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The last chapter
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
The Newton number or kissing number problem is a local variant of the sphere packing problem. Fix a sphere of radius a in Euclidean n-space. The Newton number τn is the maximal number of spheres of radius a which touch the given sphere and do not penetrate each other. Because of scaling, this does not depend on the radius a. Consider an n-dimensional integer lattice. Each point has the same number of closest neighbors. This number is then a lower bound on τn. The E8 lattice and the Leech lattice are exceptional even in this respect. In fact, we saw that each point in the E8 lattice has 240 neighbors. It can be proved that higher kissing numbers are impossible: τ8 = 240.
Propagation in Periodic Media: Bloch Waves and Evanescent Waves
Published in Didier Felbacq, Guy Bouchitté, Metamaterials Modeling and Design, 2017
Didier Felbacq, Frédéric Zolla
The structure is defined by repeating periodically an elementary cell Y along its basis vectors ai, where according to the dimension i belongs to {1}, {1, 2}, or {1, 2, 3}. The underlying structure is thus an integer lattice with basis ai . In one dimension, the metamaterial is simply characterized by its period [0, d[. In higher dimensions, the period is made of all the points M such that: OM = xiai, xi ∊ [0, 1[ (where a sum is implied over each pair of repeated index). Generically, a vector belonging to the lattice is denoted by T=niai $ {\mathbf{T}} = n^{i} {\mathbf{a}}_{i} $ with integer coefficients ni. We also define the so-called reciprocal lattice, which is a lattice whose basis vectors aiare defined by: ai·aj=2πδji $ {\mathbf{a}}^{i} \cdot {\mathbf{a}}_{j} = 2\pi \delta_{j}^{i} $ . where δij $ \delta_{i}^{j} $ is the Kronecker symbol. The Brillouin zone Y* is defined as the set of points P such that OP=yiai, $ = y_{i} {\mathbf{a}}^{i} , $ yi ∊ [ - 1/2, 1/2 [. Generically, a vector belonging to the reciprocal lattice is denoted by G = niaifor some integers ni.
Sliced Rotated Sphere Packing Designs
Published in Technometrics, 2019
A set of points in is called a lattice if it forms a group. The lattice points consist of linear combinations of p basis vectors with integer coefficients. We call a p × p matrix a generator matrix of the lattice if its rows are the basis vectors. As an example, the set of integer vectors, , is called the p-dimensional integer lattice, which can be generated from the p-dimensional identity matrix. Two important properties of lattices are their densities and thicknesses. If we place identical balls in centered at the lattice points, then the maximum radius of the balls such that no two balls overlap is called the packing radius of the lattice, and the minimum radius of the balls such that the union of overlapped balls cover is called the covering radius of the lattice. The Voronoi cell of a point x0 in a lattice L is the region The density and thickness of a lattice is the volume of one ball with packing and covering radius, respectively, divided by the volume of one Voronoi cell. Lattice-based designs with highest possible density and lowest possible thickness have asymptotically optimal separation distance and fill distance, respectively.