A minimal surface is arguably the simplest example of an equilibrium system in three dimensions: the energetics are entirely local and free of parameters. Given this status, it becomes interesting to explore how minimal surfaces realize particular states of order.

Crystalline minimal surfaces, that is surfaces with a translational symmetry, were discovered by the 19th century mathematician H. A. Schwarz and continue to be an active area of research. Shown on the left is a fragment of a conjectured quasicrystalline minimal surface. Here quasiperiodicity arises because the icosahedral symmetry of the surface is incompatible with ordinary periodicity. The rigorous constructions that work for periodic surfaces fail for this quasicrystalline example: all evidence of its existence is numerical.

My former students Qing Sheng and Chris Kimmer used a dimensional reduction scheme to test (numerically) the viability of this surface. In brief, the idea is to take a codimension-1 periodic surface in six dimensions, analogous to one of the cubic surfaces discovered by Schwarz, and stretch it in a carefully chosen three dimensional subspace (specified by icosahedral symmetry). As in lower-dimensional examples where it can be verified rigorously, we expect, that in the limit of infinite stretch, the surface degenerates into the product of a flat 3-plane and a genuine minimal 2-surface. Suitable sections of the surface, orthogonal to the 3-plane, then produce something that is minimal as well as quasiperiodic.

The numerical procedure always falls short of achieving infinite stretch. It appears, however, that a large fraction of the surface topology (roughly within the pictured sphere and repeated quasiperiodically in space) is fixed after a modest amount of stretch. The manner in which these well established pieces fit together (if at all) is an outstanding problem.