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eebo-0121

Charles

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Graphic arts --- History of civilization --- Christian special devotions --- 76.046.3 --- 769.4 --- 76 <492> "17" --- 76 LECUYER, PHILIPPUS J. --- #GROL:SEMI-246:2 --- #GMML:Maria --- Religieuze voorstellingen in de prentkunst --- Prentenverzamelingen: doodsprentjes; santjes; bidprentjes; devotieprentjes --- Grafische kunsten. Grafiek. Prentkunst--Nederland--18e eeuw. Periode 1700-1799 --- Grafische kunsten. Grafiek. Prentkunst--LECUYER, PHILIPPUS J. --- 76 LECUYER, PHILIPPUS J. Grafische kunsten. Grafiek. Prentkunst--LECUYER, PHILIPPUS J. --- 76 <492> "17" Grafische kunsten. Grafiek. Prentkunst--Nederland--18e eeuw. Periode 1700-1799 --- 769.4 Prentenverzamelingen: doodsprentjes; santjes; bidprentjes; devotieprentjes --- 76.046.3 Religieuze voorstellingen in de prentkunst --- Holy cards --- Netherlands --- Belgium --- Popular culture --- 76 LECUYER, PHILIPPUS J --- Grafische kunsten. Grafiek. Prentkunst--LECUYER, PHILIPPUS J

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In one sense, the problem of finding the densest packing of congruent circles in a square is easy to understand: it is a matter of positioning a given number of equal circles in such a way that the circles fit fully in a square without overlapping. But on closer inspection, this problem reveals itself to be an interesting challenge of discrete and computational geometry with all its surprising structural forms and regularities. As the number of circles to be packed increases, solving a circle packing problem rapidly becomes rather difficult. To give an example of the difficulty of some problems, consider that in several cases there even exists a circle in an optimal packing that can be moved slightly while retaining the optimality. Such free circles (or "rattles”) mean that there exist not only a continuum of optimal solutions, but the measure of the set of optimal solutions is positive! This book summarizes results achieved in solving the circle packing problem over the past few years, providing the reader with a comprehensive view of both theoretical and computational achievements. Typically illustrations of problem solutions are shown, elegantly displaying the results obtained. Beyond the theoretically challenging character of the problem, the solution methods developed in the book also have many practical applications. Direct applications include cutting out congruent two-dimensional objects from an expensive material, or locating points within a square in such a way that the shortest distance between them is maximal. Circle packing problems are closely related to the "obnoxious facility location” problems, to the Tammes problem, and less closely related to the Kissing Number Problem. The emerging computational algorithms can also be helpful in other hard-to-solve optimization problems like molecule conformation. The wider scientific community has already been involved in checking the codes and has helped in having the computational proofs accepted. Since the codes can be worked with directly, they will enable the reader to improve on them and solve problem instances that still remain challenging, or to use them as a starting point for solving related application problems. Audience This book will appeal to those interested in discrete geometrical problems and their efficient solution techniques. Operations research and optimization experts will also find it worth reading as a case study of how the utilization of the problem structure and specialities made it possible to find verified solutions of previously hopeless high-dimensional nonlinear optimization problems with nonlinear constraints.

Discrete geometry --- Circle packing --- Data processing. --- Packing, Circle --- Geometry --- Combinatorial geometry --- Mathematical optimization. --- Computer science. --- Discrete groups. --- Optimization. --- Math Applications in Computer Science. --- Convex and Discrete Geometry. --- Arithmetic and Logic Structures. --- Computational Science and Engineering. --- Groups, Discrete --- Infinite groups --- Informatics --- Science --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Discrete mathematics --- 519.8 --- 519.8 Operational research --- Operational research --- Data processing --- Computer science—Mathematics. --- Convex geometry . --- Discrete geometry. --- Arithmetic and logic units, Computer. --- Computer mathematics. --- Computer mathematics --- Electronic data processing --- Mathematics --- Arithmetic and logic units, Computer --- Computer arithmetic --- Electronic digital computers --- Circuits