Clamped, a stack counts in pairs

Four clicks make sixteen lengths built a register out of paper. A Kresling tube of four courses, each made a little taller than the one below, has sixteen ways to rest — each course clicked shut or left open — and with its ends free to turn, pressing and pulling along its axis can put it into every one of them. The reason was that the courses never talk to each other. Under one axial force with free ends, every course carries the whole force and no torque, so each switches at its own two thresholds and the stack is four independent switches wired to one input.

The essay ended on the case that breaks that independence. Clamp the ends against turning and a course that shuts, turning as it does, has to be paid for by the rest of the stack, which must turn back by the same amount between the same two ends. The courses then couple through the total twist, and the clean memory of independent switches might not survive. It also suggested the repair: alternate the courses’ hands, so that a right-handed click and a left-handed click cancel. Both are answered below, by simulating the stack course by course under a slowly changing force, and the answer is cleaner than either suggestion.

A click is a fixed turn #

Everything about the clamped stack follows from one fact about a single course, and it is worth deriving before anything is simulated.

A Kresling course is two regular polygons joined by triangles, and each triangle has two diagonals. A twisted tube has two heights found the course’s second state in closed form: if the course was made with its top turned by $\theta_0$, its second state — the one in which both diagonals have their own lengths again — is turned by $\theta_1 = \pi + \alpha - \theta_0$, where $\alpha$ is the angle a polygon’s side subtends at its centre. The height drops out. The difference of the two diagonals’ squares depends on the turn alone, so the turn of the second state does too.

So the angle a course turns through when it clicks is

$$\theta_1 - \theta_0 = \pi + \alpha - 2\theta_0,$$

the same for every course made with the same turn, however tall. For six-sided courses made at eighty degrees it is eighty degrees exactly.

That is the quantity the clamp acts on. With free ends, a clicking course simply rotates the part of the stack above it by eighty degrees and nothing resists. With the ends clamped, the sum of all the courses’ turns is fixed, and a click that adds eighty degrees somewhere must be removed somewhere else. A Kresling course is stiff against being turned away from its two states — turning it stretches one diagonal and compresses the other — so eighty degrees is not something the open courses can absorb by flexing.

One hand: no click at all #

The simplest clamped stack has every course of the same hand, as in the earlier essay. Pressed with its ends free, it shuts its courses one at a time, tallest first, at forces of 28.1, 32.5, 37.8 and 44.3 thousandths of a diagonal’s axial stiffness.

Clamped, the same stack does not click. Pressed to sixty thousandths — a third more than the force that shuts its stiffest course free — and in the full survey to a hundred, every course stays open. Each course is compressed and twisted a little, the stack shortens elastically, and nothing snaps. A course on the point of clicking would have to turn eighty degrees, every other course would have to turn back, every other course is of the same hand and would be turned away from its own made state toward nothing, and the energy that costs is more than the press can supply before the click.

So the clamped single-handed stack, which was the case the earlier essay worried would lose its clean memory, loses its memory entirely. It has one state, all open, and it is a spring.

Two hands: clicks in pairs #

Alternate the hands — right, left, right, left from the bottom — and the clamp has somewhere to send the turn. A right-handed course clicking turns eighty degrees one way, and a left-handed course clicking turns eighty degrees the other.

The press figure shows what happens. The clamped alternating stack shuts its courses two at a time, at 30.4 thousandths and at 41.3. The first pair is the two tallest courses — the right-handed one made at 1.6 and the left-handed one at 1.75 — and the second pair is the two shortest. Neither course of a pair can click alone, and once the force is enough for both, both go at once and the turn each makes is exactly cancelled by the other’s.

The forces at which the pairs click are not either member’s own. The first pair clicks at 30.4, between the tallest course’s free force of 28.1 and the next one’s 32.5; the second at 41.3, between 37.8 and 44.3. The taller course of each pair waits for the shorter, and the shorter is helped by the taller: neither can turn alone, and the coupled pair has a barrier between the two members’ own.

Only balanced words #

Pressing is half of a register; the other half is pulling, and a word is reachable if some sequence of presses and pulls from all open leaves the stack resting in it. The survey does that for each stack: from every word reached, the force is raised slowly to each value up to a hundred thousandths and lowered back to zero, both ways, until no new word appears.

