A twisted tube has two heights

Patterns nobody designed began with a thin cylinder crushed along its axis, folding itself into a lattice of diamonds that came out of the buckling rather than out of anybody’s choice. Twist the same cylinder about its axis instead of crushing it and it does something just as orderly. It falls into a helix of slanted triangles, each course of the tube a ring of them, and the top of each course turned against the bottom. Folded in paper on purpose, the pattern is named after Biruta Kresling, who studied it in twisted paper tubes and in the folding of plants from the 1990s. A Kresling tube has a property the Yoshimura’s does not: pushed shut, it stays shut. Pulled open, it stays open. It clicks.

The usual description of that click is a mechanism with two positions. A mechanism that closes on itself lists the Kresling among closed sheets that have “a rigid or near-rigid motion”, and treats the twist on collapse as what the closure demands. The geometry of one course says otherwise. A Kresling course has no motion at all. It has exactly two shapes in which every crease is its own length, the closed form says where both are, and every shape between them stretches some creases and shortens others. The click is not a fold. It is the paper being forced through a strained state between two unstrained ones, and the strain is what holds each end.

Two numbers and two lengths #

A course is two regular polygons with $n$ sides and circumradius $R$, one above the other at height $h$, the top turned by $\theta$ about the axis, and the gap between them filled by $2n$ triangles. Every triangle has one polygon edge as a side and two diagonals as its others. The long diagonal joins a bottom corner to the top corner turned $\theta$ from it. The short one joins the next bottom corner to the same top corner, turned $\theta - \alpha$ from it, where $\alpha = 2\pi/n$ is one sector. The polygon edges are fixed by $R$, so with every sector behaving alike the course has exactly two numbers, $h$ and $\theta$. The lengths of the diagonals follow from them:

$$b^2 = h^2 + 4R^2 \sin^2\frac{\theta}{2}, \qquad c^2 = h^2 + 4R^2 \sin^2\frac{\theta - \alpha}{2}.$$

That is two numbers held to two lengths. A count of freedoms would expect the creases to pin the course to isolated states rather than to a path, and they do. That is the opposite of a sheet with one freedom, where the Miura’s creases leave exactly one way to move. The interesting part is that the isolated states can be written down. Subtract the two equations and the height cancels:

$$b^2 - c^2 = 4R^2 \sin\frac{\alpha}{2}\,\sin\!\left(\theta - \frac{\alpha}{2}\right).$$

So any state keeping both diagonals at their lengths has the same value of $\sin(\theta - \alpha/2)$ as the state it was made in. There are two turns with that sine: the one it was made at, $\theta_0$, and $\theta_1 = \pi + \alpha - \theta_0$. The height of the second follows from $b$:

$$h_1^2 = h_0^2 - H^2, \qquad H^2 = 4R^2 \cos\frac{\alpha}{2}\,\cos\!\left(\theta_0 - \frac{\alpha}{2}\right).$$

That is the whole theory of the click, and it has a threshold in it. $H$ is the height of the course with the turn $\theta_0$ that folds exactly flat: put $h_0 = H$ and the second state has no height at all. A course made taller than $H$ has a second state at $\sqrt{h_0^2 - H^2}$, short of flat, with nothing strained. A course made shorter has no second state in which the creases are at rest.

The tube in the opening figure is the six-sided course turned 80 degrees as made, which has $H = 1.49$ radii, built 1.2 times taller at 1.79. The formula puts its second state at 0.99 radii and a turn of 160 degrees. Both drawings are placed from the formula, and both diagonals in the second come out at their lengths in the first to nine decimal places.

The strip is the same strip #

What a folder actually holds is not the tube but the flat strip it is cut from, and the strip makes the point more plainly than the tube does. Unroll one course and it becomes a row of $2n$ triangles between two straight edges, each triangle with a polygon edge, a long diagonal and a short one for sides.

Both states in the opening figure are the upper strip, closed. A strip has three side lengths and so has one shape, and that shape closes into a tube in two ways: tall and only slightly turned, or short and turned twice as far. Nothing in the strip prefers either. The preference is decided entirely by where it was creased: a sheet folded on a tall former starts in the tall state, and a sheet folded flat and opened starts in the short one.

