A paper limits spacing, not density

The density a paper allows set a physical limit beside the crease densities these essays measure. A crease is a band a few sheet thicknesses wide, so two creases closer than a band’s width are not two creases, and a sheet of copier paper, with creases about 0.6 mm wide, carries at most 1,667 metres of crease a square metre. The densest printed pattern asks for 89, and solving the Miura family’s growth for the ceiling put the finest Miura a 170 mm sheet of copier paper can carry at 139 cells a side.

One sentence in that argument carried more weight than it was given. “A two-directional pattern at that spacing is not denser — it is the same total length divided between two families at twice the spacing each.” It is not true, and working out why moves every fineness the ceiling produced by a factor of nearly two.

Two families at the same spacing #

Draw parallel creases a distance $w$ apart across a square metre. There are $1/w$ of them per metre of width, each a metre long, so the density is $1/w$ and no two are closer than $w$. That is the ceiling.

Now draw a second family at right angles, also $w$ apart. It adds another $1/w$, so the square metre carries $2/w$. Which creases are closer than $w$? Within each family, none. Across the two families, every crease of one meets every crease of the other — at a point, at right angles — and two creases that meet are not two creases crowding into one band. They share a vertex, and at a vertex creases always come arbitrarily close, because that is what meeting means.

So the claim that a second family halves the spacing confused two things. It treated the total length as fixed and asked how it could be arranged, which is the right question for parallel lines and the wrong one for a pattern. A pattern can add length in a new direction without bringing any two creases closer than they already were, and density measures length while the paper’s own band width limits spacing.

The quantity that separates them is density times spacing. For parallel creases it is exactly one. For a square grid it is two, for a triangular grid three — roughly, the number of directions the creases run in — and it is a pure number, the same at every fineness, because density grows as one over the spacing and the product cancels the scale.

What the paper actually limits #

The limit a band width imposes is on two creases that do not share a vertex. Wherever they come closer than $w$, their bands overlap without the creases meeting, and the paper has to put two folds into ground that holds one. That can happen between parallel creases, between creases converging on different vertices, or between a crease and the far end of a crease that stops short of it.

The closest such approach can be measured exactly: every pair of interior creases with no vertex in common, the distance between them as line segments, and the smallest of those distances. On the printed shelf it runs from 18.6 mm on the tapered corrugation to 36.1 mm on the square twist.

The product of density and closest spacing is then the number the first figure prints. On six printed patterns with a pair of creases that do not meet it runs from 0.77 to 2.02, and three are above one. The Yoshimura carries twice the crease its own closest spacing would allow parallel lines, because its creases run in three directions and cross at every vertex. The Miura and the hexagon twist are just under one, which on patches this small is the boundary talking: a sheet’s outermost creases stop at its edge and do not get their full length.

The Miura family, measured #

A family makes the point cleanly because the boundary’s share shrinks as the cells multiply. At four cells a side the Miura’s product is 1.22, at eight 1.55, and at twenty-four 1.80, and the closest approach measured in units of a cell has stopped moving — 0.925 of a cell, within a per cent of its value at sixteen. What remains of the climb is the boundary’s share of the crease length falling away as one over the cell count.

That settled constant is what the fineness calculation needed. If the closest approach is 0.925 of a cell and cells are $S/n$ across on a sheet of side $S$, two creases that do not meet come within $w$ when

$$n = \frac{0.925\,S}{w}.$$

On a 170 mm sheet of copier paper, with $w = 0.6$ mm, that is 262 cells a side. The density bound gave 139. The Miura’s cells at its true limit are 0.65 mm across, barely wider than a crease, which is what a limit set by crease width ought to look like; at 139 cells they were 1.22 mm, twice as wide as the thing supposedly crowding them.

The ratio between the two bounds is the same on every paper, 1.88 to 1.90, because both bounds scale with the paper in the same way and their ratio is the family’s product of density and spacing. The paper chooses how fine; the pattern chooses by how much the density bound was wrong.

