In my previous blog post, I provided a brief history and explanation of the math of minimal surfaces. Now, in this post, I will give a plethora of examples of minimal surfaces as well as illustrate the relationship between minimal surfaces and soap films and demonstrate applications to 3D printing. Some of the examples will include images from Maple where I plotted the surface and colored it by its mean curvature, while other examples were taken from online because the minimal surfaces were too difficult to plot. Overall, this post is meant to be a visual presentation of minimal surfaces.
The first nontrivial minimal surface was discovered by Leonhard Euler in 1744, which is now known as the catenoid (Dillen & Verstraelen, 2000, p.207).
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| The catenoid on the right is colored by its mean curvature. It is a single color because the mean curvature is zero at every point. |
In order to plot the catenoid in Maple, I used the
uv-parameterization that
Weisstein provided on Wolfram Mathworld:
Following the catenoid was the helicoid in 1776, discovered by Jean-Baptiste Meusier (Dillen & Verstraelen, 2000, p.207).
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| The helicoid on the right is colored by its mean curvature. |
I used the following parameterization of the helicoid given in Oprea's book (p.66).
After the helicoid, it was a while before another minimal surface was introduced. In 1834, Heinrich Scherk introduced not only one, but two minimal surfaces. His surfaces are sometimes referred to as Scherk’s First Surface and Scherk’s Second Surface or they are also known as Scherk’s Doubly Periodic Surface and Scherk’s Singly Periodic Surface (Weisstein, Scherk’s minimal surface). By using the parameterization given in Oprea's book (p. 192), I was able to plot a piece of Scherk's Doubly Periodic Surface.
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| This is only one part of Scherk's Doubly Periodic Surface. |
In 1855, Eugѐne Charles Catalan was studying a minimal surface that is now known as Catalan’s minimal surface (Catalan’s surface).
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| On the right, Catalan's surface is colored by its mean curvature. |
I used the following parameterization from Oprea's book (p. 197) to plot Catalan's surface in Maple.
Several years later, in 1861, Edmond Bour introduced another minimal surface, Bour’s minimal surface (Minimal Surface, 2017).
In 1863, Alfred Enneper was studying a minimal surface, which was eventually named Enneper’s surface (Minimal Surface, 2017). The uv-parameterization for Enneper's surface was given in Oprea's book (p.199) as
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| On the right, Enneper's surface is colored by its mean curvature. |
The next minimal surface to be introduced was Riemann’s minimal surface. Like most of the other minimal surfaces, Riemann’s minimal surface was named after its discoverer, and in this case, it was Bernhard Riemann. Riemann’s minimal surface, however, was not published until after Riemann’s death in 1866 (Zeleny, 2014b).
In 1867, Hermann Amandus Schwarz also discovered two minimal surfaces: Schwarz P Surface and Schwarz H surface (Weber, Schwarz P-surface; Weber, Schwarz H-surface).
Henneberg’s minimal surface was discovered by Lebrecht Hennerberg in 1876 (Zeleny, 2014a). The parameterization of this surface was also provided in Oprea's book (p. 197) as
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| Henneberg's surface on the right is colored by its mean curvature. |
In 1904, Herbert William Richmond first described what is now known as Richmond’s minimal surface (Gallery of Minimal Surfaces).
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| Richmond's surface is colored by its mean curvature on the right. |
This surface can also be plotted using polar coordinates. I used the uv-parameterization that is provided on Richmond's Minimal Surface website to obtain the plot above.
Then came the gyroid in 1970. It was a NASA scientist, Alan Schoen who discovered the gyroid as a minimal surface (Weisstein, Gyroid).
Soon after came Costa’s minimal surface in 1982. Celso José da Costa wrote down the equation for this minimal surface, and at that time, many mathematicians did not believe that Costa’s minimal surface existed (Weber, The Costa Surface).
Also, in 1982, the Chen-Gackstatter minimal surface was discovered by two mathematicians, Chi Chen Cheng and Fritz Gackstatter. This minimal surface was derived from Enneper’s surface (Barile).
In 1983, the trinoid was discovered by L.P.M Jorge and W. Meeks III (Weisstein, Trinoid).
Lastly, in 1990, Sven Lidin and Stefan Larsson discovered the lidinoid, which is a minimal surface that resembles the gyroid (The Lidinoid Surface).
More minimal surfaces exist; some of them are different versions of the ones above. For example, the Costa-Hoffman-Meeks surfaces are generalizations of the Costa's surface. This Virtual Math Museum has a spectacular display of various minimal surfaces for those who are interested in exploring more.
