Four clicks make sixteen lengths

A twisted tube has two heights found that one course of a Kresling tube has exactly two shapes in which every crease is its own length. The second exists when the course was made taller than the one that folds flat, and the creases strain on the way between them. It ended on the question one course cannot answer. A tube is a stack of courses, and pressing a stack is not pressing a course.

The answer turns out to depend on one curve, and the curve has a peak in it that nothing about a single course would suggest. Under one shared force, with its ends free to turn, a stack of courses is a row of independent switches, each with its own force to shut and its own force to open. A row of four switches has sixteen states. Whether pressing and pulling can reach all sixteen, or only five, depends on whether the stack opens in the order it shut or in the reverse. That depends on which side of the peak its courses were made.

Why the courses do not talk to each other #

Stack courses one on another, each sharing a polygon with the next, and push on the two ends along the axis. Every course carries the whole force, as links in a chain carry the whole pull. If the ends are free to turn, no course carries any torque, so each course takes whatever turn suits it at whatever height the force leaves it. That is exactly the relaxed course of the earlier essay, pressed alone. A course does not know what its neighbours are doing. It knows only the force.

So a course’s behaviour under the stack’s force is a property of the course. It shuts when the press exceeds a threshold and opens when the pull exceeds another, and nothing else in the stack moves either threshold. The stack is a set of switches wired to one input, and that is a system with a long history. It is the model Ferenc Preisach proposed in 1935 for magnetic hysteresis: a material as a population of tiny independent two-state elements, each flipping up at one field and down at another. A Kresling stack under a shared force is that model, built of paper, with the elements visible. A mechanism that closes on itself listed the Kresling as the tube used where a click is wanted; a stack is where the clicks start to combine.

The two forces a course switches at #

The force a course pushes back with is the slope of its energy against its height. With the turn relaxed the slope has an exact form, summed over the two diagonals of every sector: each diagonal’s strain times its height divided by its length. Plotted from the shut state to the open one it is a hump and a trough.

Pressed from its own height, the course resists with a force that rises to a peak and falls to zero at the top of the barrier. Past that, the course pulls itself shut. That peak is the force to shut it, 44.2 thousandths of a diagonal’s axial stiffness for this course. Pulled from its shut height, it resists with a force that peaks on the other side of the barrier, and that peak is the force to open it, 29.8 thousandths.

The two are different, and they have to be. The hump and the trough are the slopes of two different flanks of the barrier, and the flank toward the shut state is the shallower one. A course is easier to open than to shut, by about a third here. That asymmetry holds for every course measured here, from just above the flat-folding height to twice it. The question that decides everything below is how the two thresholds change from one course to the next.

One threshold peaks and the other does not #

Make the courses of a stack at slightly different heights, all with the same turn. Both forces change with the height as made, but not in the same way.

The force to shut falls the whole way. A taller course has a lower and broader barrier, and its peak press falls from 60 thousandths just above the flat-folding height to 28 at 1.75 times it. So in any stack the tallest course shuts first and the shortest last. Every graded stack shuts in that one order.

The force to open does not fall the whole way. It rises from 28 thousandths just above the flat-folding height to a peak of 30.5 at 1.18 times it, then falls. The two effects pull against each other. A course made just above the flat-folding height has a second state almost flat, so it is pulled open from a deep, nearly flat fold where the trough is shallow. A tall course has a low barrier everywhere. Between them the trough is deepest.

So the order a stack opens in depends on where its courses sit. Below the peak, the shortest opens first, the reverse of the order they shut in. Above it, the tallest opens first, the same order. Drag the dial and the marked courses slide along both curves together. Their shutting order never changes, while their opening order turns over as they cross the peak.

The peak moves with the turn, from 1.14 times the flat-folding height for a course turned 70 degrees to 1.18 at 80, 1.24 at 90 and 1.35 at 100. It sits where the earlier essay found the stroke still large, a third to a half of the course’s height, which is where a designer would want a working tube to be.

Last in, first out, or first in, first out #

The two orders sound like a detail. They are the whole difference between a tube that counts and one that stores.

