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Tilings and isoperimetrical shapes. I: Square lattice. (English) Zbl 1042.52019

A polyomino is a finite set (not necessary connected) of unit squares centered on the grid \(\mathbb Z^n\). The authors are interested in the following question (Q): Suppose we are given a positive integer \(n\). What are the polyominoes of perimeter \(n\) with maximum area?
This question is related to an isoperimetrical problem: What is the least perimeter of a polyomino of a given area?
The authors solve (Q) for the 2-dimensional case, and propose an application of this result in order to solve a Golomb-type pentomino exclusion problem for the plane: Find the minimum density of unit squares to be placed on the plane to exclude all polyominoes of a fixed area.
The characterization of isoperimetrical shapes allows to solve this problem for some values of the fixed area.

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
05B45 Combinatorial aspects of tessellation and tiling problems
05B50 Polyominoes
52A40 Inequalities and extremum problems involving convexity in convex geometry
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References:

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