In mathematics, a perfect lattice (or perfect 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 a
Euclidean vector 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 s ...
, that is completely determined by the set ''S'' of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of ''S''. Perfect lattices were introduced by . A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by .
proved that a lattice is extreme if and only if it is both perfect and
eutactic.
The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by
1, 1, 1, 2, 3, 7, 33, 10916 . summarize the properties of perfect lattices of dimension up to 7.
verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.
References
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*{{Citation , last1=Voronoi , first1=G. , 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=1908 , issue=133 , pages=97–178, url-access=subscription
External links
List of perfect lattices in dimension 8
Quadratic forms