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Tilings and isoperimetrical shapes. II: Hexagonal lattice. (English) Zbl 1042.52018

Let \(\mathcal H\) be the hexagonal lattice. A polyhexe is a finite set (not necessarily connected) of unit hexagons placed on \(\mathcal H\).
The author solves the following problem in \(\mathcal H\): Suppose we are given a positive integer \(n\). What are the polyhexes of perimeter \(n\) with maximal area?
A variant of the isoperimetric inequalities in lattices is considered. A precise description of the polyominoes on the honeycomb lattice which maximize their area for a fixed perimeter is given. From this characterization, the ‘explicit’ values of the smallest perimeter of a polyhexe with a fixed area are given. This characterization is also applied to the problem: Find the minimum density of unit hexagon to be placed on the honeycomb lattice so that to exclude all polyhexes of a fixed area. This problem is solved for some values of the fixed area.
For part I see [S. Gravier and Ch. Payan, ibid., 63–77 (2001; Zbl 1042.52019)].

MSC:

52C20 Tilings in \(2\) dimensions (aspects of discrete geometry)
52A40 Inequalities and extremum problems involving convexity in convex geometry

Citations:

Zbl 1042.52019
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References:

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