When you dip a closed wire frame into soapy water and pull it out, the soap film you see is a *minimal surface*. Finding a formula for this surface is generally a very difficult problem, and one of the earliest successes was achieved by Hermann Amandus Schwarz, using four edges of a regular tetrahedron as the contour.

One feature of minimal surfaces is that they can be extended across a straight line by means of a 180 degree rotation about that line. Doing so for the tetrahedral patch above generates the *diamond surface*, named so because it has the symmetries of the diamond cubic crystal structure.

One can explore this crystal structure together with the diamond surface at a much more elementary level. Start with two sides of a regular tetrahedron. They constitute a blue rhombus that has been folded along the shorter diagonal. Rotate this blue rhombus by 180 degrees about any of its four edges to obtain a second (red) rhombus that is attached to it, like in the image above.

Keep extending this surface, always by rotating a bent rhombus about a boundary edge by 180 degrees. Above you see how you can obtain a hexagonal shape, and below an annulus.

Again the surface will extend indefinitely. The following piece is a fundamental piece in the sense that mere translations of it will produce the whole infinite surface.

There is another way to render the bent rhombi by replacing each rhombus by a circular ribbon which ends at the far corners of the rhombus. I learned about this from Alison Martin at the *Shape-Up* conference in Berlin, 2015.

So what we are trying here is to visualize an abstract algorithm (extend by rotating about an edge) that can reapplied to varying geometric contexts.