---
title: "Non-Euclidean Plate Interactive Lab | Sandbox Physics"
description: "Why does a disc that grows into a curved geometry stay flat when thick but turn into a dome when thin?"
canonical_url: "https://sandboxphysics.com/labs/non-euclidean-plate/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/non-euclidean-plate/"
last_modified: "2026-10-31"
---

Geometric elasticity · Growth and shape

# Non-Euclidean Plate

**Why does a disc that grows into a curved geometry stay flat when thick but turn into a dome when thin?** Growth can ask a sheet for a curved geometry that no flat sheet can have, such as rings that grow too short for their radius. A thick disc stays flat and carries the mismatch as stretching. Below a threshold thickness bending is cheaper, so the disc buckles into a dome with almost no stretching left. The thinner the disc, the closer that dome comes to a spherical cap.

Growth can give a sheet a target geometry that no flat sheet has, here that of a sphere. Grow the disc and watch: a thick one stays flat and stretches, a thin one buckles into a dome that is nearly free of stretching. The buckling thickness, the energies and the dome height are set against the stretch-free cap and the published threshold.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The disc is the axisymmetric non-Euclidean plate of Efrati, Sharon and Kupferman (J. Mech. Phys. Solids, 2009): an annulus of geodesic radii 0.1 to 1.1 whose target metric has constant Gaussian curvature K, with the reduced energy of their Eq. 4.2 and the prefactor set to one, as in their Fig. 4.1. It is 80 radial elements, and each frame of the growth is an equilibrium found by Newton's method. Only shapes with rotational symmetry are allowed, so wrinkles, dimples and saddles of negative curvature are out of reach, and growth is imposed rather than computed. The probe is a stiff spring on the rise of the inner rim; after it lets go, the disc relaxes by overdamped gradient flow, a path to the nearest equilibrium rather than the motion of a real sheet with inertia.

Orange is stretching and blue bending, green their sum, all computed on the disc; cream dashes are the flat disc and the stretch-free cap.

More numbers and the saved record

Stretching content of the flat disc $0.0143$

Bending content of the stretch-free cap $0.541$

Crossover thickness where the two are equal $0.163$

Buckling thickness over the square root of K $0.299$

Elastic energy of the grown disc $2.69\times10^{-5}$

Height of the grown dome $0.541$

Height of the stretch-free cap $0.541$

Radial elements $80$

The JSON record holds the parameters, every equilibrium of the growth and of the thickness sweep, the buckling thresholds, and, kept apart, the flat-disc and stretch-free limits and the published threshold.

[Course: how thin things bend, buckle and snap](https://sandboxphysics.com/stories/geometric-elasticity/)

Physics tutorial

## How does growth make a flat disc curve?

**Background** Leaves, flowers and gel sheets that swell unevenly are given a geometry by their growth rather than a shape. When that geometry is curved, no flat sheet can have it, and the sheet must either stretch or bend away from the plane.

**Why it matters** Every element of the disc is told the same lengths whatever its thickness. Yet a thick disc stays flat and a thin one becomes a dome: thickness decides only which of the two costs, stretching or bending, is cheaper to pay.

Start with the essentials

- **Focus question**: Why does a disc that grows into a curved geometry stay flat when thick but turn into a dome when thin?
- **One-sentence intuition**: Stretching costs energy in proportion to the thickness and bending in proportion to its cube. A thick disc pays for the curved geometry by stretching in the plane; once the disc is thinner than a threshold, bending into a near-isometric dome is cheaper, and the stretching left over falls much faster than the bending as the disc thins.

### Core mathematical model

#### The target geometry

$$
d\bar{s}^{2}=dr^{2}+\Phi(r)^{2}\,d\theta^{2},\qquad \Phi(r)=\frac{\sin(\sqrt{K}\,r)}{\sqrt{K}},\qquad K=-\frac{\Phi''}{\Phi}
$$

Lengths along a radius are kept, but a ring at geodesic radius $r$ should have circumference $2\pi\Phi(r)$, shorter than $2\pi r$. That is the metric of a sphere of radius $1/\sqrt{K}$ (Efrati et al., Sec. 4).

#### Stretching plus bending

$$
E=\int_{r_{0}}^{r_{1}}\bigl(h\,w_{S}+h^{3}\,w_{B}\bigr)\,\Phi\,dr,\qquad w_{S}=\epsilon_{1}^{2}+\epsilon_{2}^{2}+2\nu\,\epsilon_{1}\epsilon_{2}
$$

Efrati et al., Eq. 4.2, for a surface of revolution with radius $\phi(r)$ and slope $\zeta$. The strains $\epsilon_{1}=\phi'^{2}(1+\zeta^{2})-1$ and $\epsilon_{2}=\phi^{2}/\Phi^{2}-1$ measure the mismatch with the target lengths; $w_{B}$ is the bending density. The disc here is the annulus $0.1\le r\le 1.1$.

