The cylinder the pattern chooses

A Yoshimura pattern folds into a tube. The obvious question about a tube is how fat it is, and the obvious answer is that it depends on the paper: a bigger sheet, a bigger cylinder.

That answer is wrong, and it is wrong in an unusually clean way. The diameter is written into the crease pattern. A sheet twice as tall, ruled with the same columns, gives a tube twice as long and exactly as fat.

A course goes round exactly once #

The pattern is diamonds in offset rows, and the horizontal creases that separate one row from the next are its courses. Every course runs from one edge of the sheet to the other, cut into as many segments as the pattern has columns.

Folding does not change the length of any of those segments. It changes the angle between consecutive ones, and when the sheet becomes a tube the chain of segments closes on itself: the course is a closed polygon around the cross-section.

So the perimeter of that polygon is the total length of the course, which is the width of the sheet. There is nothing left to choose. Ruling the columns spends the sheet’s width on the circumference, and no folding afterwards can spend it again.

This is a different kind of statement from the ones that usually get made about a corrugation. How much smaller a pattern folds is a ratio, and a ratio survives scaling: a Yoshimura folds thirty-two times smaller whatever size it is drawn at. The circumference is not a ratio. It is a length, in the sheet’s own units, and it is fixed.

The radius is one number and there is no second one #

If the columns are equal, the closed course is a regular polygon of N equal sides, and a regular polygon of a given side has exactly one radius. For a side w it is w divided by twice the sine of π over N.

At six columns on a unit sheet the side is 0.166667 and the radius comes out at 0.166667 as well. That equality is a fact about hexagons rather than about paper — a regular hexagon’s radius is its side — and it is worth saying so before it starts to look like a discovery.

Doubling the column count changed the radius by three per cent. That is the shape of the whole result: the perimeter is pinned, so the radius of a polygon with that perimeter is pinned too, and the only thing the column count controls is how close the polygon is to the circle through its corners.

It is tempting to skip the polygon and say the radius is the sheet’s width divided by 2π. That gives 0.159155, and it is wrong by 4.72% at six columns. A regular polygon is shorter than its circumscribed circle, and the shortfall is not negligible at the column counts anybody actually folds.

The pattern library checks that shortfall against a closed form the construction never uses: to second order it is one sixth of the square of π over N, which is 4.57% at six columns and 1.645% at ten, against 4.72% and 1.66% measured. Two routes to the same number, and the figure refuses to draw if they disagree by more than a quarter.

The radius has a floor the columns cannot reach #

The polygon formula is worth turning round, because written in terms of the sheet rather than the side it says what the column count is actually worth.

With $N$ equal columns on a sheet of width $W$, each side is $W/N$ and the closed course’s radius is

$$R = \frac{W}{2N\sin(\pi/N)}.$$

Two of its values are exact. At six columns $2\sin 30° = 1$, so $R = W/6$ — the hexagon whose radius is its side. At ten, $2\sin 18° = 1/\varphi$, so $R = \varphi W / 10$: the golden ratio, arriving from a decagon rather than from anything about paper.

As $N$ grows, $N\sin(\pi/N) \to \pi$ and the radius falls monotonically to

$$R_\infty = \frac{W}{2\pi} = 0.15915\,W.$$

So there is a floor, and no column count reaches it. A Yoshimura on a sheet of a given width cannot make a tube thinner than that sheet’s width over $2\pi$, because the course has to go round once and a circle is the shortest closed curve enclosing a given radius.

Which prices the lever. Six columns sit 4.7% above the floor and ten sit 1.6% above it, so the entire range available to the column count is under five per cent of the diameter — and going from six columns to a hundred buys less than a twentieth of the tube’s width while quadrupling the creasing.

The sheet’s width is therefore not one lever among several. It is the only one, and the column count is a choice about how polygonal the tube looks.

What a larger sheet actually buys #

Adding columns makes the tube fatter and, on a sheet of fixed width, makes it fatter by very little. Adding rows makes it longer and does nothing else at all.

That is the sense in which the pattern chooses the cylinder. A designer with a diameter to hit chooses the column pitch; a designer with a length to hit chooses the row count; and the two choices do not interfere, which is a rare and useful property in this subject.