The Relationship Between Minimal Surfaces and Soap Films
The soap films created by submerging some type of frame, usually a wire frame, into soap solution are minimal surfaces. In the second chapter of Explorations in Complex Analysis, Michael Dorff further elaborated on the connection between minimal surfaces and soap films by describing the Steiner Problem. Imagine that four houses form a square:
A road with the least length has to be constructed to connect the four houses. How should the road be built? These are some possible solutions:
The correct solution has a length of 2.7 miles:
As Dorff stated in his paper, the Steiner Problem minimizes distance, a 1-dimensional object, in a plane, which is a 2-dimensional object; meanwhile, soap films minimize area, a 2-dimensional object, in space, which is 3-dimensional. Therefore, if you dip a cube in soap solution, the outcome of the soap film is going to look like the solution to the Steiner Problem given above.
As a force pulls molecules toward the center of water, the force creates surface tension, which is what minimizes surface area of the shape. Because soap solution has a lower surface tension than water, it allows soap films to form, creating a minimal surface. Here are some more examples of soap films as minimal surfaces:
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| A helicoid |
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| A part of Scherk's Doubly Periodic Surface. |
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| It was difficult to capture a good picture of the catenoid, so the image on the right is from soapbubbledk. |
Applications to 3D Printing
Viewing minimal surfaces on paper or on a computer is a completely different learning experience than having the minimal surface in your own hands. I found it more exciting to study minimal surfaces by examining soap films and by 3D printing them. Some minimal surfaces are not difficult to 3D print, such as the helicoid. In Mathematica, I simply plotted the helicoid using its uv-parameterization, and I exported it as an STL file. The 3D printed helicoid is shown below:
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| The supports are still attached to the helicoid. |
In Tinkercad, I was able to assemble a helix with a handle so that it can be used to create the helicoid soap film. This process was also not difficult. I first adjusted the built-in helix to ensure that the helix was thick enough. I did the same to another helix and grouped the helicies together to obtain one big helix. To make the handle, I significantly stretched and slimmed down a cylinder and I attached the handle to the helix by using another short and slimmed down cylinder. The final result is displayed below:
Unfortunately, I did not have enough time this summer to 3D print more minimal surfaces, but there are plenty of beautiful examples online:
In making this post, I realized that minimal surfaces are a form of art. Not only is the math fascinating, but minimal surfaces as soap films and 3D printed minimal surfaces are exquisite in their own way.
References
Barile, M. Chen-Gackstatter surfaces. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/Chen-GackstatterSurfaces.html
Brilleslyper, M. A., Dorff, M. J., Mcdougall, J. M., Rolf, J. S., & Schaubroeck, L. E. (2012). Explorations in complex analysis. Washington D.C: Mathematical Association of America.
Catalan’s surface. Wolfram Demonstrations Project. Retrieved from https://www.wolframcloud.com/objects/demonstrations/CatalansSurface-source.nb
Dillen, F. J. E., & Verstraelen, L. C. A. (2000). Handbook of differential geometry, volume 1. Amsterdam, The Netherlands: North Holland.
Gallery of Minimal Surfaces. Retrieved from http://profs.scienze.univr.it/~baldo/tjs/
Minimal Surface. (2017). Retrieved August 6, 2019, from https://www.mathcurve.com/surfaces.gb/minimale/minimale.shtml
Oprea, J. (2000). The mathematics of soap films: explorations with Maple. Providence, RI: American Mathematical Society.
Richmond's minimal surface. Retrieved from https://www.math.hmc.edu/~gu/curves_and_surfaces/surfaces/richmond.html
The lidinoid surface. The Scientific Graphics Project. Retrieved from http://www.msri.org/publications/sgp/jim/papers/morphbysymmetry/lidinoid/index.html
Weber, M. Schwarz H-surfaces. [Web blog post]. Minimal surfaces. Retrieved from https://minimalsurfaces.blog/home/repository/triply-periodic/schwarz-h-surfaces/
Weber, M. Schwarz P-surface. [Web blog post]. Minimal surfaces. Retrieved from https://minimalsurfaces.blog/home/repository/triply-periodic/schwarz-p-surface/
Weber, M. The Costa surface. [Web blog post]. Minimal surfaces. Retrieved from https://minimalsurfaces.blog/home/repository/tori/the-costa-surface/
Weisstein, E. W. Catenoid. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/Catenoid.html
Weisstein, E. W. Gyroid. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/Gyroid.html
Weisstein, E. W. Scherk’s minimal surface. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/ScherksMinimalSurfaces.html
Weisstein, E. W. Trinoid. Wolfram Mathworld. Retrieved from http://mathworld.wolfram.com/Trinoid.html
Zeleny, E. (2014a). Henneberg’s minimal surface. Wolfram Demonstrations Project. Retrieved from http://demonstrations.wolfram.com/HennebergsMinimalSurface/
Zeleny, E. (2014b). Riemann’s minimal surface. Wolfram Demonstrations Project. Retrieved from http://demonstrations.wolfram.com/RiemannsMinimalSurface/
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