Abstract
This paper considers the task of finding the smallest circle into which one can pack a fixed number of non-overlapping unit squares that are free to rotate. Due to the rotation angles, the packing of unit squares into a container is considerably harder to solve than their circle packing counterparts. Therefore, optimal arrangements were so far proved to be optimal only for one or two unit squares. By a computer-assisted method based on interval arithmetic techniques, we solve the case of three squares and find rigorous enclosures for every optimal arrangement of this problem. We model the relation between the squares and the circle as a constraint satisfaction problem (CSP) and found every box that may contain a solution inside a given upper bound of the radius. Due to symmetries in the search domain, general purpose interval methods are far too slow to solve the CSP directly. To overcome this difficulty, we split the problem into a set of subproblems by systematically adding constraints to the center of each square. Our proof requires the solution of 6, 43 and 12 subproblems with 1, 2 and 3 unit squares respectively. In principle, the method proposed in this paper generalizes to any number of squares.
Highlights
Let S1, . . . , Sn be n open unit squares and denote by Cr the closed circle of radius r centered at the origin
At the i-th iteration, we look at every possible combination of i triangles which can accommodate i unit squares into a circle with the radius at most r n
Definition 1 [standard constraint satisfaction problem (SCSP)] Let r > 0 be an upper bound for the radius of the smallest circle into which one can pack n non-overlapping unit squares and s be a scaling factor
Summary
Let S1, . . . , Sn be n open unit squares and denote by Cr the closed circle of radius r centered at the origin. Markót studied the packing of circles into a square from the interval analysis point of view in a series of papers [15,16,24]. Recent contributions in the packing of unit squares into a square include new bounds for the wasted area [6], the optimality proof for the cases n = 5, . Friedman [9] maintains a list of proved and best-known values for the packing of unit squares into circles, triangles, L-shapes, and pentagons.
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