On sublattices of the hexagonal lattice
نویسندگان
چکیده
منابع مشابه
On sublattices of the hexagonal lattice
How many sublattices of index N are there in the planar hexagonal lattice? Which of them are the best from the point of view of packing density, signal-to-noise ratio, or energy? We answer the first question completely and give partial answers to the other questions.
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We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, ...
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Kepler’s Conjecture, recently proved by T. Hales, states that the densest packing of spheres in 3-space has spheres centered along the face-centered cubic (fcc) lattice. A lattice is a free Z-module formed by taking the span of a collection of linearly independent vectors in R over the integers. The two-dimensional analogue of Kepler’s Conjecture, proved by L. F. Toth in 1940, states that the d...
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In this note we give a simple proof of the classical fact that the hexagonal lattice gives the highest density circle packing among all lattices in R2. With the benefit of hindsight, we show that the problem can be restricted to the important class of well-rounded lattices, on which the density function takes a particularly simple form. Our proof emphasizes the role of well-rounded lattices for...
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ژورنال
عنوان ژورنال: Discrete Mathematics
سال: 1997
ISSN: 0012-365X
DOI: 10.1016/0012-365x(95)00354-y