
The contact-point idea of Fig 6 is the lab's ladder. Run 094 asked whether moving the outer contacts from rows 1 and 7 to rows 2 and 6 changes which pattern sharpens faster. It does not: with either set of rows the ladder puts T and X in the opposite order to the real carpet.