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Minimal surfaces: some pictures


In this document you will find many pictures of minimal surfaces. The program used for the computation of the minimal surfaces is surf (manu-fatto). The 3D rendering engine used is povray. Every picture is available in a high-resolution version

Introduction

Minimal surfaces are intended to be surfaces that minimize their own area. A classical problem is the following (Plateau's Problem): what is the surface which has the least area among all surfaces with a given closed curve as boundary?

A fine thing about minimal surfaces is that they are easily obtained in real-life. In fact, if you construct a closed curve with a wire, and then put it in a soap solution then you will get a soap film (a surface) wich has the wire as boundary. This surface is a minimal surface which solves Plateu's Problem.

What follows is a tracking shot of minimal surfaces.

Minimal surfaces in different homology classes

Looking for minima of area in the class of surfaces with a given boundary, we may restrict ourself in the subclass of surfaces with a given homology. In this way it is sometimes possible to get surfaces which are connected, or orientable, or regular... When we restrict ourself to a given homological class, we get a locally minimal surface, so that also these minima may be obtained as soap films.

For example, if we take two circles as boundary, we get two different local minimizers a disconnected one and a connected one: two disk or the cathenoid.


Considering the following closed curve:
we get two local area minimizers: a non orientable surface (the Möbius band) and an orientable one. Notice that both these surfaces are regular.


You get an example of a singular minimal surface if you consider the following boundary curve:
The global minimizer, in this case, is a surface which has a circumference of singular points (there are three surfaces joining in such points with an angle of 120 degrees):
we get local minimers if we impose the surface to be regular (but not oriented): or oriented:



In the following example we see a minimal surface with infinite genus. Notice that the boundary is again a continuous closed curve.

Parametric surfaces

To give a mathematical form to the Plateau's problem it is necessary to choose a good definition of what a surface is. A possibility is to define a surface as the image of a continuous map over a certain two-dimensional domain (parametric surface).

For example this minimal surface (helicoid) can be parameterized over a disk-type domain.

Notice that with every choice of the domain we can get different solutions.

For example these surfaces have the same boundary. The surface on the left is the least area surface among all parametric surfaces wich have a disk as base domain. While to the right, we have the minimal surface that we obtain as a soap film. Note that the function describing the parametric surface is not injective (the surface intersects itself) while the "true" minimal surface is regular in all points.

Odd examples

It is possible to solve the Plateau's problem on a boundary which is not regular. For example, this is the minimal surface with boundary the edges of a cube. Such surface is also interesting because you can find both types of singular points that can be found in minimal surfaces. In fact there are some points (along 12 segments) of the surface in which three foils meet (with an angle of 120 degrees) and there are four points (the vertices of the square in the center) where four foils meet with an angle of about 108 degrees.

In this example we see a surface which retracts on its boundary.
It is possible (but this is no way intuitive) to deform this surface, always letting the boundary fix, up to let it coincide with its boundary. So, in a certain sense, this surface is not a global minimum of area since it may be deformed to a surface with null area.



This example is maybe only a curiosity. We know that given any closed curve it is possible to find a surface with least area which has this curve as boundary. So, what is the shape of a surface which has as boundary a knotted curve? In this example we see the minimal surface with boundary the trefoil knot, which is the simplest, non trivial, knot.


In the last example we see that it is not always obvious to understand what is the shape of the minimal surface with a given boundary. Here we consider as boundary the union of three disjoint circumferences. The minimal surface obtained is a surface "omeomorphic" to a disk with two holes and a handle.


9 luglio 2003: aggiunta l'elica cilindrica e altre piccole revisioni.
7 aprile 2003: aggiunto il link al codice sorgente.
8 marzo 2003: revisione completa, creazione delle immagini mediante POVRAY.

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Last updated: lun mag 15 2006