Take a stack whose courses shut in the order D, C, B, A, the tallest first. A press to any level shuts every course whose threshold lies below it, so a press can only ever add courses from the front of that queue. A pull to any level opens every course whose opening threshold lies below it, so it can only take courses away from the front of the opening order. Starting from all open, the reachable states are those that two moves can build: add a prefix of the shutting order, or remove a prefix of the opening order.

If the opening order is the shutting order reversed, A, B, C, D, then a pull always removes the courses shut last. The shut set is always a prefix of the shutting order: nothing shut, D, D and C, D and C and B, all four. That is five states out of sixteen, and the stack is a counter. It can be stepped up and down, but it always holds the tallest courses shut and the shortest open.

If the opening order is the shutting order itself, D first both ways, then a pull removes the courses that were shut first. Press to shut D, C and B, pull to open D, and the stack holds C and B with D open, which no counter can do. Alternating presses and pulls of shrinking size build any set at all. All sixteen states are reachable.

The search behind the figure starts from all open and applies the two moves until nothing new appears. It agrees with the argument at every size tried: a stack of $n$ courses graded below the peak reaches $n + 1$ of its $2^n$ states, and graded above it reaches all $2^n$, checked from two courses to eight. For four that is five against sixteen. For eight it is nine against 256.

The stacks in the figure were chosen so that every state also has its own length, since the four strokes, 0.70, 0.60, 0.52 and 0.47 radii for the upper stack, add up differently in every combination. Sixteen states are therefore sixteen lengths, from 9.10 radii with everything open down to 6.81 with everything shut. A tube that has to be set to a length, as an antenna mast or a bellows does, can be set to any of them by load alone.

A stack straddling the peak sits between the two. Drag the dial through it and the count goes from 5 below the peak through 7 and 10 to 16 above it. That is the count for opening orders that are neither the shutting order nor its reverse.

Writing a word #

In the upper stack every word is reachable, and the way to reach a given one is short enough to write down. It is the same move that takes the memory out of a magnet: a load that swings back and forth with a shrinking amplitude.

To write 1010, A and C shut and B and D open, press until A shuts. Every course with a lower threshold shuts too, so all four are shut. Pull just hard enough to open B. Since opening goes tallest first, D, C and B open and A stays shut. Press just hard enough to shut C, which shuts D with it. Pull just hard enough to open D alone. Each swing is smaller than the last, and each one fixes one boundary between a shut run and an open run in the word. The number of swings is the number of such runs, counted from the top course that has to be shut, and no word of four courses needs more than four.

This is the Preisach model’s characteristic behaviour, and it brings the model’s best-known property with it. The stack has return-point memory: a load that wanders and comes back to an earlier extreme returns the stack to exactly the state it had there, because the state depends only on the history’s surviving extremes, and returning to an earlier one wipes out everything that happened in between. Nothing here needed a controller or a latch, and the word is held with no load at all, the way a course of the earlier essay held its second height. That is the property paper that folds itself found hardest to get: a choice between outcomes made by the structure rather than by whatever drives it.

The order is the fragile part #

The thresholds are numbers with gaps between them, and the count depends on the order of the gaps, not their size. That makes some gaps matter more than others.

In the upper stack the forces to shut are 44.2, 37.7, 32.4 and 28.0 thousandths, 13 to 17 per cent apart. The forces to open are 29.8, 27.7, 25.3 and 22.8, 7 to 10 per cent apart. Near the peak the opening forces are much closer: across 1.02 to 1.14 they run 28.4, 29.5, 30.1 and 30.5, and the last two differ by 1.3 per cent. The opening order is always the one at risk, because the curve it is read from is flat where the decision is made. A real stack whose courses’ stiffnesses scatter by a few per cent will keep its shutting order and may shuffle its opening order, and with it the set of words it can reach. A tolerance is a direction found that a mesh forgives some errors and not others; here the forgiven errors are the ones along the steep curve.

That gives a design rule a designer can use. To build a counter, make the courses well below the peak, where shutting and opening both run steeply in opposite directions. To build a register, make them well above it. Courses at 1.30 to 1.75 times the flat-folding height keep their opening order through a scatter of several per cent. Avoid the peak itself, where the opening forces are nearly equal and the stack’s behaviour is set by its manufacturing errors rather than its design.

