Zhang, Y. Liu, S. Chen, Packing different-sized circles into a rectangular container using simulated annealing algorithm, in ENFORMATIKA (2004), pp.Maths week treasure hunt. Deng, An effective hybrid algorithm for the problem of packing circles into a larger containing circle. H. Wang, W. Huang, Q. Zhang, D. Xu, An improved algorithm for the packing of unequal circles within a larger containing circle. (Also, if the rectangle is only 2 m r units tall, we can alternate columns with m and m 1 circles. A rectangle packing problem is considered, where the goal is to suitably arrange a subset of given rectangles within a container so that the. Casado, I. Garcia, New Approaches to Circle Packing in a Square (Springer, New York, 2007) So if you want the triangular packing to have m circles in each column, and n columns, then the rectangle must be at least ( 2 m + 1) r units tall and ( 2 + ( n 1) 3) r units long. New Trends in Equilibrium Systems (Kluwer, Boston, 2000), pp. 1–15 Garcia, Equal circle packing in a square I – Problem setting and bounds for optimal solutions. Stoyan, Y. Yaskov, G. Scheithauer, Packing of various radii solid spheres into a parallelepiped. Stoyan, Y. Yaskov, A mathematical model and a solution method for the problem of placing various-sized circles into a strip. Yaskov, Mathematical model and solution method of optimization problem of placement of rectangles and circles taking into account special constraints. I do think that for large cases the hexagonal. However, 3 circles do have space in that with the following setup: Again, this does not really answer the general case, but shows that even in small cases the hexagonal packing may not find the optimum. I. Stewart, Wie viele kreisförmige Kekse passen auf ein Kuchenblech? Spektrum der Wissenschaft 3, 112–114 (1999) Choosing b 1.6 and l 2.6 won't allow any of those setups to fit in more than 2 circles. Romanova, G. Scheithauer, On the global minimum in a balanced circular packing problem. Rebennack, Computing tight bounds via piecewise linear functions through the example of circle cutting problems. R. Peikert, Dichteste Packungen von gleichen Kreisen in einem Quadrat. N. Oler, An inequality in the geometry of numbers. Wright, Numerical Optimization (Springer, New York, 1999) N. Mladenovic, F. Plastria, D. Urosevic, Reformulation descent applied to circle packing. H. Melissen, Densest packing of six equal circles in a square. Graham, Minimum perimeter rectangles that enclose congruent non-overlapping circles. H. Isermann, Heuristiken zur Lösung des zweidimensionalen Packproblems für Rundgefäße. Xu, New heuristics for packing unequal circles into a circular container. Huang, M. Chen, Note on: an improved algorithm for the packing of unequal circles within a larger containing circle. M. Hifi, R. M’Hallah, Strip generation algorithms for constrained two-dimensional two-staged cutting problems. M. Hifi, R. M’Hallah, Approximate algorithms for constrained circular cutting problems.
Östergård, Dense packings of congruent circles in a circle. If youre trying different values, you can record them to the. Lamar, Packing different-sized circles into a rectangular container. Ed Southall solvemymaths posed this nice problem, with the small radius 6 cm, area 243. Dowsland, Optimising the palletisation of cylinders in cases. Y. Cui, Generating optimal multi-segment cutting patterns for circular blanks in the manufactoring of electric motors. Y. Cui, Dynamic programming algorithms for the optimal cutting of equal rectangles. Pinter, Solving circle packing problems by global optimization: numerical results and industrial applications. Therefore, there is a natural problem about the packings of rectangles: Can.
Szabo, T. Csendes, Equal circle packing in a square II – New results for up to 100 circles using the TAMSASS-PECS algorithm, in New Trends in Equilibrium Systems (Kluwer, Boston, 2000), pp. 1–16 The circle packing theorem 12 says that for any finite planar graph G. P. Bose, P. Morin, A. Vigneron, Packing two disks into a polygonal environment. Sobral, Minimizing the object dimensions in circle and sphere packing problems. S. Bespamyatnikh, Packing two disks in a polygon. G. Belov, G. Scheithauer, Setup and open-stacks minimization in one-dimensional stock cutting.