The lower strip is the reason the Kresling belongs beside the Yoshimura rather than among the deployables. A course of the Yoshimura is the Kresling course turned by exactly half a sector, where its top corners sit over the middles of the bottom edges. There the two diagonals are equal and the strip is a row of isosceles triangles. The cylinder the pattern chooses found that the Yoshimura’s columns close into a polygon whose size the paper does not choose. This is the same family seen from its other free number. The Yoshimura is the member with no lean. The Kresling is every member with one, and what the lean buys is the second state.

What pressing the course costs #

A state between the two has no shape in which both diagonals are at their lengths, so it has to strain them. The usual model makes each diagonal a bar of the same stiffness, storing $\tfrac12 \ell_0 \varepsilon^2$ for a strain $\varepsilon$ on a rest length $\ell_0$, and lets the course turn freely at every height to whatever turn stores least. Pressing the course from its own height toward flat then traces one curve of energy against height. The dial moves how tall the course was made, in multiples of $H$.

At 1.2 times the threshold the curve does what the closed form promised. It starts at zero, rises to a barrier at 1.47 radii, and falls back to exactly zero at 0.99. Past that it rises steeply, because pushing the course on toward flat strains both diagonals again. The barrier is what makes each end stable: a course at either height has to be pushed over it to reach the other.

Drag the dial below 1 and the second zero becomes a well that does not reach zero. At 0.85 the course still has a barrier, at 0.78 radii, and past it the energy falls to a well at flat, with strain left over in the creases. The course clicks flat and stays there, held by strain it cannot release. This is the regime the closed form does not see, because it only asks where the strain is zero. Drag further and the well at flat disappears. Pressed flat, the course springs back.

The edge of that middle regime also has a closed form, though a different kind. At the relaxed flat state, whether the energy curves up or down as the course is lifted is the sign of $\varepsilon_b/\ell_b + \varepsilon_c/\ell_c$. Whether the turn is in balance is the sign of $\varepsilon_b \cos(\theta/2) + \varepsilon_c\cos((\theta-\alpha)/2)$. Both can vanish together with nonzero strains only where $\sin(\theta - \alpha) = \sin\theta$, which is a turn of $(\pi + \alpha)/2$. At that turn the condition reads

$$\frac{1}{b_0} + \frac{1}{c_0} = \frac{1}{2R\sin\frac{\pi + \alpha}{4}} + \frac{1}{2R\sin\frac{\pi - \alpha}{4}},$$

and solving it for the height gives the lowest course that stays flat: 0.98 radii for the six-sided course turned 80 degrees, against an $H$ of 1.49.

The two lines are different kinds of statement, and the difference matters to anybody using them. $H$ is geometry. It asks only whether a shape exists with every crease at its length, and it holds whatever the creases are made of. The held-flat line is mechanics. It assumes the two diagonals are equally stiff, and a material whose short diagonal was stiffer than its long would move it. The first is a fact about the pattern, the second a fact about the pattern in a particular material.

Where on the strain the click happens #

The energy curve hides what the two diagonals are doing, and they are doing opposite things.

The long diagonal is stretched and the short one compressed at every height between the two states, and neither changes sign. At the top of the barrier the long diagonal is 3.11 per cent over length and the short one 2.44 per cent under. The short one’s worst, 2.60 per cent, comes a little earlier in the press. The course, free to turn, splits the misfit between the two diagonals in whatever ratio stores least. It turns from 80 degrees at the start to 120 at the barrier and 160 at the end, and the turn is how it does the splitting.

For paper those numbers are large. Paper tears at a few per cent in tension, and it does not shorten by 2.6 per cent along a line without buckling out of its plane. A paper Kresling tube does not do what the bar model says, and it cannot. The triangles bend out of their own planes, so the panels take up the misfit by curving rather than the diagonals by stretching. The bar model is the standard account of this tube for the same reason the barrier is real: it gives the right number of stable states and puts them in the right places, and its strains say how much misfit the panels have to absorb, not how they absorb it. A real tube’s barrier is lower than the bars say and its clicks softer.

