In differential geometry, the Schwarz minimal surfaces are periodic minimal surfaces originally described by Hermann Schwarz.
In the 1880s Schwarz and his student E. R. Neovius described periodic minimal surfaces.[1][2] They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces.[3]
The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the boundary lines also produce valid minimal surfaces that can be continuously joined to the original solution. If a minimal surface meets a plane at right angles, then the mirror image in the plane can also be joined to the surface. Hence given a suitable initial polygon inscribed in a unit cell periodic surfaces can be constructed.[4]
The Schwarz surfaces have topological genus 3, the minimal genus of triply periodic minimal surfaces.[5]
They have been considered as models for periodic nanostructures in block copolymers, electrostatic equipotential surfaces in crystals,[6] and hypothetical negatively curved graphite phases.[7]
^H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.
^Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)[1]
^Hermann Karcher, Konrad Polthier, "Construction of Triply Periodic Minimal Surfaces", Phil. Trans. R. Soc. Lond. A 16 September 1996 vol. 354 no. 1715 2077–2104
^"Alan Schoen geometry".
^Mackay, Alan L. (April 1985). "Periodic minimal surfaces". Nature. 314 (6012): 604–606. Bibcode:1985Natur.314..604M. doi:10.1038/314604a0. S2CID 4267918.
^Terrones, H.; Mackay, A. L. (December 1994). "Negatively curved graphite and triply periodic minimal surfaces". Journal of Mathematical Chemistry. 15 (1): 183–195. doi:10.1007/BF01277558. S2CID 123561096.
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