With free ends the stack reaches all sixteen words, which is the earlier essay’s count. Clamped, every word any stack reaches has as many right-handed courses shut as left-handed. That is the condition for the clicks’ turns to cancel, so every reachable word is a word in which the ends have not moved and every diagonal is at its own length — a stress-free state, held without any torque at the clamps.

Within that condition the stacks do not reach every balanced word. A stack of two of each hand has six balanced words — all open, all shut, and the four ways to shut one course of each hand — and every arrangement tried reaches exactly four of them. Alternating right, left, right, left; right, right, left, left; right, left, left, right: in each, the stack reaches all open, all shut, the two tallest of opposite hands shut, and the two shortest of opposite hands shut. The pairs are fixed by rank within each hand: the tallest right-handed course clicks with the tallest left-handed one, the shortest with the shortest, wherever they sit in the stack.

So a clamped stack of two of each hand is a register of two switches, each switch a pair, and two switches have four words. A stack of one hand has no pairs and reaches one. A stack of three right-handed courses and one left-handed one is not settled: it can only ever pair its one left-handed course with one of the three, and two ways of simulating the slow press disagree about which — one finds a single pairing and two words, the other three pairings and four — so it is left out of the figure and taken up below.

Why the pairs are fixed by rank #

The pairing has a plain reason in the thresholds the earlier essay measured. With free ends the force to shut a course falls the whole way as the course is made taller, so in each hand the tallest course is the one closest to clicking. When the press rises, the first click that can happen is the one whose two members are both closest to their own thresholds, and in each hand that is the tallest. Pulling runs the same way on the stacks measured here, which sit above the peak of the opening force, where taller courses also open first.

The same argument says what would change it. A stack graded below the opening force’s peak opens its shortest courses first; its clamped pairs might form one way pressing and another way pulling, and that is exactly the case in which the earlier essay found a free stack reduced from sixteen words to five. Nothing below measures it, and the reason is given in the section on what the model leaves out.

A register that never turns its ends #

The earlier essay’s suggestion was a stack whose hands are chosen so that every reachable word has zero net twist, which would be a register that does not need free ends at all. Clamping finds that stack automatically, and for every arrangement of hands at once. Clamped, a stack can only reach words of zero net twist, because those are the only words in which a clicked course has been paid for.

What the clamp charges for that is capacity. Four courses free make sixteen lengths; four courses clamped make four, because a pair of courses of opposite hand behaves as one switch. In engineering terms that is a trade between a tube that needs a bearing at each end and stores four bits, and one that can be fixed rigidly at both ends and stores two. The tube that gets built catalogued the tubes engineering makes from folded patterns; a deployable mast or a bellows with fixed end fittings is the clamped case, and the register it can be is the paired one.

Interacting switches, and the ones that cancel #

The free stack was the textbook model of hysteresis built in paper: a population of independent two-state elements, each flipping up at one input and down at another, the model Ferenc Preisach proposed in 1935 for magnetism. The clamped stack is the case that model leaves out, elements that interact, and it is worth saying which kind of interaction this is, because the physics of hysteresis has more than one.

The common kind is a mean field: each element’s thresholds are shifted by how many of the others have flipped, so that flips encourage or discourage further flips and the memory acquires avalanches. That is not what the clamp does. The clamp is a conservation law. The total turn is fixed, each click carries a fixed quantum of turn, and so the only flips allowed are combinations whose quanta add to nothing. A course’s thresholds are not merely shifted by its neighbours; a single course cannot flip at all, at any force tried, unless a partner of the opposite hand flips with it.

That makes the clamped stack a closer relative of a system with a conserved charge than of a magnet. The reachable words are the neutral ones, a pair of opposite hands is the smallest neutral combination, and the count of reachable words is the count of ways the neutral pairs can be switched. Patterns nobody designed found the Kresling pattern as what a twisted cylinder buckles into on its own; under a clamp the same physics that chose the pattern also chooses which of its states can be visited, and it chooses by a rule that makes no reference to energy at all, only to the sum of the turns.

Where the simulation stops #

The stacks are four courses. Six courses of alternating hands were also simulated, and the words they reach do not settle as the force is stepped more finely: at one step size seven words, at half of it six, and among them a word pairing two courses that the four-course rule would not pair. With three pairs, the coupling between pairs evidently matters, and nothing here resolves it. The four-course results do settle, and are reported only because they do.

