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A gyroid is an infinitely connected triply periodic minimal surface discovered by Alan Schoen in 1970.〔Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)().〕〔David Hoffman, Computing Minimal Surfaces. In Global Theory of Minimal Surfaces: Proceedings of the Clay Mathematics Institute 2001 Summer School, Mathematical Sciences Research Institute, Berkeley, California, June 25-July 27, 2001 American Mathematical Society 2005.〕 ==History and properties== The gyroid is the unique non-trivial embedded member of the associate family of the Schwarz P and D surfaces with angle of association approximately 38.01°. The gyroid is similar to the Lidinoid. The gyroid was discovered in 1970 by Alan Schoen, then a scientist at NASA. He calculated the angle of association in his NASA Technical Report and gave a convincing demonstration but did not provide a proof of embeddedness (although he did provide pictures of intricate plastic models). Schoen notes that the gyroid contains neither straight lines nor planar symmetries. Karcher 〔Hermann Karcher's The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions, Manuscripta Math. 64, 291-357 (1989)〕 gave a different, more contemporary treatment of the surface in 1989 using the conjugate surface construction. In 1996 Große-Brauckmann and Wohlgemuth 〔Karsten Große-Brauckmann and Meinhard Wohlgemuth, The gyroid is embedded and has constant mean curvature companions, Calc. Var. Partial Differential Equations 4 (1996), no. 6, 499–523.〕 proved that it is embedded, and in 1997 Große-Brauckmann provided CMC variants of the gyroid and made further numerical investigations about the volume fractions of the minimal and CMC gyroids. The gyroid separates space into two identical labyrinths of passages. The gyroid has space group ''Ia'd''. Channels run through the gyroid labyrinths in the (100) and (111) directions; passages emerge at 70.5 degree angles to any given channel as it is traversed, the direction at which they do so gyrating down the channel, giving rise to the name "gyroid". One way to visualize the surface is to picture the “square catenoids” of the P surface (these are formed by two squares in parallel planes, with a nearly circular waist); rotation about the edges of the square generate the P surface. In the associate family, these square catenoids “open up” (similar to the way the catenoid “opens up” to a helicoid) to form gyrating ribbons, then finally become the Schwarz D surface. For one value of the associate family parameter the gyrating ribbons lie in precisely the locations required to have an embedded surface. The gyroid refers to the member that is in the associate family of the Schwarz P surface, but in fact the gyroid exists in several families which preserve various symmetries of the surface; a more complete discussion of families of these minimal surfaces appears in the entry on triply periodic minimal surfaces. Curiously, like some other triply periodic minimal surfaces, the gyroid surface can be trigonometrically approximated by the equation: : The gyroid structure is closely related to the K4 crystal (Laves' graph of girth ten).〔T. Sunada, Crystals that nature might miss creating, Notices of the AMS, 55(2008), 208-215〕 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「Gyroid」の詳細全文を読む スポンサード リンク
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