Which designs click at all #

Every point in the plane of turn as made against height as made is a design, and the two closed-form curves divide that plane into the three behaviours the dial showed. The dots are designs sampled across it, each classified by pressing it and reading its own energy curve rather than by the formulas.

All 117 sampled designs fall on the side of the curves they should. The vertical line at one sector is a third boundary, and it is not about strain at all. The second state’s turn is $\pi + \alpha - \theta_0$, so a course made at less than one sector of turn has a second state turned by more than half a turn. A triangle with corners at angles $0$, $\alpha$ and more than $\pi$ about the axis contains the axis, and by the symmetry every triangle of its kind meets the axis at the same height. The panels would have to pass through one another. A brute-force test of every pair of triangles agrees with that criterion on 1,260 generic designs of four to ten sides, with no exceptions. So the second state exists on the page and not in the tube for every course turned less than one sector, and the clicking designs are a band between one sector and a quarter turn plus half a sector, beneath the curve $H$.

The top of the band falls to nothing at a quarter turn plus half a sector, 120 degrees for six sides. There $H$ is zero: a course turned that far folds flat from any height, because its second state is already flat-folded. The band is widest near one sector, where a course has to be tallest, about 1.7 radii for six sides, before it clicks to a second height rather than to flat.

The stroke is paid for in strain #

A designer of the tubes that get built, a bellows or a mast, wants two things from a click: a long stroke, so that the closed tube is much shorter than the open one, and a small barrier, so that the creases survive being clicked. The closed form says how the stroke depends on the height as made. As a share of the course’s height it is $1 - \sqrt{1 - (H/h_0)^2}$, which is all of it at $h_0 = H$, 45 per cent at 1.2 times $H$, 13 per cent at twice $H$ and 3 per cent at four times.

Every curve rises with the stroke, and the turn as made sets how steeply. A course turned 70 degrees, near the bottom of the band, needs 5.9 per cent strain for a stroke all the way to flat. Turned 80 degrees it needs 4.2, at 90 degrees 2.6, and at 100 degrees 1.3. A small stroke is cheap on all of them, and that is the trap in the design: the cheap clicks are the ones that barely shorten the tube.

The turn, read off the band, buys strain at the price of height. A course turned 100 degrees clicks with a fifth of the strain of one turned 70, but its flat-folding height is 1.09 radii against 1.63, so a tube of the same length needs half as many courses again. That trade between a short, gentle course and a tall, stiff one is invisible in the strip. Its proportions are a few lengths, and the bistable band, the barrier and the stroke all follow from them.

Pressed straight, the symmetric course twists #

The strip figure put the Yoshimura’s course inside the family, as the member with no lean. Its own second state is turned by $\pi + \alpha/2$, more than half a turn, so it lies through the axis. The Yoshimura’s course has no second height a tube can reach. What it does have shows up when it is pressed straight down, the way the crushed can was, and left free to turn.

It keeps its turn while it is shallowly pressed, because both diagonals are equal and shortening them equally is the cheapest response. At 1.23 radii that stops being true. A course that turns one way lengthens one diagonal and shortens the other, and past that height turning stores less than squeezing both. So the symmetric course chooses a hand, in either direction with equal right. Pressed further, it follows its chosen branch toward a strained flat state turned about 154 degrees.

That is the Kresling appearing from inside the Yoshimura. The twist a Kresling course is made with is the twist a symmetric course falls into when it is pressed hard enough, and the handedness is decided by whichever way a small imperfection tips it. The numbers say why one is a click and the other a crush. The worst strain on the way is 23 per cent, against the Kresling design’s 3.1, because the symmetric course has to pay to break its own symmetry before it can pay to move. Paper does not survive 23 per cent. It creases, and the extra creases are the crumpling a crumple has no tail counted, which is why a crushed can stays crushed and a Kresling tube can be clicked again.

Hunt and Ario put the twist of a crushed cylinder and the Kresling pattern side by side in 2005, under the title “Twist buckling and the foldable cylinder: an exercise in origami”. Guest and Pellegrino’s papers on the folding of triangulated cylinders, from 1994, had worked out the geometry of the twisted course and found that it folds only by deforming its panels. What the closed forms add is the boundary of that statement, drawn on the design plane.

