## Minimal Yet Interesting Surfaces

Some days you just run across a shape you never heard of before and that’s interesting. Matthias Weber of The Inner Frame gave me one last night. In a string of essays Weber shows a figure which comes up from minimal surface theory. This is a study of making a shape that fits to some given boundary while keeping a property called “mean curvature” equal to zero. This is how mathematicians make it sound all academic when they talk about soap bubbles in wire frames.

This is from a particular kind of surface developed in the 1860s by Alfred Enneper, whom I admit I never heard of before either. It’s just outside my specialty. But he was a student of Peter Gustav Lejeune Dirichlet, who’s just all *over* partial differential equations and Fourier series. Enneper and Karl Weierstrauss — whose name is all over analysis — described a way to describe these surfaces, using differential geometry. Once again I’m sad I don’t know that field more, as it produces such compelling pictures.

Here Weber introduces the surface, complete with a craft project! If you’d like you can cut out and fit together a wonderful exotic little surface. The second essay looking at some shapes with similar properties, and at what you get by stacking these surfaces. The third part extends this even farther, to the part of mathematics that’s just Googie architecture. I hope you enjoy.

## KnotTheorist 7:25 pm

onTuesday, 21 June, 2016 Permalink |What fun and funky surfaces!

By the way, since you mentioned it, what is your area of specialty?

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## Joseph Nebus 9:14 pm

onTuesday, 21 June, 2016 Permalink |Glad you like it.

My particular specialty was in Monte Carlo methods, which are numerical techniques for finding quite good approximations to a solution. And particularly, in using them to find equilibriums of a viscosity-free fluid’s flow. You can treat planetary atmospheres as viscosity-free for some problems without making an insufferably large error.

I came to that after the failure of a project in graph theory, so I’m more conversant with that than my thesis and scattered few papers would suggest.

But I came to the fluid flow problem by accident, basically. My advisor was particularly interested in it, while I hadn’t given planetary atmospheres much thought. So I have a remarkably scattershot knowledge of my own specialty! … Or more fairly, my first area of specialty is Monte Carlo methods, with graph theory a secondary interest, and fluid mechanics something I plundered for my own purposes. And I keep wanting to know mechanics better.

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## xianyouhoule 7:16 am

onSunday, 26 June, 2016 Permalink |Thanks for your share,I have learned some great things from your posts.

What’s application of minimal surface theory??I don’t hear about Its any application before.

and I am curious about Monte Carlo method(I don’t even hear about it ),What’s its core idea and application??

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## Joseph Nebus 5:43 am

onTuesday, 28 June, 2016 Permalink |Thank you. I’m glad you’ve liked.

The obvious application of minimal surface theory is that it describes the shape that soap bubbles, or other lightweight, self-adhesive fluids, take on. That might not seem like it’s very exciting. But the equations that describe this shape also describe ways to look at physics problems. These are ways that let us talk about what equilibriums are like, and how small changes in the setup change the kind of behavior shown.

Monte Carlo methods are a bunch of related tools. They’re all built on the idea of using probability, randomly-evaluated solutions, to work out approximate solutions to complicated questions. My specialty was in the Metropolis-Hastings algorithm. That’s used for problems where it’s hard to find the best answer, but it’s easy to tell whether one answer is better or worse than another.

You start with a randomly-generated guess at the answer. And then you make a randomly-generated change in the answer. If you’ve made the answer better, you accept that change. If you’ve made the answer worse, usually you reject the change, but sometimes you do accept it anyway. Sometimes you can get to a much better answer by way of a start that looks unpromising, which is why we sometimes take a bad move. Do this over and over. And you’ll get to very good answers after all.

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## xianyouhoule 12:47 pm

onTuesday, 28 June, 2016 Permalink |Thank you so much

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