The waterbomb, and a factor that agrees #

The waterbomb is a harder test, because its creases run in four directions rather than two and it lays more length into each cell. Its closest approach is simpler than the Miura’s: two parallel diagonals, exactly half a cell apart, at every size measured. A pattern with four directions might be expected to carry four times the ceiling at its own spacing. It does not, and the reason is that its spacing is half the Miura’s to begin with — the parallel diagonals are the price of the extra directions.

The waterbomb’s product settles at 1.89, and its spacing bound on copier paper is 141 cells a side against a density bound of 74. The ratio is 1.91 to 1.92 on the four papers.

That the two families land within a few per cent of each other is the part worth noticing. A Miura has straight creases in one direction and zigzags in the other; a waterbomb has horizontals, verticals and two diagonals. What they share is that each is worth about two families of parallel lines at its own closest spacing, and that is the whole of what the paper cares about. The Miura gets its two by running two directions at full spacing; the waterbomb gets its two by running four directions and paying for them with a spacing of half a cell.

Where the two families’ 1.9 comes from #

The constant can be read off each pattern’s cell without measuring any sheet.

A Miura cell carries a straight crease one cell long and a slanted crease a little longer, because the zigzag leans: with the lean these patterns are drawn at, the interior cell carries about 2.04 cell-widths of crease for each cell of area. Its closest pair of creases that do not meet are two neighbouring slanted creases in the same row, and they are 0.925 of a cell apart — a little less than a cell, because leaning lines one cell apart horizontally are nearer than that measured square across. Multiplied, $2.04 \times 0.925 \approx 1.89$, which is where the Miura’s measured product is climbing to as its boundary’s share falls away; at twenty-four cells it has reached 1.80.

A waterbomb cell carries far more crease — horizontals, verticals and both diagonals, 3.80 cell-widths of crease a cell — and its closest pair are parallel diagonals half a cell apart. Multiplied, $3.80 \times 0.5 = 1.90$, and the measured product is already 1.89 at twenty-four cells, because the waterbomb’s boundary costs it less.

So the agreement is not a coincidence of the two numbers measured, but it is not a law either. It is two different routes to about two parallel families’ worth of crease: one pattern with two directions at nearly full spacing, one with four directions at half. A pattern that kept four directions at full spacing — a grid of horizontals, verticals and diagonals meeting only at shared lattice points, if its diagonals could be laid that way — would reach four, and there is no bound on the product short of how many directions can pass through common points.

Why a denser family still meets the paper sooner #

The density a paper allows concluded that a denser family meets the material at a coarser cell count, and that survives, for a different reason. The waterbomb’s limit on copier paper is 141 cells against the Miura’s 262, and the ratio between those, 0.54, is the ratio of their closest approaches per cell — 0.5 against 0.925. What makes the waterbomb meet the paper sooner is its half-cell spacing, not its density, and it happens that the two readings agree because both families’ products are near 1.9.

They need not agree. A pattern with creases in three directions at full spacing — a triangular grid, or the Yoshimura at 2.02 on the shelf — would carry more density than either family at the same fineness, and meet the paper no sooner than a pattern of parallel lines at the same spacing. The density bound would call it the densest and the first to fail, and it would be neither.

What the correction does to the shelf #

The printed shelf was already far inside the ceiling, and it is further inside the right bound. The densest pattern, the waterbomb tessellation at 89 metres a square metre, has its closest non-meeting pair 20 mm apart; copier paper’s crease is 0.6 mm wide. Its creases could be printed thirty-three times closer before the paper ran out of room, and the whole shelf thirty-one times smaller, where the density reading said nineteen.

That does not change the conclusion that the material is not what limits the printed patterns. What it changes is the margin, and which pattern is nearest. On the density reading the waterbomb and the Yoshimura were nearest the ceiling. On the spacing reading the nearest is the tapered corrugation, whose closest pair is 18.6 mm apart, and which was in the middle of the density table. The shortest crease is not a crease found creases on a generated patch shorter than light’s wavelength; those, not any printed pattern, are where spacing actually runs out.

