منابع مشابه
Improving Dense Packings of Equal Disks in a Square
We describe a new numerical procedure for generating dense packings of disks and spheres inside various geometric shapes. We believe that in some of the smaller cases, these packings are in fact optimal. When applied to the previously studied cases of packing n equal disks in a square, the procedure confirms all the previous record packings [NO1] [NO2] [GL], except for n = 32, 37, 48, and 50 di...
متن کاملRepeated Patterns of Dense Packings of Equal Disks in a Square
We examine sequences of dense packings of n congruent non-overlapping disks inside a square which follow specific patterns as n increases along certain values, n = n(1), n(2), ...n(k), .... Extending and improving previous work of Nurmela and Österg̊ard [NO] where previous patterns for n = n(k) of the form k, k − 1, k − 3, k(k + 1), and 4k + k were observed, we identify new patterns for n = k − ...
متن کاملCurved Hexagonal Packings of Equal Disks in a Circle
Abstract. For each k ≥ 1 and corresponding hexagonal number h(k) = 3k(k + 1)+ 1, we introduce m(k) = max{(k − 1)!/2, 1} packings of h(k) equal disks inside a circle which we call the curved hexagonal packings. The curved hexagonal packing of 7 disks (k = 1, m(1) = 1) is well known and one of the 19 disks (k = 2, m(2) = 1) has been previously conjectured to be optimal. New curved hexagonal packi...
متن کاملMore Optimal Packings of Equal Circles in a Square
The problem of nding the maximum radius of n non-overlapping equal circles in a unit square is considered. A computer-aided method for proving global optimality of such packings is presented. This method is based on recent results by De Groot, Monagan, Peik-ert, and WWrtz. As an example, it is shown how the method can be used to get an optimality proof for the case n = 7, which has not earlier ...
متن کاملDense Packings of Equal Disks in an Equilateral Triangle: from 22 to 34 and Beyond
Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up to 34 disks. For each n in the range 22 ≤ n ≤ 34 we present what we believe to be the densest possible packing of n equal disks in an equilateral triangle. For these n we also list the second, often the third and some...
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ژورنال
عنوان ژورنال: Discrete & Computational Geometry
سال: 2017
ISSN: 0179-5376,1432-0444
DOI: 10.1007/s00454-016-9843-x