The stack is the tall one. The courses here are made at 1.3 to 1.75 times their flat-folding height, above the peak of the opening force. Courses made just above the flat-folding height, the earlier essay’s short stack, behave differently in this simulation even with free ends. Pressed by a force rather than held at a height, they flatten past their second state toward a turn of half a revolution, and the free short stack reaches two or three words, depending on how finely the force is stepped, where the earlier essay’s threshold count found five. That is a disagreement between two models of the same course about what happens past the second state, and until it is settled the short stack’s clamped behaviour is not reported.

And the clamp is perfect. A real clamp has some torsional give, and a slightly compliant clamp would allow a single course to click at some higher force, at the cost of storing the turn in the clamp. Between a perfectly rigid clamp and free ends there is a whole family of stacks whose capacity runs from four words to sixteen, and a sheet with one freedom is the reminder that what a structure can do is set as much by what holds its boundary as by its creases.

Bars, not paper #

The model is the bar model every essay on this tube has used: each course a ring of identical sectors, each sector’s two diagonals elastic bars of equal stiffness, its polygon edges rigid and its creases free hinges. The panels do not bend, which is the idealisation a twisted tube has two heights named; a real Kresling tube’s panels bend and buckle, and a clamped stack, whose courses are twisted against their will while the press rises, may relieve that twist by panel bending the bars cannot represent. The force is applied slowly enough that the stack is always at rest, and a click is a jump from a state that has stopped existing to the nearest one that has not.

Simulated, and checked against the free count #

Nothing about thresholds is assumed for the clamped stack. At every force the whole stack — every course’s height and turn — is brought to rest by Newton steps from where it was, with the clamped stack’s turns required to add to what they were as made. A word is reached if a press or a pull, followed by release to zero, leaves the stack resting in it.

The same simulation with the ends free is required to reach all sixteen words of the free stack, which is the earlier essay’s threshold count, made a different way. The clamped stacks are required to reach only balanced words, the stack of one hand only all open, and each stack of two of each hand exactly four words — the same four when the force is stepped twice as finely, which the six-course stack does not pass. The press figure requires the free stack to click four times and the clamped alternating one twice, each pair between its members’ free forces. The clamped stack drawn is required to hold its word with its total turn unchanged and every diagonal at its own length. The turn a click makes is required to be $\pi + \alpha - 2\theta_0$ at every height.

Still open: three pairs, and the short stack #

The unbalanced stack comes first. Three right-handed courses and one left-handed one have three possible pairs, and whether the press picks one of them or can be steered into any depends on exactly the part of the simulation that two methods disagree on: what the stack does in the instant after a course becomes unstable. A dynamics, rather than a sequence of rests, would decide it.

The six-course stack is the next. Its reachable words move with the force step, which means the order in which its pairs click depends on details the four-course stack never exposes: with three pairs, a click in one pair shifts the twist the others feel, and whether they still pair by rank or start to pair across ranks is exactly the interaction the earlier essay asked about. A simulation that follows the stack through each click with a proper dynamics, rather than settling it by steps, would say which.

The short stack is the second. The disagreement between pressing by force and the threshold count is a statement about what a course does past its second state, and that is the model, not the stack. A course whose triangles are allowed to bend, or one held to heights rather than loads, would say which account of the short stack is right; until then its threshold count of five and its simulated two or three are answers to slightly different questions.

Sideways, a corrugation has one resting state found that a corrugation folding without strain has no second well, and the only pattern that moves that rigid motion is the rare case among patterns. The clamped Kresling stack is a third kind of object: a pattern with strain-free states that it cannot reach one at a time, only in combinations that leave its boundary where it was. The cylinder the pattern chooses found that a found pattern chooses its own circumference; here the boundary condition chooses which of the pattern’s states exist at all.

The habit worth carrying is about switches that share a boundary. Before counting a system’s states as a product of its parts, find the quantity each part’s switch changes, and ask whether the boundary lets it change. Here it was a turn, the same for every course; with the ends free it went nowhere and cost nothing, and with the ends clamped it had to be cancelled, and the system’s states became the combinations in which it was.