How far from the nearest reference

A fold needs something to align: a crease is specified by bringing existing features of the paper together, so before anything has been folded the only features are the four corners and the four edges. Everything else is built.

The natural way to measure that building is to count. One fold from the four corners and four edges produces twelve distinct fold lines, which cross at nine points. A second round produces ninety-two lines and five hundred and sixty-five points. The sheet decides which points exist counts them on rectangles of different proportions; cheap where it reaches counts how many arrive at each depth.

Counting is the wrong measure for one of the two things a folder wants, and it is worth saying which.

Two questions a reference answers #

A folder asks references for two different things.

Name a place. Fold the corner to the point one third along the edge is a request for a point with a description, and what matters is whether such a point exists and how it is reached. Counting answers this: more references means more nameable places, and the numbers a fold reaches is the account of which ones.

Crease here. A designer with a pattern in hand needs a fold at a particular place, decided by the design rather than by what happens to be constructible. What matters then is not how many references there are but whether one is near enough to the place the crease has to go.

The second question has an exact measure and it is not the count. It is the largest distance from anywhere on the sheet to the nearest reference — the radius of the biggest disc that can be placed on the paper without covering one.

The two numbers, side by side #

With no folds the sheet has four references — its corners — and the worst-covered point is the centre, at 0.707 of a sheet from the nearest.

With one fold it has nine references and twelve lines, and the worst-covered point is the centre of a quadrant, at 0.354.

With two folds it has five hundred and sixty-five references and ninety-two lines, and the worst-covered point is at 0.089.

So from no folds to two the count of references multiplies by a hundred and forty-one and the reach improves by a factor of eight. From one fold to two it is sixty-three times the points for four times the reach.

The quantities do not move together and they cannot: a disc of radius r placed anywhere with no reference in it needs at least as many points as would be required to cover the sheet with discs of that radius, which grows as one over r squared. Halving the covering radius therefore costs at least four times the points, and in practice much more, because the reachable points are not placed to cover anything.

What a covering radius is for, in one example #

Take a design that wants a crease three-sevenths of the way along an edge. A seventh is constructible — there is an exact procedure — so the naming question is answered and the count is not the obstacle.

Now take a design that wants a crease at 0.3712 of the way along, because that is where the optimiser put it. There is no procedure and no name. What matters is how far the nearest reference is, and at two folds the answer might be anything up to 0.089 of a sheet — thirteen millimetres on a hundred-and-fifty-millimetre square, which is a visible error in a finished model.

Three more rounds of folding would help, and this is where the count becomes actively misleading: the third round produces tens of thousands of lines, and the covering radius improves by rather less than the count suggests, and the references that arrive are mostly too close together to tell apart anyway.

Where the worst place is, and how it moves #

The worst-covered point is not where intuition puts it, and it moves.

With one fold it is at a quarter and a quarter — the middle of a quadrant, which is exactly where the diagonals and the midlines are furthest from anything. That is symmetric and unsurprising.

With two folds it is at 0.932 and 0.932: near a corner, not in the middle. The centre of the sheet, which was the worst place before any fold, has become one of the best covered — the second round of folds crosses there constantly — and the sparse region has migrated outward to a band near the corners where few of the new lines reach.

That reverses a result this collection has stated in other terms. Cheap where it reaches found that references arrive most readily on the edges of the sheet, because an edge is a line that was there before any fold was made — and the edge was there first sharpened it: of the points reachable in two folds, those on the sheet’s edge arrive a whole fold earlier on average than those inside it.

Both are about the edges. The worst-covered spot at two folds is near a corner, a little way in from both edges, which is the one part of the sheet that is close to two edges and reached by neither’s constructions.

What the number is, exactly #

The covering radius is computed on a grid and the grid is stated rather than hidden.

The true worst point of the plane is at most half a cell diagonal from the worst point of the grid, so the true radius lies between what is reported and that number plus half a cell diagonal. At the grid used the bracket is 0.089 to 0.092 at two folds and 0.354 to 0.357 at one.

