In mathematics, a eutactic lattice (or eutactic form) is a
lattice in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
whose minimal vectors form a
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 th ...
. 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
Perfect commonly refers to:
* Perfection, completeness, excellence
* Perfect (grammar), a grammatical category in some languages
Perfect may also refer to:
Film
* Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama
* Perfect (2018 f ...
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 language, English: ''Journal for Pure and Applied Mathematics'').
History
The journal wa ...
, issn=0075-4102 , volume=133 , pages=97–178 , issue=133
Quadratic forms