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In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in ℝ3 that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries. Monoclinic and triclinic examples are certain to exist, but have proven hard to parametrise.〔http://epinet.anu.edu.au/mathematics/minimal_surfaces〕 TPMS are of relevance in natural science. TPMS have been observed as biological membranes,〔Yuru Deng and Mark Mieczkowski. Three-dimensional periodic cubic membrane structure in the mitochondria of amoebae Chaos carolinensis. Protoplasma, 203(1-2):16–25, 1998.〕 as block copolymers,〔Novel Morphologies of Block Copolymer Blends via Hydrogen Bonding. Jiang, S., Gopfert, A., and Abetz, V. Macromolecules, 36, 16, 6171–6177, 2003〕 equipotential surfaces in crystals〔Alan L. Mackay, "Periodic minimal surfaces", Physica B+C, Volume 131, Issues 1–3, August 1985, Pages 300–305〕 etc. They have also been of interest in architecture, design and art. == Properties == Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant).〔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 ()〕 All connected TPMS have genus ≥ 3,〔 and in every lattice there exist orientable embedded TPMS of every genus ≥3.〔Martin Traizet: On the genus of triply periodic minimal surfaces. Journal of Diff. Geom. 79, 243–275 (2008) ()〕 Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths). If they are congruent the surface is said to be a balance surface.〔()〕 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「Triply periodic minimal surface」の詳細全文を読む スポンサード リンク
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