What the picture cannot show #

A tube held between rigid plates. If the stack’s ends cannot turn, its courses no longer relax independently. The total twist is fixed, a course that shuts turns the others, and the switches interact. The independence on which the whole count rests is a consequence of free ends, and many testing rigs and many applications clamp them.

A press by displacement rather than force. A testing machine sets the length and reads the force. Under that control a course that passes its peak does not snap freely; it shortens while the others lengthen to keep the total, and the stack follows a sawtooth rather than jumping. The states the stack can rest in are the same, but which of them a given history reaches is a different question, and the switch model answers it only for a load.

Paper. The thresholds are computed for diagonals as bars of one stiffness and triangles that do not bend, as before. A paper course bends its panels and its creases have their own stiffness, which lowers both thresholds and could move the peak, and a held fold slowly relaxes toward wherever it is held, so a word left in a stack for a long time would move its own thresholds. The structure of the argument survives any change that keeps one threshold monotone and the other peaked. Whether paper keeps that shape is a measurement.

The idealisations underneath #

Each course is the symmetric, relaxed course of the earlier essay: rigid regular polygons, every sector alike, the two diagonals bars of one axial stiffness storing $\tfrac12\ell_0\varepsilon^2$, and the turn free at every height. Forces are in units of that stiffness and lengths in units of the polygon’s radius.

The stack is in series with free ends and is loaded slowly, so every course carries the same force and no torque, and a course that passes its threshold completes its click before the load moves again. Dynamics, inertia and the noise of a real click are left out.

How the claims were checked #

The forces are the exact slope of each course’s relaxed energy, $\sum \varepsilon\, h/\ell$ over its diagonals by the envelope theorem, evaluated along a press of 400 to 600 steps, with no finite difference anywhere. The force to shut is the largest compression before the barrier and the force to open the largest tension after it. The force to shut is checked to fall at every one of 71 heights as made from 1.01 to 1.91 times the flat-folding height, and the force to open to have an interior peak.

The reachable states come from a breadth-first search from all courses open, applying the two moves until nothing new appears. Its counts are checked against the argument, $n + 1$ and $2^n$, for stacks of two to eight. For the two stacks drawn it finds five words and sixteen, and the sixteen lengths of each stack are all different.

The writing sequence is simulated threshold by threshold. It must end holding the word asked for, with each swing of the load smaller than the last of the same sign.

Still open: what a clamped stack can still reach #

The free-ended stack is the clean case, and it is clean because its courses are independent. Clamp the ends against turning and the courses couple through the total twist. A course that shuts turns by a hundred degrees or more, and the others must take up the difference. Its click then moves every other course’s thresholds, and the stack is no longer a row of independent switches. Interacting switches are the general case in the physics of hysteresis, and interactions are what can break the clean memory structure described above. The arithmetic to find out what survives here is the same energy summed over courses with one more constraint, and the count of reachable words is the measurement it would produce.

The other direction is the courses’ handedness. Every course here turns the same way. Stacks are often built with alternate courses of opposite hand, so that the tube’s ends do not rotate against each other. Under free ends that changes nothing in the count, but under clamped ends it changes everything, because opposite-handed clicks cancel each other’s twist. A stack whose hands are chosen so that every reachable word has zero net twist would be a register that does not need free ends at all.

Sideways, a corrugation has one resting state found that a strain-free corrugation has a single well, and the tube that gets built catalogued the tubes engineering makes. The Kresling stack adds one that engineering wants and paper supplies cheaply: a tube with as many stable lengths as the subsets of its courses, set by load alone. The only pattern that moves found rigid motion to be the rare case, and the cylinder the pattern chooses found that the circumference of a found pattern is not chosen. Here the found pattern turns out to choose something else: a stack’s memory is decided by which side of a peak its heights sit on.

The habit worth carrying is about systems built from switches. Before counting a system’s states, ask in what order its parts switch each way. The states a system has are a product of its parts. The states it can reach are a property of two orders, and a single peak in one curve was enough to turn sixteen into five.