The Miura is the standing counter-example, and comparing them sharpens what is being claimed. Its folded footprint is a function of how far it has been closed, and that dependence is the whole reason it behaves like a mechanism. A Yoshimura tube is not exempt from having a motion — it shortens as the diamonds flatten — but the closure of the course is a yes-or-no event, and on the far side of it the cross-section is a polygon whose sides are the pattern’s own.

The widths are free and the heights are not #

There is a second asymmetry in the pattern, on a different axis from the first, and it is the more surprising of the two.

Kawasaki’s condition at an interior Yoshimura vertex is an alternating sum over its six sectors. Work out what that sum contains and the column widths cancel out of it entirely. The row heights do not: they are the whole of it.

So a Yoshimura may be tapered round its cylinder and may not be tapered along it. A cone made by widening the columns is a legal crease pattern; a cone made by deepening the rows is not, and the difference is invisible in a drawing.

Kawasaki is a statement about angles alone. It does not mention which creases are mountains and which are valleys, so no relabelling rescues a pattern that fails it — the refusal is final rather than an invitation to try a different assignment. Maekawa’s difference of two and the big-little-big lemma are both about the assignment; Kawasaki is not, and that is why it is the one that settles this.

This is the essay’s surprising connection, and it is not a resemblance but the same inequality. A leaf packing into a bud faces exactly this constraint, on a flat sheet, in a pattern with no cylinder anywhere in it.

A corrugated leaf narrows toward its tip and does it in the column widths, because those are the ones the condition does not contain. The Yoshimura widens round its circumference and does it in the same place, for the same reason. The direction that is free on the leaf is the direction that goes round the tube — one is a shape and the other is a circumference, and the geometry cannot tell them apart.

A second condition on the same number #

Uniform rows satisfy Kawasaki whatever their height, which sounds like freedom and is not. The height has a ceiling, and a different theorem imposes it.

The regular Yoshimura vertex is six sectors of 60°, which means the diamonds are pairs of equilateral triangles and the row height is the equilateral height of a column: the column width times the square root of three, halved. Raise the row height above that and the diagonals steepen, and the sector between two diagonals at a course becomes strictly smaller than both of its neighbours.

On the standard assignment — courses mountain, diagonals valley — that small sector is flanked by two valleys, and the big-little-big lemma forbids exactly this. The paper would have to fold two layers the same way through a wedge that has run out of room.

The numbers are worth quoting because they separate the two conditions cleanly. On the pattern in the taper figure, with six columns widening by half and the row height taken from the narrowest column, all twenty-two interior vertices pass everything. Raise the row height to the mean column’s equilateral pitch — 0.14434 against 0.11547 — and Kawasaki still holds exactly at every vertex, both alternating sums a straight angle to the last digit the arithmetic carries, while big-little-big fails at ten of the twenty-two, on a sector of 49.58° with a valley on each side. Take the widest column’s pitch instead and twenty of the twenty-two fail.

So the row height is constrained twice over, by two conditions that object to different things. Kawasaki objects to the height varying; big-little-big objects to it being raised. And the second is assignment-dependent while the first is not, so a pattern that fails big-little-big is refused for this labelling and might survive another, whereas a pattern that fails Kawasaki is finished.

Which conditions were checked, and how #

The ring in the first figure is not drawn from the radius. It is built the other way round, and the order matters.

The course’s segments are found by their assignment and their position in the crease pattern, not by asking for a row of a given length. Three things are then required of them before anything is drawn. There must be exactly as many segments as the pattern has columns, or the course came out in the wrong number of pieces and the ring is not the polygon the count claims. The longest and the shortest must agree to a part in a million million, or the ring has no single radius to report. And their sum must equal the sheet’s width to the same precision, or the ring does not close on the paper it was cut from.

Only then is the radius computed from one side. The polygon is built from that radius, and every one of its chords is measured back against the pattern’s own segment — a second route to a length already in hand, so that an arithmetic slip in the radius shows up as a mismatch rather than as a picture. The perimeter of the built ring is compared with the course a third time.

The last check is the one that could most easily have been left out. The circle through the ring’s corners is longer than the ring, by an amount that has a closed form; the construction never uses that form, and the measured excess is required to sit inside a band around it. A radius belonging to some other polygon would satisfy every earlier check and fail this one.