What the picture cannot show #

Whether a real tube is bistable at the height the band gives. The band is drawn for bars of equal stiffness and rigid polygons, and a paper course is neither: its triangles bend, its creases have a stiffness of their own that the model leaves out, and a held fold slowly changes what that stiffness is aimed at, and its polygon edges are creases that can open. The stress-free curve $H$ survives all of that, because it is a statement about lengths. The held-flat curve and every strain figure do not, and a paper tube will click with less effort and hold less firmly than the bars say.

How several courses share a press. Each course here is drawn and pressed alone. A tube of many courses pressed at its ends clicks one course at a time rather than all at once, because each course’s barrier is a maximum and the course that reaches it first goes first. The order in which a stack clicks, and whether courses of opposite hand cancel the tube’s net twist, is a question about the stack that one course cannot answer.

Contact on the way. The admissibility test asks whether panels pass through each other in the two end states. It does not follow the whole press, and a design near the edge of the band could meet itself on the way between two states that are both clear.

The idealisations underneath #

Every sector behaves alike, and the polygons stay regular. That is the assumption that makes a state two numbers, and it is the standard one. A course free to lose its symmetry has more ways to move and could find a cheaper path between the two states. The barrier drawn here is therefore an upper bound on the barrier of a symmetric press, and the two end states are exact.

The diagonals are bars of one stiffness storing $\tfrac12\ell_0\varepsilon^2$, and nothing else stores energy. Folding the creases costs nothing, the polygon edges are rigid, and the triangles are flat. Everything is measured in units of the polygon’s circumradius.

How the claims were checked #

The second state is placed from the closed form and checked by recomputing both diagonals there: they agree with the design’s to $10^{-9}$, and a test of every pair of triangles finds none passing through another in either state.

The three behaviours are read from each design’s own energy curve, computed by pressing it in 260 steps and minimising over the turn at every height by a scan and a golden-section search. A design counts as having a second state if the closed form gives one, as held flat if its curve has a maximum above its flat end, and as springing back otherwise. Every one of 117 designs across the six-sided band lands where the two closed-form curves put it. The held-flat curve is also found by bisecting on the energy curves directly, at three turns, and it agrees with the closed form to within the step of the press.

The admissibility rule, that a second state is reachable only from a turn of more than one sector, was checked against a brute-force test of every pair of triangles on 1,260 generic designs of four to ten sides, with no disagreement.

The symmetric course is pressed with its turn searched only on one side of half a sector, and its mirror is drawn by reflection. The straight path it leaves is checked to store more energy from the turning height on.

Still open: the order a stack clicks in #

A Kresling tube is a stack of courses, and pressing the stack is not pressing a course. Each course’s barrier is a maximum of its own energy curve, so under a slowly rising load the stack clicks one course at a time, and which goes first is decided by imperfections, as the symmetric course’s hand was. A stack whose courses differ slightly in height as made would click in a definite order, tallest-barrier last, and that order is a design variable. It would turn a tube that clicks as a whole into one that steps through a sequence of lengths, which is what a deployable mast or a programmable bellows would want. The energy of a stack is the sum of its courses’ curves at a shared load, and finding the order is a calculation rather than a search.

The other direction is the panel bending the bars leave out. Replacing each triangle’s stretch with a bend along its long diagonal is the next model up, and it would say how much of the 3.1 per cent the paper’s own bending absorbs. Whether the band moves when it does is a question about the held-flat curve only, since $H$ cannot move. Sideways, a corrugation has one resting state found that a corrugation folding without strain has no second well to rest in, and the only pattern that moves found that a rigid motion is the rare case among patterns. The Kresling is the clean example of the other case: a pattern that cannot fold without strain, and is useful because it cannot.

The habit worth carrying is about the word mechanism. Before describing a structure’s two states as positions of a motion, count its freedoms against its constraints. Two numbers held to two lengths is a structure with isolated states, and the path between them is paid for. Here that payment is the click, and it explains why the click holds.