Geometric elasticity · Origami mechanics

Kresling Origami Cell

Why does a twisted origami tube click shut when pressed, and stay shut? A Kresling cell is a paper tube folded into slanted triangles, so pressing it down also twists it and stretches its creases. If the cell has a second shape in which every crease is back at its natural length, it pulls itself into that folded shape and stays there. Otherwise it springs back, or folds flat until its plates touch. A cell started at a right-angle twist barely resists the first push.

A Kresling cell is a tube folded into slanted triangles, so pressing it also twists it. Press the top plate: a bistable cell resists, then pulls itself into a folded state where no crease is stretched, while a monostable cell springs back. The rest height and twist are set against the closed-form second state.

Shapes in which no crease is stretched $2$

Height where the pressed cell comes to rest $49.2\,\mathrm{mm}$

Twist of the top plate at rest $100^{\circ}$

Stiffness against the first push $0.853\,nk$

Model assumptions

The cell is the truss model of Yasuda, Tachi, Lee and Yang (Nat. Commun. 8, 962, 2017): two regular pentagons of circumradius 90 mm joined by ten creases, each a linear spring of constant k, with the energy of their Methods Eqs. 1 and 2 and no calibration angle. The panels carry no load; they are drawn only to make the folding readable. The top plate has unit mass, the moment of inertia of a uniform disc and viscous damping, all assumptions of this Lab, so times are in units of the square root of mass over k, and a stiff contact stops the plates 9 mm apart; a cell whose energy keeps falling as it shortens rests against that contact. The paper reports the zero-stiffness cell 0.26 % and 0.18 % as stiff as the monostable and bistable cells; this model gives 0.11 % and 0.080 %. The paper's fourth prototype, which bifurcates, is left out: without the calibration angle its untwisted branch stays stable in this model.

Bright is high energy. Green is the valley, the lowest energy at each compression; yellow is the path the cell took, blue your own push; cream rings are the closed-form stress-free states.

More numbers and the saved record

Height of the second stress-free state, closed form

49.2\,\mathrm{mm}

Twist of the second stress-free state, closed form

100^{\circ}

Compression at which the gripper lets go

0.233\,h_{0}

Highest stored energy on the way in

0.0221\,k h_{0}^{2}

Stiffness against the first push with the twist held

1.65\,nk

Rest length of a valley crease

150\,\mathrm{mm}

Rest length of a mountain crease

186\,\mathrm{mm}

Damping ratio of the top plate

0.1

The JSON record holds the parameters, the simulated path with the gripper force, the energy valley, and, kept apart, the closed-form second state and stiffness and the published prototypes.

Physics tutorial

How can a folded tube have two resting shapes?

BackgroundKresling folding turns a paper tube into a ring of slanted triangles. Pressed along its axis, the tube shortens and twists at the same time, which is why it is used in deployable structures, crawling robots and mechanical memory cells.

Why it mattersEvery crease starts at its natural length, and pressing the cell stretches some of them. Whether the cell clicks shut or springs back depends only on the geometry: is there a second shape in which every crease is at its natural length again?

Start with the essentials

Focus question: Why does a twisted origami tube click shut when pressed, and stay shut?
One-sentence intuition: The ten creases have only two different lengths, and two equations fix two unknowns, height and twist. So a cell either has a second stress-free shape or none. If it has one, the stored energy rises on the way in and falls back to zero there, and the cell clicks into it and stays. If it has none, the cell springs back unless its energy keeps falling until the plates touch. Where that second shape merges with the first, the first push meets almost no resistance.

Core mathematical model

Crease lengths and stored energy

a^{2}=H^{2}+4R^{2}\sin^{2}\!\Bigl(\frac{\theta}{2}-\frac{\pi}{2n}\Bigr),\quad b^{2}=H^{2}+4R^{2}\sin^{2}\!\Bigl(\frac{\theta}{2}+\frac{\pi}{2n}\Bigr),\quad U=\frac{nk}{2}\bigl[(a-a_{0})^{2}+(b-b_{0})^{2}\bigr]

With height $H=h_{0}-u$ and twist $\theta=\theta_{0}+\varphi$ , the valley crease $a$ and the mountain crease $b$ follow from the two polygons of circumradius $R$ (Yasuda et al., Methods, Eqs. 1 and 2, without the calibration angle). Each of the $2n$ creases is a spring of constant $k$ .

The second stress-free state

\varphi_{2}=\pi-2\theta_{0},\qquad H_{2}^{2}=h_{0}^{2}+4R^{2}\Bigl[\sin^{2}\!\Bigl(\frac{\theta_{0}}{2}-\frac{\pi}{2n}\Bigr)-\cos^{2}\!\Bigl(\frac{\theta_{0}}{2}+\frac{\pi}{2n}\Bigr)\Bigr]

Because $b^{2}-a^{2}=4R^{2}\sin\theta\,\sin(\pi/n)$ , both creases return to their rest lengths only where $\sin\theta=\sin\theta_{0}$ , which gives the second twist; $a=a_{0}$ then gives the height. The state exists when $H_{2}^{2}>0$ . For $h_{0}=150\,\mathrm{mm}$ and $\theta_{0}=40^{\circ}$ it is $H_{2}=49.2\,\mathrm{mm}$ at $\varphi_{2}=100^{\circ}$ .

