via John Baez

The

A

If you make a minimal surface this way, it will have edges along the wire frame. A minimal surface

sqrt(x^2 + y^2) = c cosh(x/c)

for your favorite constant c. A

z = c θ

for some constant c. You can see a helicoid here - and see how it can continuously deform into a catenoid:

• Eric Weisstein, Helicoid, http://mathworld.wolfram.com/Helicoid.html

In 1970, A. H. Schoen discovered another minimal surface: the gyroid! This one is very different. It’s

• Eric Weisstein, Gyroid, from Mathworld, http://mathworld.wolfram.com/Gyroid.html

Starting around 1987, people started using computers to find

• Ken Brakke, Triply periodic minimal surfaces, http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html

How do you find a minimal surface? For starters, a minimal surface needs to be locally saddle-shaped. More precisely, it has

**gyroid**is a surface that chops space into two parts. This is a portion of it, made into an elegant sculpture by Bathsheba Grossman.A

**minimal surface**is a surface in ordinary 3d space that can’t reduce its area by changing shape slightly. You can create a minimal surface by building a wire frame and then creating a soap film on it. As long as the soap film doesn’t actually enclose any air, it will try to minimize its area - so it will end up being a minimal surface.If you make a minimal surface this way, it will have edges along the wire frame. A minimal surface

*without*edges is called**complete**. For a long time, the only known complete minimal surfaces that didn’t intersect themselves were the plane, the catenoid, and the helicoid. You get a**catenoid**by taking an infinitely long chain and letting it hang to form a curve called a**catenary**, and then turning the curve around to form a surface of revolution with equation like this:sqrt(x^2 + y^2) = c cosh(x/c)

for your favorite constant c. A

**helicoid**is like a spiral staircase; in cylindrical coordinates it’s given by the equationz = c θ

for some constant c. You can see a helicoid here - and see how it can continuously deform into a catenoid:

• Eric Weisstein, Helicoid, http://mathworld.wolfram.com/Helicoid.html

In 1970, A. H. Schoen discovered another minimal surface: the gyroid! This one is very different. It’s

**triply periodic**, meaning that it repeats itself over and over as we move in 3 different directions in space, like a crystal. He was working for NASA, and his idea was to use it for building ultra-light, super-strong structures:• Eric Weisstein, Gyroid, from Mathworld, http://mathworld.wolfram.com/Gyroid.html

Starting around 1987, people started using computers to find

*lots*minimal surfaces. You can see a bunch of triply periodic ones here:• Ken Brakke, Triply periodic minimal surfaces, http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html

How do you find a minimal surface? For starters, a minimal surface needs to be locally saddle-shaped. More precisely, it has

**zero mean curvature**: at any point, if it curves one way along one principal axis of curvature, it has to curve an equal and opposite amount along the perpendicular axis. Supposedly this was proved by Euler. If we write this requirement as an equation, we get a second-order nonlinear differential equation called**Lagrange’s equation**. So, finding new minimal surfaces amounts to finding new solutions of this equation.- 618mx likes this
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