Half a rim

The comparison this collection has been making is between two objects: a rectangle of a repeating pattern cut out of the plane, and the same rectangle with its opposite edges declared to be the same edge. The first has an edge and the second does not, and the difference between them is four free letters per cell of rim.

Two objects give one measurement. A measurement is a difference between two states, and a difference establishes that something costs without establishing what it costs per. Everything said so far about the rim has therefore been of the form there is a cost and here is its total, and every attempt to say the cost is a rate has been an extrapolation from a single interval.

The middle of the scale removes the extrapolation. Glue one pair of the rectangle’s edges and leave the other pair alone, and the result is a cylinder: a sheet with two edges instead of four or none.

What is being held still #

The value of the comparison depends entirely on what is not changing, so it is worth listing.

The drawing is one drawing. The same pattern, the same rectangle, the same creases in the same places. Nothing is redrawn between the four sheets, and nothing is scaled.

The vertices are the same vertices. The rectangle’s corner is chosen so that its edges miss every vertex of the pattern, which means a crease crossing an edge does so at an ordinary point of a crease rather than at a place where creases meet. Identifying the edges cannot then create a vertex or destroy one, and the number is checked at every gluing rather than assumed: sixteen on the Miura’s two-by-two cell, on all four sheets.

The conditions asked at those vertices are identical, because the conditions read the angles and letters at a point and every point is unchanged.

What moves is the rim, and with it two counts that depend on it: how many free letters there are, and how many panels.

Why a crease at the edge is two creases #

The letter count is the one the search cares about, and the reason it moves is worth being concrete about.

A crease of the pattern that runs off the right-hand edge of the rectangle runs back on at the left, one cell over — it is one crease of the tessellation and the rectangle shows it as two pieces. On the cut sheet those two pieces are two separate creases, each free to be a mountain or a valley independently of the other. On the glued sheet they are one crease and take one letter.

So gluing a pair of edges removes exactly as many free letters as there are creases that pair of edges divides. On the two-by-two square twist cell that is four per pair: forty free letters cut out, thirty-six with one pair glued, thirty-two with both.

The panels behave the same way and not quite #

Panels do the same thing with one correction, and the correction is the corner.

A panel touching the left edge and its partner touching the right are the same panel of the pattern, so gluing that pair merges them in pairs. Gluing both pairs merges in pairs twice — and the four pieces at the corners of the rectangle are all one panel, so they merge into one rather than into two.

That is why the panel counts do not add the way the letters do. On the square twist’s two-by-two cell they run twenty-five, twenty, twenty, sixteen: five saved by one gluing, five by the other, and nine rather than ten by both.

The corner is the one place in the whole construction where the two directions interact, and it caused an actual defect: a crease running exactly through a corner of the rectangle has no partner on any one of the four edges that meet there, and Euler’s count is what noticed.

What the search pays #

The letters are what a search over letterings chooses among, so removing them ought to make the search cheaper. It does not, and the reason is instructive.

The total number of nodes goes down as the rim goes, because there are fewer letters to decide. The cost per panel goes up, because the panel count falls faster than the node count does. And per panel is the right measure, since the four sheets do not hold the same number of panels and a total would be reporting the panel count under another name.

Underneath that is the reading an earlier measurement arrived at from one end of the scale: the rim is slack, and slack is what makes a search cheap. A crease the rim divides is two letters free to disagree with each other, and a letter free to disagree is a letter the search can set without consequence. Take the freedom away and every remaining letter matters more.

What the middle case adds #

A reader might reasonably ask what the cylinder buys that the two ends did not.

It buys attribution. Two objects differing in four things support the claim that those four things together cost something. Three objects differing one at a time support the claim that each of them costs something separately, and the cylinder differs from the cut sheet in exactly one pair of edges.

It buys linearity, which is a claim of its own: what one pair saves is what the other pair saves and the two add. That cannot be checked with two points, because two points lie on a line whatever they are.

And it buys a sheet the collection did not have. A cylinder is a genuinely different object from both a square and a torus, it is the shape every folded tube in engineering actually has, and until now nothing here could build one.

The two cylinders are not one sheet #

There is a trap in the phrase half the rim and the figures above have already stepped round it.

Gluing the left and right edges and gluing the top and bottom give two sheets with the same Euler characteristic, the same number of edges left, and the same kind of shape. They are not the same object, because the drawing on them is not isotropic.

On the Miura, one direction crosses one crease per period and the other crosses four. On the elongated twist tessellation, gluing across the pleats saves ten letters per cell and gluing along them saves two. So half the rim is a description of the topology and not of the cost, and the two halves are different sizes.

That is taken up separately, because it turns out to change the search cost by more than an order of magnitude on some tilings and by nothing at all on others.

What has not changed #

Three things, and stating them is the point of a controlled comparison.

The pattern is the same pattern on all four sheets. Anything true of the tessellation is true on each of them.

The vertex conditions hold or fail identically. A drawing that is developable and satisfies Kawasaki at every vertex does so on every gluing of it, because the conditions are at points and no point moved.

The printed sheet is unchanged. Every one of these objects can be printed as a flat rectangle and folded, because a rectangle is what all four of them are drawn as; the gluing is a sentence about the rectangle rather than a change to it.

The numbers, on one cell #

It helps to have a single case written out completely, small enough that every count can be checked by eye.

Take the square twist tessellation and a rectangle holding two of its periods in each direction. Cut out of the plane it has sixteen interior vertices, forty free letters and twenty-five panels. Glued in one direction: sixteen vertices, thirty-six letters, twenty panels. Glued in the other: the same, because the square twist is symmetric under exchanging the two directions. Glued both ways: sixteen vertices, thirty-two letters, sixteen panels.

