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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.〔H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.〕〔E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal achen", ''Akad. Abhandlungen'', Helsingfors, 1883.〕 They were later named by Alan Schoen in his seminal report that described the gyroid and other triply periodic minimal surfaces.〔Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)()〕 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.〔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〕 The Schwarz surfaces have topological genus 3, the minimal genus of triply periodic minimal surfaces.〔http://schoengeometry.com/e-tpms.html〕 They have been considered as models for periodic nanostructures in block copolymers, electrostatic equipotential surfaces in crystals.,〔Alan L. Mackay, "Periodic minimal surfaces", ''Physica B+C'', Volume 131, Issues 1–3, August 1985, Pages 300–305〕 and hypothetical negatively curved graphite phases.〔H. Terrones and A. L. Mackay, "Negatively curved graphite and triply periodic minimal surfaces", ''Journal of Mathematical Chemistry'', Volume 15, Number 1 (1994), 183–195, DOI:10.1007/BF01277558〕 == Schwarz P ("Primitive") == Schoen named this surface 'primitive' because it has two interwined congruent labyrinths, each with the shape of an inflated tubular version of the simple cubic lattice. While the standard P surface has cubic symmetry the unit cell can be any rectangular box, producing a family of minimal surfaces with the same topology.〔W. H. Meeks. The theory of triply-periodic minimal surfaces. Indiana University Math. Journal, 39 (3):877{936, 1990.〕 It can be approximated by the implicit surface :.〔http://archive.msri.org/about/sgp/jim/geom/level/library/triper/index.html〕 The P surface has been considered for prototyping tissue scaffolds with a high surface-to-volume ratio and porosity.〔Jaemin Shin, Sungki Kim, Darae Jeong, Hyun Geun Lee, Dongsun Lee, Joong Yeon Lim, and Junseok Kim, Finite Element Analysis of Schwarz P Surface Pore Geometries for Tissue-Engineered Scaffolds, Mathematical Problems in Engineering, Volume 2012, Article ID 694194, doi:10.1155/2012/694194〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Schwarz minimal surface」の詳細全文を読む スポンサード リンク
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