Packing Equal Circles in a Square: I. Solution Properties
نویسندگان
چکیده
In this paper some properties of optimal solutions for the problem of packing n equal circles into the unit square will be derived. In particular, properties, which must be satissed by at least one optimal solution of the problem and stating the intuitive fact that as many circles as possible should touch the boundary of the unit square, will be introduced.
منابع مشابه
Packing Circles in a Square A Review and New Results
There are many interesting optimization problems associated with the packing and covering of objects in a closed volume or bounded surface Typical examples arise in classical physics or chemistry where questions of the kind what does the densest packing of atoms or molecules look like when a crystal or macro molecule is formed with the lowest energy Also engineering and information science conf...
متن کاملPacking up to 50 Equal Circles in a Square
The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 [Bol04]) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of vacuitas (in Latin, the gap in the triangle outside the circles). It is interesting that these packings are not always optimal in s...
متن کاملThe packing of circles on a hemisphere
The problem of the closest packing of N equal nonoverlapping circles on a sphere has been of interest in geometry, chemistry, biology, engineering and optimization. The problem of packing N equal nonoverlapping circles on a hemisphere is a different problem of the same type and is also of interest in various fields such as detecting signals in a multisource neighbourhood and measuring solar rad...
متن کاملPacking equal circles in a square: a deterministic global optimization approach
In this paper the problem of packing n equal circles into the unit square will be considered. Starting from a general rectangular branch-and-bound algorithm, many tools, which exploit the special structure of the problem, will be introduced and discussed. Computational results will be presented and, in particular, the optimality within a given tolerance of best known solutions in the literature...
متن کاملA better packing of ten equal circles in a square
Let S be a square of side s in the Euclidean plane. A pucking of circles in S is nothing else but a finite family of circular disks included in S whose interiors are pairwise disjoints. A natural problem related with such packings is the description of the densest ones; in particular, what is the greatest value of the common radius r of n circles forming a packing of S? Clearly, the centres of ...
متن کامل