Read down the letter column — forty, thirty-six, thirty-six, thirty-two — and the steps are four and four. Read down the panel column — twenty-five, twenty, twenty, sixteen — and the steps are five and five and then four, the missing one being the corner.

Read the vertex column and nothing happens at all, which is the control and the reason the rest of the table means anything.

Euler’s number, $V - E + F$, comes out one, nought, nought, nought. That is the characteristic of a disc, a cylinder, a cylinder and a torus, and it is computed from the quotient rather than looked up.

Why the vertex count is the control and not a coincidence #

The claim that identifying edges cannot make or destroy a vertex is doing a great deal of work above, and it is true by construction rather than by luck.

The corner of the rectangle is not placed at a round number. It is searched for: the two coordinates are chosen independently so that the rectangle’s edges fall in the largest gap between the drawing’s own features in each direction, which puts every edge clear of every vertex. A crease crossing an edge therefore crosses it at an ordinary interior point of that crease, and identifying the two ends of that crossing joins two half-creases into one crease rather than joining anything at a vertex.

If the search ever failed — if a tessellation had no rectangle whose edges could miss its vertices — the construction would refuse rather than produce a cell with a vertex sitting on its boundary. There is such a case in the collection: a crease can run exactly through a corner, where four edges meet, and that one had to be found and repaired.

The result is that the comparison is genuinely controlled, in the strong sense that the only thing differing between the four sheets is the sentence saying which boundary points are the same point.

What a folder would notice #

None of this is abstract in the way it might look, and a person with paper meets the middle case constantly.

A paper tube is a cylinder. A drinking straw, a rolled poster, a length of tubing: each is a rectangle whose two long edges have been joined, and each is a sheet with two edges rather than four. Every engineered folded tube — and there are many — is a rectangle of corrugation with one pair of its edges glued, which is exactly the object above.

What a folder does not meet is the torus. Joining both pairs of a rectangle’s edges cannot be done with paper without stretching it: the result is a doughnut, and a doughnut made from a flat rectangle without stretching is not a thing that fits in space. The torus in this collection is a mathematical object throughout, described by a drawing and a rule, and it is folded only in the sense that its panels have positions.

So the middle rung of the scale is the one that is physically real, and it is the one that had not been built.

The cost per panel, read carefully #

The rising cost per panel is the least intuitive number in the table and it repays a paragraph.

A search over letterings explores a tree: pick a crease, try a letter, propagate what that forces, and back up when a contradiction appears. The nodes of that tree are the units the cost is counted in.

Cut out of the plane, a two-by-two Miura rectangle has twenty-two letters to decide over fifteen panels, and it settles in fifteen nodes — one per panel, which is the reading the collection found on cut patches across five families. Glue one pair of edges and it has eighteen letters over ten panels and settles in eleven nodes. Glue both and it has sixteen letters over eight panels and settles in ten.

Eleven nodes is fewer than fifteen. Eleven over ten panels is more than fifteen over fifteen.

Both readings are correct and they answer different questions. How long does this take is answered by the total. How hard is this object, size for size is answered by the ratio, and the ratio is what makes patterns of different sizes comparable at all.

The ratio going up says something specific: the letters that were removed were the cheap ones. A letter on a crease the rim divides can be set without agreeing with anything, because its partner on the other side of the rectangle is a separate letter. Remove that freedom and every remaining letter has to agree with something.

One thing this does not settle #

The three rungs of the scale are three sheets and they are not three amounts of rim.

A rectangle’s four edges come in two pairs and each pair is glued or not, so what is being varied is a count of glued pairs. It happens that each pair carries the same length of rim on a square cell, and on a cell that is twice as long as it is wide it does not, and no cell here is that shape because every cell is a tessellation’s own period.

So the linearity below is linearity in the number of glued pairs, and whether it is also linearity in the length of rim is a question this comparison cannot separate. It would take rectangles of the same pattern at different aspect ratios, and building those means choosing a cell that is not the pattern’s period, which brings its own difficulties: a rectangle that is not a period does not glue at all, because the drawing does not match up across the join.

That is a real limitation and it is recorded rather than argued round.

What this measurement is for #

Two things, and they are both about what comes next.

The first is that any claim of the form the rim costs so much now has an arithmetic behind it rather than an interval. The cost is per glued pair, it is the number of creases that pair divides, and it can be computed for a drawing without running anything.

The second is that the cylinder is the sheet on which several questions in this collection change character. The order the layers come in needs a bottom panel, and a bottom panel lives on the rim: take half the rim away and half the candidates go, and the order still has a least element until the last of the rim goes with it. That is a middle case for a question that previously had only two answers, and it is the subject of its own argument.

Where the scale ends #

The rim can be removed in halves and it cannot be removed in quarters.

An edge of the rectangle is glued to the opposite edge or it is not; there is no gluing a third of an edge, because the identification has to be consistent along the whole of it or the pattern will not match up where the gluing stops. So the scale has exactly three rungs — four edges, two, none — and the cylinder is the only middle it has.

That is enough for the linearity claim, which needs three points, and it is not enough for anything finer. A reader wanting to know whether the cost is linear in the length of rim rather than in the number of glued pairs will not find it settled here: the two coincide on every rectangle, because a rectangle’s opposite edges are the same length.

Rectangles with different aspect ratios would separate them, and every cell in this collection is the tessellation’s own period rather than a shape somebody chose. That one is left where it stands.