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Eutactic Lattice
In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients ''c''''i'' such that (''v'', ''v'') = Σ''c''''i''(''v'', ''m''''i'')2 where the sum is over the minimal vectors ''m''''i''. "Eutactic" is derived from the Greek language, and means "well-situated" or "well-arranged". proved that a lattice is extreme if and only if it is both perfect and eutactic. summarize the properties of eutactic lattices of dimension up to 7. References * ** * * * *{{Citation , last1=Voronoi , first1=G. , authorlink=Georgy Voronoy , title=Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premier Mémoire: Sur quelques propriétés des formes quadratiques positives parfaites , url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002166534 , language=French , doi=10.1515/crll.1908.133.97 , year=1908 , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (group)
In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb^n. For any basis of \mathbb^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. A lattice may be viewed as a regula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eutactic Star
In Euclidean geometry, a eutactic star is a geometrical figure in a Euclidean space. A star is a figure consisting of any number of opposing pairs of Euclidean vector, vectors (or arms) issuing from a central origin. A star is eutactic if it is the orthogonal projection (mathematics), projection of plus and minus the set of standard basis vectors (i.e., the vertices of a cross-polytope) from a higher-dimensional space onto a Euclidean subspace, subspace. Such stars were called "eutactic" – meaning "well-situated" or "well-arranged" – by because, for a common Scalar (mathematics), scalar multiple, their vectors are projections of an orthonormal basis. Definition A ''star'' is here defined as a set of 2''s'' vectors ''A'' = ±a1, ..., ±a''s'' issuing from a particular origin in a Euclidean space of dimension ''n'' ≤ ''s''. A star is eutactic if the a''i'' are the projections onto ''n'' dimensions of a set of mutually perpendicular equal vectors b1, .. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
