---
title: "Kresling folds and their stress-free shapes"
description: "A folded tube stays shut if its creases allow a second shape at natural length. Derive that shape, when it exists, and why a 90 degree start has no stiffness."
canonical_url: "https://sandboxphysics.com/stories/geometric-elasticity/kresling-folds-and-stress-free-states/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/geometric-elasticity/kresling-folds-and-stress-free-states/"
last_modified: "2026-10-30"
---

[Geometric elasticity · A course with experiments](https://sandboxphysics.com/stories/geometric-elasticity/)

L3 · Chapter 7

# Kresling folds and their stress-free shapes

A Kresling cell is two polygons joined by triangles, and Yasuda et al. (2017) replace its paper panels with springs along the creases. Each crease length depends on both the height and the twist, so more than one shape can put every crease back at its natural length. Working this out, a second such shape with every crease at its natural length exists at a twist of $\pi-2\theta_{0}$ whenever the height formula gives a real number. Such a cell is bistable; without that shape it springs back, or collapses until its plates touch. At $\theta_{0}=90^{\circ}$ the two shapes merge and the first push meets no stiffness.

Sandbox Physics · 2026-10-30

In this chapter

[All eight chapters](https://sandboxphysics.com/stories/geometric-elasticity/)

Before you start

Chapter 3’s idea that a pushed structure leaves its pusher where the holding force reaches zero; the law of cosines and the identity $\sin^{2}\beta-\sin^{2}\alpha=\sin(\alpha+\beta)\sin(\beta-\alpha)$; and partial derivatives of a function of two variables.

## A tube of springs

Fold a strip of paper into a ring of triangles and close it, and you get a tube that shortens as it twists. Yasuda et al. call it triangulated cylindrical origami. Paper panels warp and wear as the tube folds, so they keep only the creases: $n$ valley creases of length $a$ and $n$ mountain creases of length $b$, each a linear spring of stiffness $k$, between two $n$-sided polygons on a circle of radius $R$.

The cell has two coordinates: the compression $u$ from the starting height $h_{0}$, and the extra twist $\phi$ of the top plate from the starting angle $\theta_{0}$. Each crease is the hypotenuse of a right triangle whose legs are the height and a chord of the circle, which gives Eq. 1 of the paper’s Methods. The stored energy is the sum of the spring energies.

$$
a^{2}=(h_{0}-u)^{2}+4R^{2}\sin^{2}\alpha,\quad b^{2}=(h_{0}-u)^{2}+4R^{2}\sin^{2}\beta,\quad \alpha,\beta=\frac{\theta_{0}+\phi}{2}\mp\frac{\pi}{2n},\quad U=\frac{nk}{2}\bigl[(a-a_{0})^{2}+(b-b_{0})^{2}\bigr]
$$

## The second stress-free shape

A shape with no stored energy needs $a=a_{0}$ and $b=b_{0}$ at once. Subtracting the two crease formulas removes the height: $b^{2}-a^{2}=4R^{2}\sin(\alpha+\beta)\sin(\pi/n)$. This must keep its starting value, so $\sin(\alpha+\beta)$ must too. Besides the start, the only way is $\alpha+\beta=\pi-\theta_{0}$, a total twist of $\pi-\theta_{0}$ and an extra twist $\phi_{2}=\pi-2\theta_{0}$.

Putting this twist back into either crease gives the height $H_{2}$. The shape exists only if $H_{2}^{2}>0$. Zero is the lowest energy the springs can have, so the shape is then a second minimum: the cell is bistable, and geometry alone holds it shut.

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

## Three kinds of cell

Yasuda et al. build cells with $R=90\ \mathrm{mm}$ and $n=5$. With $(h_{0},\theta_{0})=(150\ \mathrm{mm},40^{\circ})$ their energy along the valley has two minima; with $(90\ \mathrm{mm},46^{\circ})$ it only rises. The formula agrees: the first has $H_{2}=49.2\ \mathrm{mm}$, the second $H_{2}^{2}=-10108\ \mathrm{mm^{2}}$, so no second shape.

