en>ChmL7 at 00:56, 31 January 2014

2014-01-31T00:56:36Z

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	It'~~s true that warrior members of this Upstate New York nation occasionally came this way to raid settlements, but it~~'s ~~likely that the path here was blazed~~ by ~~local nations such as~~ the ~~Pocumtucks~~.<br>~~Contact the managers of your local grocery stores to let them know about your coming project and ask for their involvement~~. ~~Collect stacks of paper grocery bags from~~ the ~~stores~~ and ~~distribute them to your fourth grade students~~.<br>http://~~www.islandgreenclean~~.com/~~cheap~~/~~uggs~~.~~asp?p=133~~ <br /> http://~~www.islandgreenclean~~.com/~~cheap~~/~~uggs~~. ~~If you liked this short article~~ and ~~you would like to obtain far more information pertaining to [http~~:~~//www~~.~~bendtrapclub.com~~/~~cheap~~/~~ugg~~.~~asp Cheap Uggs Boots~~] ~~kindly pay~~ a ~~visit to our web~~-~~site~~. ~~asp?p~~=21 <br /> http://~~www~~.~~islandgreenclean~~.~~com~~/~~cheap~~/~~uggs~~.~~asp?p=41~~ <br /> ~~http~~://~~www~~.~~islandgreenclean~~.~~com~~/~~cheap/uggs~~.~~asp?p=159~~ <br /> http://www.~~islandgreenclean~~.~~com~~/~~cheap~~/~~uggs.asp?p=72 <br~~ /~~> <br>http:~~//~~klec~~.~~org/?document_srl=941172<br>~~http://~~vetopia~~.~~dothome~~.~~co.kr~~/xe/~~?document_srl~~=~~1353997~~		In [[differential geometry]], the '''Schwarz minimal surfaces''' are [[Triply periodic minimal surface\|periodic]] [[minimal surface]]s originally described by [[Hermann Schwarz]].

			In the 1880s Schwarz and his student E. R. Neovius described periodic [[minimal surface]]s.<ref>H. A. Schwarz, Gesammelte Mathematische Abhandlungen, Springer, Berlin, 1933.</ref><ref>E. R. Neovius, "Bestimmung zweier spezieller periodischer Minimal achen", ''Akad. Abhandlungen'', Helsingfors, 1883.</ref> They were later named by Alan Schoen in his seminal report that described the [[gyroid]] and other triply periodic minimal surfaces.<ref>Alan H. Schoen, Infinite periodic minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)[http://schoengeometry.com/19700020472_1970020472%5B1%5D.pdf]</ref>

			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.<ref>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</ref>

			The Schwarz surfaces have [[Genus (mathematics)\|topological genus]] 3, the minimal genus of triply periodic minimal surfaces.<ref>http://schoengeometry.com/e_tpms.html</ref>

			They have been considered as models for periodic [[nanostructures]] in [[block copolymer]]s, electrostatic equipotential surfaces in crystals.,<ref>Alan L. Mackay, "Periodic minimal surfaces", ''Physica B+C'', Volume 131, Issues 1–3, August 1985, Pages 300–305</ref> and hypothetical negatively curved graphite phases.<ref>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</ref>

			== Schwarz P ("Primitive") ==
			[[File:Schwarz P Surface.png\|thumb\|Schwarz P surface]]

			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.<ref>W. H. Meeks. The theory of triply-periodic minimal surfaces. Indiana University Math. Journal, 39 (3):877{936, 1990.</ref>

			It can be approximated by the implicit surface
			:<math> \cos(x) + \cos(y) + \cos(z) = 0 \ </math>.<ref>http://archive.msri.org/about/sgp/jim/geom/level/library/triper/index.html</ref>

			The P surface has been considered for prototyping [[tissue engineering\|tissue scaffolds]] with a high surface-to-volume ratio and porosity.<ref>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</ref>
			{{clear}}

			== Schwarz D ("Diamond") ==
			[[File:Schwarz D Surface.png\|thumb\|Schwarz D surface]]
			Schoen named this surface 'diamond' because has two intertwined congruent labyrinths, each having the shape of an inflated tubular version of the [[Diamond lattice\|diamond bond structure]]. It is sometimes called the F surface in the literature.

			It can be approximated by the implicit surface

			: <math>\sin(x)\sin(y)\sin(z) + \sin(x)\cos(y)\cos(z) + \cos(x)\sin(y)\cos(z) + \cos(x)\cos(y)\sin(z) = 0.\ </math>

			An exact expression exists in terms of [[elliptic integrals]], based on the [[Weierstrass representation]].<ref>Paul J.F. Gandy, Djurdje Cvijović, Alan L. Mackay, Jacek Klinowski, Exact computation of the triply periodic D (`diamond') minimal surface, Chemical Physics Letters, Volume 314, Issues 5–6, 10 December 1999, Pages 543–551</ref>

			{{clear}}

			== Schwarz H ("Hexagonal") ==
			[[File:Schwarz H Surface.png\|thumb\|Schwarz H surface]]
			The H surface is similar to a [[catenoid]] with a triangular boundary, allowing it to tile space.

			{{clear}}

			== Schwarz CLP ("Crossed layers of parallels") ==
			[[File:Schwarz CLP Surface.png\|thumb\|Schwarz CLP surface]]

			{{clear}}

			== Illustrations ==
			* http://www.susqu.edu/brakke/evolver/examples/periodic/periodic.html
			* http://www.indiana.edu/~minimal/archive/Triply/genus3.html

			== References ==
			{{reflist}}

			[[Category:Differential geometry]]
			[[Category:Minimal surfaces]]

en>Michael Hardy at 17:56, 18 August 2012

2012-08-18T17:56:05Z

New page

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Flat pseudospectral method - Revision history

en>ChmL7 at 00:56, 31 January 2014

en>Michael Hardy at 17:56, 18 August 2012