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Zbl 1042.52018
Gravier, Sylvain
Tilings and isoperimetrical shapes. II: Hexagonal lattice.
(English)
[J] Acta Univ. Palacki. Olomuc., Fac. Rerum Nat., Math. 40, 79-92 (2001). ISSN 0231-9721

Let $\Cal H$ be the hexagonal lattice. A polyhexe is a finite set (not necessarily connected) of unit hexagons placed on $\Cal H$. \par The author solves the following problem in $\Cal H$: Suppose we are given a positive integer $n$. What are the polyhexes of perimeter $n$ with maximal area? \par 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. \par For part I see [{\it S. Gravier} and {\it Ch. Payan}, ibid., 63--77 (2001; Zbl 1042.52019)].
[Serguey M. Pokas (Odessa)]
MSC 2000:
*52C20 Tilings in 2 dimensions (discrete geometry)
52A40 Geometric inequalities, etc. (convex geometry)

Keywords: minimizing the area for a given perimeter; tiling; polyhexe; isoperimetrical inequality; hexagonal lattice; graphs; optimal shapes; polyominoes on the honeycomb lattice; minimum density of unit hexagon; Colomb type problem

Citations: Zbl 1042.52019

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