In mathematics, a eutactic lattice (or eutactic form) is a
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an ornam ...
in
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 ther ...
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
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*{{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 , journal=
Journal für die reine und angewandte Mathematik
''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics'').
History
The journal was founded by Augus ...
, issn=0075-4102 , volume=133 , pages=97–178 , issue=133
Quadratic forms