The bracket is reported because a covering radius computed on a grid is exactly the sort of number that could be quietly wrong in either direction, and because a figure printing 0.0888 without saying what it was measured on is claiming four digits it does not have.

What the corner is doing #

The migration of the worst spot to a corner deserves following, because it says something about the shape of the sheet rather than about the constructions.

A corner is where two edges meet, and both edges are references from the start. So the corner itself is extremely well served — it is a reference, and every construction that names either edge passes near it. What is not well served is the region a little way in from both edges along the diagonal: far enough from each edge that folds along them do not reach, and not on the diagonal, which is the one line that does go there.

At one fold the diagonals and midlines are the only lines, so the sparse regions are the four quadrant centres, symmetrically. At two folds the new lines are mostly constructed from crossings that already lie on the diagonals and midlines, so they reinforce the middle and leave a band near each corner comparatively bare.

That is a consequence of the square in particular. The square is a choice, and a sheet with a different proportion has its lines and its gaps somewhere else — a fact worth knowing before quoting any of these numbers about paper that is not square.

What the corners are worth, re-read #

There is a result here that now needs qualifying, and the qualification is small but real.

The corner is worth four times the middle measures what a sheet’s corners are worth to a design — a corner of the paper reaches further into a base than the same area in the middle, because it is bounded by two edges and can become a long flap. That is about paper, not about references.

Cheap where it reaches measures what the edges are worth to construction — a reference on an edge arrives a fold earlier on average than one inside. That is about references, and it is not contradicted here: an edge reference is still cheap.

What the covering radius adds is that the region near a corner, an eighth of a sheet in from both edges, is the worst-served part of the paper at two folds. Cheap on the edges, well-covered in the middle, and a gap in between. None of the three statements is about the same quantity and all three have been stated as facts about corners.

Why the reachable points are so badly placed #

Five hundred and sixty-five points on a unit square, if spread evenly, would put a reference within about 0.03 of anywhere. They manage 0.089, which is three times worse.

The reason is that they are not points but crossings of lines, and lines through a square cross where the lines happen to concentrate. The constructions produce families of lines through the same few places — every fold through a corner, every bisector of the same pair of edges — so the crossings pile up along the diagonals and the midlines and thin out in between.

Closer than a crease is wide measured the other consequence of that piling up: references so close together that a folder cannot tell them apart, which at a hundred and fifty millimetres is a real limit rather than an abstraction. So the same clustering that makes the covering radius poor makes a large fraction of the points useless individually.

Both are the same fact, and stating either as the sheet has five hundred and sixty-five references is what hides it.

The number a folder could carry #

There is a version of this a folder could use directly, and it is the covering radius expressed in millimetres of a real sheet.

At a hundred and fifty millimetres square: no folds, the worst place is a hundred and six millimetres from the nearest reference. One fold, fifty-three. Two folds, thirteen. A sixteenth grid, four and a half.

Those are the numbers that say what a construction is worth to somebody holding paper, and they are the ones a count of five hundred and sixty-five never conveys. Thirteen millimetres is about the width of a thumb, and a crease placed by eye thirteen millimetres from the nearest thing to align to is not a crease anybody would call exact.

Exact is not accurate is the standing account of the difference between a construction being right and a fold being placed well. This is the same distinction one level up: a reference set can be exact everywhere it exists and still leave a thumb’s width of paper with nothing in it.

What a designer does instead #

The practical answer, and it is why exact construction and design have largely separated.

A designer needing a crease at an arbitrary place does not construct it. They fold to a grid — sixteenths or thirty-seconds of the sheet, obtained by repeated halving — and then design on the grid rather than on the paper. Designing on a grid is the whole of box pleating, and its point is precisely that a grid has a covering radius by construction: on a sixteenth grid the worst place is a cell centre, half a cell diagonal from the four points around it, which is $\sqrt2/32 = 0.044$ — half the axiom closure’s radius at two folds.

