By S. Hyde, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, S. Andersson, K. Larsson
This ebook develops the thesis that constitution and serve as in a number of condensed platforms - from the atomic assemblies in inorganic frameworks and natural molecules, via molecular self-assemblies to proteins - may be unified while curvature and floor geometry are taken including molecular form and forces. An staggering number of man made and organic assemblies should be adequately modelled and understood when it comes to hyperbolic surfaces, whose richness and wonder are just now being published via utilized mathematicians, physicists, chemists and crystallographers. those surfaces, frequently with reference to periodic minimum surfaces, weave and twist via house, carving out interconnected labyrinths whose diversity of topologies and symmetries problem the resourceful powers.
The ebook bargains an outline of those constructions and structural changes, convincingly demonstrating their ubiquity in covalent frameworks from zeolites used for cracking oil and toxins keep an eye on to enzymes and structural proteins, thermotropic and lyotropic bicontinuous mesophases shaped by means of surfactants, detergents and lipids, man made block copolymer and protein networks, in addition to organic cellphone assemblies, from muscular tissues to membranes in prokaryotic and eukaryotic cells. The relation among constitution and serve as is analysed when it comes to the formerly ignored
hidden variables of curvature and topology. hence, the catalytic task of zeolites and enzymes, the very best fabric homes of interpenetrating networks in microstructured polymer composites, the shipping necessities in cells, the transmission of nerve signs and the folding of DNA could be extra simply understood within the gentle of this.
The textual content is liberally sprinkled with figures and color plates, making it obtainable to either the start graduate scholar and researchers in condensed topic physics and chemistry, mineralogists, crystallographers and biologists.
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Extra resources for The Language of shape: the role of curvature in condensed matter--physics, chemistry, and biology
Then F decomposes into two components. iIltersections points 8H with the rays starting from the inner points of H and having the opposite direction from the ray L+. ut, 18 Chapter 1. Convex Bodies and Hypersurfaces Thus every point A E [)H has a neighborhood U C [)H, which is projected one-to-one in a hyperplane. Let W(G) be the set of convex hypersurfaces in En+l which project orthogonally and one-to-one onto a convex open domain G C P. Let Xl, ... , X n, X n+ 1 = z be Cartesian coordinates in En+!
Moreover, the point Ak (k = 1, 2, ... , m) is a vertex of P if and only if Ak does not belong to CO(AI U ... U A k - l U Ak+l U ... U Am U V). 6 can be an (n + 1)-convex polyhedral angle. The last theorem has a natural generalization if V is any k-convex solid polyhedral angle in En+l. The proof of this generalization will be left as a useful exercise. 3 Approximation of Closed Convex Hypersurfaces by Closed Convex Polyhedra Let S be any closed convex hypersurface in En+l. Denote by F the bounded solid convex body such that S = BF.
The proof of this generalization will be left as a useful exercise. 3 Approximation of Closed Convex Hypersurfaces by Closed Convex Polyhedra Let S be any closed convex hypersurface in En+l. Denote by F the bounded solid convex body such that S = BF. Remember that closed n-convex polyhedra are considered as closed convex hypersurfaces in En+l. The closed nconvex polyhedron P is said to be inscribed in the closed hypersurface S if all its vertices belong to S. 7. There exists a sequence of closed n-convex polyhedra inscribed in any convex hypersurface S which converges to S.
The Language of shape: the role of curvature in condensed matter--physics, chemistry, and biology by S. Hyde, Z. Blum, T. Landh, S. Lidin, B.W. Ninham, S. Andersson, K. Larsson