A third outcome sits outside the formula. With no second shape the energy may still fall all the way down the valley, and then the cell collapses until its plates meet. The springs alone would let the plates pass through each other into a mirror cell, which no real tube can do. The Lab therefore adds an assumption of its own, outside the paper’s truss: a stiff contact that keeps the plates 0.1 R, 9 mm, apart.

## A start with no stiffness

Hold the twist free and push. The cell’s first resistance is the axial stiffness at the start, the ratio of the energy’s Hessian determinant to its twist entry. Writing the crease derivatives out, the numerator is proportional to $(\sin2\alpha_{0}-\sin2\beta_{0})^{2}$, which vanishes when $\alpha_{0}+\beta_{0}=\pi/2$, that is at $\theta_{0}=\pi/2$, whatever $h_{0}$, $R$, $n$ or $k$. Yasuda et al. state that this zero-stiffness mode does not depend on $k$ or $h_{0}$ and give the proof in their supplement.

The second shape explains it. At $\theta_{0}=90^{\circ}$, $\phi_{2}=0$ and $H_{2}=h_{0}$: the two stress-free shapes coincide, and the energy near the start grows as the fourth power of the compression. At $92^{\circ}$ they are 3.2 mm and $4^{\circ}$ apart, and the stiffness is $6.86\times10^{-4}\,nk$, 0.112 % of the monostable cell’s 0.610 nk and 0.080 % of the bistable cell’s 0.853 nk. The paper reports 0.26 % and 0.18 % for its prototypes, which are built with a calibration angle of $9.7^{\circ}$; both pairs of numbers are tiny.

$$
\frac{k_{\mathrm{axial}}}{nk}=\frac{\bigl(a_{u}b_{\phi}-a_{\phi}b_{u}\bigr)^{2}}{a_{\phi}^{2}+b_{\phi}^{2}}\ \propto\ \bigl(\sin2\alpha_{0}-\sin2\beta_{0}\bigr)^{2}=0\quad\text{at}\quad\theta_{0}=\frac{\pi}{2}
$$

## Folding until a sheet meets itself

Diamant and Witten (2011) meet the same ending in a different sheet. Their film floating on a liquid buckles into a fold that deepens as it is pushed. Its exact shape keeps working past the point where the fold touches itself, at a pressure of about 0.040 and an end displacement of about 5.6 in their units, but beyond it the solution passes through itself and is unphysical.

In both cases the elastic energy alone does not know that a sheet cannot cross itself. A model has to add that rule, and where the motion ends then depends on it.

Worked example

## The cell that clicks shut

Take $h_{0}=150\ \mathrm{mm}$, $\theta_{0}=40^{\circ}$, $n=5$, $R=90\ \mathrm{mm}$. Then $\pi/2n=18^{\circ}$, and the creases start at $a_{0}=(150^{2}+32400\sin^{2}2^{\circ})^{1/2}=150.13\ \mathrm{mm}$ and $b_{0}=(150^{2}+32400\sin^{2}38^{\circ})^{1/2}=186.50\ \mathrm{mm}$.

The bracket in $H_{2}^{2}$ is $\sin^{2}2^{\circ}-\cos^{2}38^{\circ}=0.00122-0.62096=-0.61974$, so $H_{2}^{2}=22500-20080=2420\ \mathrm{mm^{2}}$, $H_{2}=49.2\ \mathrm{mm}$, and the top plate turns a further $180^{\circ}-80^{\circ}=100^{\circ}$. At the total twist of $140^{\circ}$, $32400\sin^{2}52^{\circ}+2420=22539=a_{0}^{2}$ and $32400\sin^{2}88^{\circ}+2420=34781=b_{0}^{2}$. In the Lab the gripper lets go at a compression of $0.233\,h_{0}=34.9\ \mathrm{mm}$, where holding would need a pull, and the cell settles at 49.2 mm and $100^{\circ}$.