#### The stretch-free cap

$$
\phi=\Phi,\qquad H_{0}=\frac{\cos\sqrt{K}r_{0}-\cos\sqrt{K}r_{1}}{\sqrt{K}},\qquad E_{B0}=\tfrac{2}{3}(1+\nu)\bigl(\cos\sqrt{K}r_{0}-\cos\sqrt{K}r_{1}\bigr)
$$

A zone of the sphere has exactly the target lengths, so its stretching is zero. Its height is $H_{0}$, and its bending content $E_{B0}$ follows because the bending density is $\tfrac{2}{3}(1+\nu)K$ everywhere on it. For $K=1$ and $\nu=0.5$, $H_{0}=0.541$ and $E_{B0}=0.541$.

#### Two thicknesses

$$
h_{C}=\sqrt{E_{S}^{\infty}/E_{B0}}\approx 0.163,\qquad h_{B}\approx 0.299
$$

Setting the flat disc's stretching $h\,E_{S}^{\infty}$ equal to the cap's bending $h^{3}E_{B0}$ gives the crossover $h_{C}$. Linear stability of the flat disc gives the buckling thickness $h_{B}$; Efrati et al. report about 0.3 and note that $h_{C}$ is smaller.

### Common difficulties

#### Thin does not mean weak here

**Typical misconception** A thin disc buckles because it is too weak to hold its shape.

**Better mental model** Every thickness is asked for the same curved lengths. Thickness only sets the price of bending against stretching, which grows as the thickness squared, so below a threshold the disc chooses to bend.

#### No equal sharing

**Typical misconception** At equilibrium stretching and bending share the energy about equally.

**Better mental model** In the energy chart, as the thickness falls from 0.037 to 0.01, the stretching per thickness falls 27-fold, from $4.1\times10^{-5}$ to $1.5\times10^{-6}$, while the bending part falls only 12-fold, from $5.5\times10^{-4}$ to $4.7\times10^{-5}$. Efrati et al. call this lack of equipartition surprising; it is why very thin sheets behave as if they cannot stretch.

#### What the model leaves out

**Typical misconception** Real leaves and gel discs look exactly like this.

**Better mental model** The disc is the axisymmetric non-Euclidean plate of Efrati, Sharon and Kupferman (J. Mech. Phys. Solids, 2009): an annulus of geodesic radii 0.1 to 1.1 whose target metric has constant Gaussian curvature K, with the reduced energy of their Eq. 4.2 and the prefactor set to one, as in their Fig. 4.1. It is 80 radial elements, and each frame of the growth is an equilibrium found by Newton's method. Only shapes with rotational symmetry are allowed, so wrinkles, dimples and saddles of negative curvature are out of reach, and growth is imposed rather than computed. The probe is a stiff spring on the rise of the inner rim; after it lets go, the disc relaxes by overdamped gradient flow, a path to the nearest equilibrium rather than the motion of a real sheet with inertia.

### Run the experiment

1. 01

   #### Grow a thin disc

   Start from Thin disc grows into a cap and watch one growth cycle.

   **What to observe:** The disc leaves the plane almost at once, at a target curvature of 0.0186, and ends as a dome 0.999 as high as the stretch-free cap. Only 7.2 % of its energy is stretching, and the color stays dark.

2. 02

   #### Make it thick

   Choose Thick disc stays flat, where the thickness is 0.45.

   **What to observe:** The disc grows all the way to K = 1 without leaving the plane and glows with stretching. Its thickness is above the buckling thickness of 0.299, so the flat state stays stable.

3. 03

   #### Sit just past the threshold

   Choose Just past the buckling threshold, with thickness 0.26.

   **What to observe:** The disc stays flat until K = 0.760, then rises gently. At K = 1 the dome is only 0.577 of the cap and stretching still holds 59.7 % of the energy.

4. 04

   #### Turn it inside out

   Choose Cap pops inside out.

   **What to observe:** The probe pushes the inner rim down. The energy climbs to about six times that of the cap, then drops by half at a dome height of about −0.21 as the outer part flips. Let go, and the disc relaxes into the mirror cap with the same energy, $8.48\times10^{-5}$.

5. 05

   #### Push the rim yourself

   Drag the yellow ring at the inner rim down through the disc, then let go.

   **What to observe:** Let go early and the dome springs back; push past the snap and it settles inside out. Your path is drawn in blue on the energy chart.

Physical conclusion

A target geometry that no plane can hold is paid for by stretching in a thick sheet and by bending in a thin one, and below the buckling thickness the bending wins almost completely.

## Related experiments and articles

Geometric elasticity · Reading path

### Geometric elasticity: how thin things bend, buckle and snap

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a thin sheet bends instead of stretching**](https://sandboxphysics.com/stories/geometric-elasticity/why-thin-sheets-bend-instead-of-stretch/)
2. [Lab **Arch Snap-Through** Press a clamped arch until it snaps](https://sandboxphysics.com/labs/arch-snap-through/)
3. [Article **Growth that no flat sheet can hold**](https://sandboxphysics.com/stories/geometric-elasticity/growth-that-no-flat-sheet-can-hold/)
4. [Article **The thickness where a disc buckles**](https://sandboxphysics.com/stories/geometric-elasticity/the-thickness-where-a-disc-buckles/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)