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Enneper’s Minimal Surface

What is minimal surface

Minimal surfaces are important in physics, chemistry, and architecture. Minimal surfaces can be regarded as surfaces of minimal surface area for given boundary conditions. For a given closed curve \(C\) in \(R^3\), the problem is to how find the surface \(S\) of least area spanning. In mathematics, minimal surfaces are defined as surfaces with zero mean curvature. In the particular case when \(C\) lies in a two-dimensional plane, the minimal surface is simply the planar region bounded by \(C\). However, the general problem of minimal surface is difficult to solve. The easiest way to physically study minimal surfaces is to look at soap films. In the late nineteenth century, the Belgian physicist Joseph Plateau conducted a number of soap film experiments. He showed that a minimal surface can be obtained in the form of a soap film stretched on a wire framework. The mathematical boundary value problem for minimal surfaces is therefore also called the Plateau problem. A minimal surface parametrized as \(S=(u,v,h(u,v))\) therefore satisfies Lagrange's equation: \[(1+h^2_v)h_{uu}-2h_uh_vh_{uv}+(1+h^2_u)h_{vv}=0\] It is not easy to solve the nonlinear, second-order partial differential equation with specified boundary conditions (determined by the given closed curve \(C\)). The problem of minimal surface is considered part of the calculus of variations. With the aid of a computer, numerical algorithms can be used to find approximate solutions. Enneper’s minimal surface can be parametrized as following: \[\left\{\begin{split} \begin{aligned} x(u, v) &= u( 1 - \frac{1}{3} u^2 + v^2 )\\ y(u, v) &= v( 1 - \frac{1}{3} v ^2 +u^2 )\\ z(u, v) &= u^2 – v^2\\ \end{aligned} \end{split}\right.\] where u and v are coordinates on a circular domain of radius \(R\), i.e. \(u= r cos\theta, v= r sin\theta, 0 \le \theta \le 2\pi, 0\le r \le R\). For \(R \lt 1\) the surface is stable and has a global area minimizer.

An example of Enneper’s minimal surface

In our example, let us take \(R=0.8\), and \(u=0.8*cos\theta\), \(v=0.8*sin\theta\), where \(0\le \theta \le 2\pi\). The boundary of the Enneper’s minimal surface is defined by the functions: \[\left\{\begin{split} \begin{aligned} x(\theta) &= 0.8*cos\theta( 1 - \frac{1}{3}*0.64*(cos\theta)^2 +0.64*(sin\theta)^2 ) \\ y(\theta) &= 0.8*sin\theta( 1 - \frac{1}{3} 0.64*(sin\theta)^2 +0.64*(cos\theta)^2) \\ z(\theta) &= 0.64*cos2\theta \\ \end{aligned} \end{split}\right.\] The shape of the frame we chose is shown in the following picture.
The bubble on our frame is an example of Enneper’s minimal surface, which shape is shown in the following picture.

Why to use this example

This example of the minimal surface was chosen for the following reasons:
  1. The mathematical form of Enneper’s minimal surface is concise and easy to understand. The Enneper’s minimal surface is special because it is a complete minimal surface in \(R^3\) for which the integral of the Gaussian curvature over the whole surface is \(-4\pi\).
  2. Minimal surfaces have wide applications, especially in architecture. Among the surfaces having the same boundary minimal surface is the surface of the least area. Its weight is therefore less and the amount of material is reduced to a minimum. Minimal surface is used for light roof constructions, form-finding models for tents, nets and air halls. In the following picture, the form of the bamboo shell structure takes on the mathematical Enneper’s minimal surface.
    http://www.iaacblog.com/programs/animated-systems-_camboo-pavilion/
In the above example, we investigate Enneper’s surface. This helps us to better understand the concept of minimal surfaces. The 3D print of boundary curve is shown in the following picture:

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