{"id":995,"date":"2021-08-23T02:08:30","date_gmt":"2021-08-23T02:08:30","guid":{"rendered":"http:\/\/summergeometry.org\/sgi2021\/?p=995"},"modified":"2021-08-23T02:08:32","modified_gmt":"2021-08-23T02:08:32","slug":"minimal-surfaces-but-periodic","status":"publish","type":"post","link":"https:\/\/summergeometry.org\/sgi2021\/minimal-surfaces-but-periodic\/","title":{"rendered":"Minimal Surfaces, But Periodic"},"content":{"rendered":"\n

By Zhecheng Wang, Zeltzyn Guadalupe Montes Rosales, and Olga Gu\u021ban<\/em><\/p>\n\n\n\n

Note: This post describes work that has occurred between August 9 and August 20. The project will continue for a third week; more details to come.<\/p>\n\n\n\n

I. Introduction<\/em><\/h4>\n\n\n\n

For the past two weeks we had the pleasure of working with Nicholas Sharp<\/a>, Etienne Vouga<\/a>, Josh Vekhter<\/a>, and Erik Amezquita<\/a>. We learned about a special type of minimal surfaces: triply-periodic<\/em> minimal surfaces<\/em>. Their name stems from their repeating pattern. Broadly speaking, a minimal surface<\/em><\/a> minimizes its surface area. This is equivalent to having zero mean curvature. A triply-periodic minimal surface<\/em><\/a> (TPMS) is a surface in \\(\\mathbb{R}^{3}\\) that is invariant under a rank-3 lattice of translations.<\/p>\n\n\n\n

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Figure 1. (Left) A Minimal Surface [source<\/a>] and (right) a TPMS [source<\/a>]. <\/figcaption><\/figure><\/div>\n\n\n\n

Let\u2019s talk about nonmanifold<\/em> surfaces. \u201cManifold\u201d is a geometry term that means: every local region of the surface looks like the plane (more formally — it is homeomorphic to a subset of Euclidean space). Non-<\/em>manifold<\/a> then allows for parts of the surface that do not look like the plane, such as T-junctions. Within the context of triangle meshes, a nonmanifold surface is a surface where more than 3 faces share an edge.<\/p>\n\n\n\n

II. What We Did<\/em><\/h4>\n\n\n\n

First, we read and studied the 1993 paper<\/a> by Pinkhall and Polthier that describes the algorithm for generating minimal surfaces. Our next goal was to generate<\/em> minimal surfaces. Initially, we used pinned (Dirichlet) boundary conditions and regular manifold shapes. After ensuring that our code worked on manifold surfaces, we tested it on non-manifold input. <\/p>\n\n\n\n

Additionally, our team members learned how to use Blender. It has been a very enjoyable process, and the work was deeply satisfying, because of the embedded mathematical ideas intertwined with the artistic components. <\/p>\n\n\n\n

III. Reading the Pinkall Paper<\/em><\/h4>\n\n\n\n

“Computing Discrete Minimal Surfaces and Their Conjugates,”<\/em> by Ulrich Pinkall and Konrad Polthier, is the classic paper on this subject; it introduces the iterative scheme we used to find minimal surfaces. Reading this paper was our first step in this project. The algorithm that finds a discrete locally area-minimizing surface is as follows:<\/p>\n\n\n\n