Stiffness against the first push

k_{\mathrm{ax}}=nk\,\frac{\bigl(a_{u}b_{\varphi}-a_{\varphi}b_{u}\bigr)^{2}}{a_{\varphi}^{2}+b_{\varphi}^{2}},\qquad k_{\mathrm{ax}}=0\iff\theta_{0}=\frac{\pi}{2}

Letting the twist relax, the stiffness is the determinant of the energy's Hessian over its twist entry; subscripts are derivatives at the rest state. The bracket vanishes when $\sin 2\alpha_{0}=\sin 2\beta_{0}$ , that is at $\theta_{0}=\pi/2$ for any height, radius, side count or $k$ , as the paper states. There the second state coincides with the first and the energy grows as the fourth power of the compression.

Common difficulties

Nothing latches

Typical misconceptionThe folded cell stays shut because the creases lock or friction holds it.

Better mental modelNothing in this model locks and there is no friction. The folded state is a second geometry in which every crease is at its natural length, so no stored energy is left to push the cell open.

Zero stiffness is not zero strength

Typical misconceptionA cell with zero initial stiffness simply collapses.

Better mental modelOnly the first slope of the force vanishes. Near the start the energy grows as the fourth power of the compression instead of the square, and further in it rises steeply, so the cell still holds a load once it is pressed in.

What the model leaves out

Typical misconceptionReal paper cells behave exactly like this truss.

Better mental modelThe cell is the truss model of Yasuda, Tachi, Lee and Yang (Nat. Commun. 8, 962, 2017): two regular pentagons of circumradius 90 mm joined by ten creases, each a linear spring of constant k, with the energy of their Methods Eqs. 1 and 2 and no calibration angle. The panels carry no load; they are drawn only to make the folding readable. The top plate has unit mass, the moment of inertia of a uniform disc and viscous damping, all assumptions of this Lab, so times are in units of the square root of mass over k, and a stiff contact stops the plates 9 mm apart; a cell whose energy keeps falling as it shortens rests against that contact. The paper reports the zero-stiffness cell 0.26 % and 0.18 % as stiff as the monostable and bistable cells; this model gives 0.11 % and 0.080 %. The paper's fourth prototype, which bifurcates, is left out: without the calibration angle its untwisted branch stays stable in this model.

Run the experiment

01
Watch a cell click shut
Start from Bistable cell clicks shut and watch one cycle.
What to observe: The gripper presses slowly while the top plate twists. At a compression of $0.233\,h_{0}$ holding would need a pull, so the gripper lets go and the cell pulls itself in. It comes to rest at 49.2 mm and 100°, the closed-form second state, with every crease back at its natural length.
02
Try a cell that cannot stay folded
Choose Monostable cell springs back, 90 mm tall with a 46° starting twist.
What to observe: The gripper presses all the way to $0.4\,h_{0}$ and the cell never pulls itself in. Released, it springs back to 90 mm: no second stress-free shape exists, so the energy only rises along the valley.
03
Find the soft start
Choose Almost no stiffness at first, with a 92° starting twist.
What to observe: The first push meets a stiffness of only $6.86\times10^{-4}\,nk$ , 0.080 % of the bistable cell's. The second stress-free state sits just 3.2 mm above the start and 4° away; at a 90° start the two merge.
04
Fold it almost flat
Choose Barely bistable: folds almost flat, 145 mm tall.
What to observe: Five millimetres less height moves the second state from 49.2 mm down to 30.7 mm. The cell still clicks shut and stays there, now much flatter.
05
Find a third ending
Choose Folds flat against itself, 140 mm tall with a 30° starting twist.
What to observe: This cell has no second stress-free shape, yet it does not spring back. Its energy keeps falling as it shortens, so it collapses until the plates touch 9 mm apart and stays pressed there, its mountain creases still stretched by 5.3 %.
06
Press it yourself
Drag the yellow handle on the top plate down, then let go.
What to observe: Let go early and the cell springs back; press past the top of the energy valley and it pulls itself shut. Your path is drawn in blue on the energy map.

Kresling Origami Cell

How can a folded tube have two resting shapes?

Core mathematical model

Crease lengths and stored energy

The second stress-free state

Stiffness against the first push

Common difficulties

Nothing latches

Zero stiffness is not zero strength

What the model leaves out

Run the experiment

Watch a cell click shut

Try a cell that cannot stay folded

Find the soft start

Fold it almost flat

Find a third ending

Press it yourself

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