$$
H_{2}=\bigl(22500-32400\times0.61974\bigr)^{1/2}=49.2\ \mathrm{mm},\qquad \phi_{2}=100^{\circ}
$$

An experiment to think through

## Predict, change, explain

1. Before you press: will a cell with $h_{0}=140\ \mathrm{mm}$ and $\theta_{0}=40^{\circ}$ click shut, spring back or fold flat? Use the condition $H_{2}^{2}>0$.
2. In the Kresling Lab play “Bistable cell clicks shut”, “Monostable cell springs back” and “Almost no stiffness at first”. Then set the height to 140 mm with the twist at 40° and press the top plate yourself.
3. Explain each ending with $H_{2}^{2}$, and why the 92° cell barely resists the first push although it is stiff further in.

[Press a folded Kresling tube](https://sandboxphysics.com/labs/kresling-origami/)

The Kresling Lab treats each crease as a linear spring in the truss model of Yasuda and colleagues and stops the plates on contact 9 mm apart.

## Try explaining it yourself

### 1. For $\theta_{0}=40^{\circ}$, $n=5$ and $R=90\ \mathrm{mm}$, what is the shortest cell that is bistable?

Read an explained answer

Bistability needs $h_{0}^{2}>32400\times0.61974=20080\ \mathrm{mm^{2}}$, so $h_{0}>141.7\ \mathrm{mm}$. The 145 mm preset is just above it, with $H_{2}=30.7\ \mathrm{mm}$; at 140 mm there is no second shape and the Lab cell collapses onto its contact.

### 2. Show that the (90 mm, 46°) cell has no second stress-free shape.

Read an explained answer

$\pi/2n=18^{\circ}$, so the bracket is $\sin^{2}5^{\circ}-\cos^{2}41^{\circ}=0.0076-0.5696=-0.5620$ and $H_{2}^{2}=8100-18208=-10108\ \mathrm{mm^{2}}$. No real height restores both creases, so the energy rises along the valley and the cell springs back.

### 3. Keep $(150\ \mathrm{mm},40^{\circ})$ but use hexagons, $n=6$. Is the cell still bistable?

Read an explained answer

Now $\pi/2n=15^{\circ}$ and the bracket is $\sin^{2}5^{\circ}-\cos^{2}35^{\circ}=-0.6634$, so $H_{2}^{2}=22500-21495=1005\ \mathrm{mm^{2}}$ and $H_{2}=31.7\ \mathrm{mm}$. Still bistable, but it folds flatter, because the bistability threshold rises to 146.6 mm.

## Sources to read alongside

1. [Diamant & Witten, Compression induced folding of a sheet: an integrable system, Physical Review Letters 107, 164302 (2011), arXiv:1107.5505v1](https://arxiv.org/abs/1107.5505v1)

   Read in full: the wrinkle wavelength of a sheet on liquid, the exact fold shapes (Eq. 12), the exactly quadratic fall of the pressure as the sheet is pushed in, and the point where the symmetric fold touches itself, at a pressure of about 0.040 in their units.

2. [Yasuda, Tachi, Lee & Yang, Origami-based tunable truss structures for non-volatile mechanical memory operation, Nature Communications 8, 962 (2017), arXiv:1702.05402v1](https://arxiv.org/abs/1702.05402v1)

   Main text and Methods read: the truss model with its two crease lengths and the stored energy (Methods, Eqs. 1 and 2), the four prototypes (90 mm, 46°), (150 mm, 40°), (140 mm, 92°) and (119 mm, 0°) with a polygon radius of 90 mm and a calibration angle of 9.7°, the zero-stiffness cell with a right-angle starting twist, the stiffness ratios 0.26 % and 0.18 %, and springs of 3.32 kN/m. The supplementary information was not read.

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