The idealisation underneath #

A crease is a band of fixed width. The width is taken as six sheet thicknesses, as before; the crease has a radius is the fuller account of why a fold occupies ground at all, and nothing here depends on the factor, since both bounds scale with it identically.

Two bands may overlap where their creases meet and nowhere else. That is the assumption that makes the spacing the limit, and it is generous at a vertex: where several creases converge, their bands overlap over a small region round the point, and paper does something there — a small cone, a bulge — that a line drawing does not. The spacing bound treats that region as free, and it is not free, which is the question it leaves.

The sheet is flat and the patterns are exact. The closest approach is measured between ideal segments on a printed pattern, not between creases a hand has put in, and closer than a crease is wide is the reminder that two references a folder is asked to distinguish run out of room long before two creases in a pattern do.

What the measurement cannot see #

It cannot see the vertex. At a vertex of degree $d$, every crease is within a crease-width of every other one near the point. A disc one crease wide centred on a degree-four vertex holds two crease-widths of crease; one on the preliminary base’s degree-eight centre holds four. The spacing bound ignores all of it by construction, because creases that share a vertex are exempt, and so it cannot say what the overlap at a vertex costs.

It measures closest approach, not how much of the pattern is near its limit. A family has one closest pair repeated everywhere; the tapered corrugation’s closest pair is at its narrow end and nowhere else. The bound says where the first overlap would appear, not how much of the sheet would be affected.

And it measures geometry, not folding. How much line is on the paper observed that laying the last crease into a sheet already carrying the others is not the same act as laying the first, and a pattern at its spacing limit would be unfoldable by hand well before its bands overlapped.

A correction that the arithmetic invited #

The density ceiling was the natural first calculation because density was the quantity every earlier measurement in this line had produced — a count is not a length and where the length sits both priced patterns in metres of crease per square metre. A ceiling in the same unit could be set beside them in one division, and it was.

What the division assumed was that density is the thing a material runs out of. A material runs out of room, and room is a spacing. Density and spacing are the same number for parallel lines and differ by a pure factor for anything else, and the factor is not small: it is the number of directions a pattern’s creases take, discounted by how close those directions force them. That is also why a crease with no vertex to belong to could have density with no vertices at all — a band of parallel creases is the one pattern for which density and spacing say the same thing.

Still open: what a vertex costs #

The spacing bound exempts every place creases meet, and every crease pattern is made of such places.

At a vertex, bands overlap over a region whose size is set by the smallest angle between creases, not by their number. Two creases a sector angle $\theta$ apart are within a band-width $w$ of each other out to a distance $w/(2\sin(\theta/2))$ from the vertex, so a vertex with a narrow sector is double-creased much further out than one with wide ones. That is a quantity computable for every vertex of every printed pattern, and it would say how much of each sheet is creased twice.

It would also give a third bound. As a pattern is refined, its vertices come closer together, and at some fineness the overlap regions of neighbouring vertices touch — at which point a vertex’s exemption is no longer local. Whether that happens before or after two non-meeting creases come within a band-width is not obvious, and on a pattern with narrow sector angles it could bind first.

Sideways from here, the product of density and spacing is a new invariant with nowhere to live yet. It is one for parallel creases and near two for both tessellations measured; what a corrugation costs prices every tessellation’s compaction against its crease length, and pricing it against this number instead would ask how much folding a pattern buys per unit of the room it uses rather than per metre of line.

The habit worth carrying is about limits stated in derived units. When a bound is quoted in a quantity that combines several things, ask which of them the material actually runs out of. A density is a length divided by an area; a paper runs out of neither, but of the distance between two lines, and a bound in the wrong unit is right only for the one arrangement where the units agree.