The trade is that a grid reaches nothing off it. Every crease is at a grid point or at forty-five degrees, and a design wanting a crease at a third of the way along has to be redesigned. That is a real constraint and box-pleated designs are shaped by it.

Two hundred and eighty-nine points against five hundred and sixty-five, and twice the coverage. That comparison is the argument for the grid stated in one line, and it is not an argument about how many places can be named — the axioms name far more, including places no grid ever reaches.

How far the points are from covering anything #

The clustering can be given a number, and the number says how much of the shortfall is the constructions’ fault rather than arithmetic’s.

Covering a unit square with discs of radius $r$ takes at least $1/(\pi r^{2})$ of them, since each covers that much area at most. Compare that floor with what each depth actually spends.

No folds: radius 0.707 needs at least one point, and the sheet carries four. One fold: radius 0.354 needs at least three, and the sheet carries nine. Two folds: radius 0.089 needs at least forty-one, and the sheet carries five hundred and sixty-five.

So the reachable set is about four times, three times and fourteen times more numerous than a perfect covering of the same radius would require.

The clustering gets worse, not better #

That sequence is the finding, and it is not the one more points would suggest.

A construction that spread its points evenly would sit near a constant multiple of the floor at every depth. This one starts at three or four times the floor and is fourteen times it after two folds — so each round of folding produces points that are less useful for coverage than the last round’s, which is what “the crossings pile up along the diagonals and the midlines” means when it is counted.

Extrapolating the trend rather than the count is what makes the essay’s refusal to compute depth three easy to accept. Even if the third round produced ten million points, at fourteen times the floor and worsening, the radius would improve by a factor somewhere near the square root of the usable fraction rather than of the count.

Which is the grid’s whole argument #

Now price the grid the same way. Two hundred and eighty-nine points at a covering radius of 0.044 need at least 163 by the area floor.

The grid is within a factor of 1.8 of a perfect covering. The axiom closure is fourteen times off it.

That is the argument for box pleating in one number, and it is a different argument from the one about counts. A grid is not chosen because it has more references — it has half as many — nor because it names more places, since it names far fewer. It is chosen because its points are placed to cover the sheet, and a construction whose points are crossings of lines through a few clusters cannot be.

Coverage is a property of arrangement, and the axioms optimise for reachability. Those are different objectives, and a set optimised for one lands fourteen times off the other.

What a third fold would buy #

The natural next question is what the ladder does past two folds, and it is a question this collection refuses to answer rather than answers badly.

The closure at depth three would be built from the ninety-two lines and five hundred and sixty-five points of depth two, and the number of alignments among those is in the tens of millions. The construction refuses past a stated cap and says which round it refused at, which is the right behaviour: a closure computed with an arbitrary subset of the available alignments is a number about the subset.

What can be said without computing it is a lower bound on how little it can help. Halving a covering radius requires at least four times the points, because discs of half the radius cover a quarter of the area each. Two folds bought a factor of four in radius for a factor of sixty-three in points, which is fifteen times worse than the bound — and the third round’s points will be no better placed than the second’s, since they are crossings of the same kinds of lines through the same clusters.

So a third fold is not going to change the picture, and the picture is that exact construction gives a folder places to fold rather than a way of folding anywhere.

Which measure to quote #

Neither, alone, and the pairing is the point.

Count is the right measure for what a construction can name: which lengths are reachable, which polygons are constructible, what folding buys over the compass. Every one of those questions is about the existence of a particular point, and coverage is irrelevant to it.

Covering radius is the right measure for what a fold is worth to a design: how close a crease can be put to where it has to go. It is the number a grid is chosen for, and it is the number that explains why designers use grids despite grids reaching so much less.

Quoting the first when the second is meant is how a set of five hundred and sixty-five references comes to sound like a fine-grained coordinate system for the sheet, which it is not.