The taper figure asserts both halves rather than either. The column-tapered pattern must pass at every interior vertex, and the row-tapered one must fail, and the failure must be Kawasaki’s specifically. A generator that only demanded the failure would be satisfied by a pattern that fails both ways, which is the shape of the mistake this site keeps finding in its own work: an assertion that has never rejected anything proves nothing.

Where the drawing stops #

The pattern is drawn flat. A Yoshimura is intrinsically cylindrical, and the lattice on the page is a development of the tube rather than the tube. The figure is honest about the connectivity and silent about the curvature, which is the same caveat the discovery essay makes and it does not go away at this rung.

The ring is assumed planar and regular. The closure argument takes the course’s segments to lie in one plane and to keep their lengths exactly. Both are idealisations, and the first is the weaker of the two: a real tube’s course wanders out of its cross-section plane by whatever the diamonds either side of it require.

The single radius survives only while the columns are equal, and the columns are precisely what the conditions leave free. The two results of this essay pull against each other: a tapered Yoshimura is a perfectly legal crease pattern and its folded course is a polygon with unequal sides, which has no one radius. Asked to measure such a pattern the figure refuses rather than averaging, and refusing is the correct answer.

Nothing here is about how the tube gets there. The closed course is a state, not a path, and the question of which motions reach it — and whether a real sheet can be driven along one without tearing — belongs to the rigid-folding account and is not touched by any measurement above.

The sheet has no thickness anywhere in this. A tube of thirty-two layers is thirty-two thicknesses of material, and what that does to a fold is a separate subject with its own arithmetic.

The vertex in the middle of an edge #

There is a feature of this pattern that has already cost this site something, and it is the same feature the closure argument runs on.

Every course goes edge to edge. So every course end is a vertex sitting in the middle of a boundary edge — on the sheet’s rim, with no boundary edge attached to it, because the rim of a sheet is four long segments and a crease arriving partway along one does not subdivide it.

That is the exact case this site’s boundary test used to misread. A vertex like that was analysed as though the paper continued past the edge of the paper, its sectors read as a partial turn, and perfectly good patterns were rejected. The Yoshimura is what found it, and it found it because every course does this rather than one.

The pleasing part is that the two facts are one fact. A course that reaches both edges is a course whose length is the sheet’s width, which is why the ring’s perimeter is what it is; and it is a course whose ends are mid-edge vertices, which is why the checker broke. The property that makes the geometry work is the property that made the machinery fail.

Who found it, and when #

The pattern is named for Yoshimura, who described the diamond lattice in the aeronautical literature of the early 1950s, in a study of thin cylinders collapsing under axial load. The report is about failure; there is no folding in it. That story, and the sharper claim that the buckling chose the mountain-and-valley assignment as well as the creases, is told at the rung below this one and is not retold here.

What is older than either the naming or the analysis is the object. A concertina paper lantern is a Yoshimura, and so is a great deal of collapsible packaging, and every one of them was made by somebody who knew perfectly well that the lantern’s diameter is set by how the paper is scored and its height by how much paper there is. The geometry above is a folk fact of the craft, and what the pattern library adds is the arithmetic and the refusals.

The deliberate use came much later and from the other direction. A tube that collapses along a known pattern and springs back is a mast that packs short and extends long, which is why the pattern turns up in deployable hardware wherever a boom has to leave a spacecraft longer than the spacecraft.

Where the ladder goes next #

The immediate continuation is the pair of numbers this pattern hides. A Yoshimura draws in by four times across and four times down, and a shrink measured along both axes turns out to separate the whole pattern library into corrugations, twists and the ones that are neither — with the Yoshimura sitting on the diagonal, drawing in equally both ways, which is not what a tube looks like it should do.

The other direction is the motion. Everything here is about a state: the course closed, the polygon regular, the radius reported. What happens between the flat sheet and the closed tube is a path, and paths are where the rigid conditions bite and where a pattern that folds can still fail to be foldable in any useful order.

And there is a question this essay deliberately leaves standing. The columns are free, so a tapered Yoshimura folds; a tapered Yoshimura’s course is an irregular polygon; an irregular polygon of a given perimeter has a family of closed shapes rather than one. What the tapered tube’s cross-section actually settles into is not decided by the flat-folding conditions at all, and finding out what does decide it is a rung further up than anything measured here.