Eutactic lattice
{{Short description|Set of points in Euclidean space}}
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 ci such that (v, v) = Σci(v, mi)2 where the sum is over the minimal vectors mi. "Eutactic" is derived from the Greek language, and means "well-situated" or "well-arranged".
{{harvtxt|Voronoi|1908}} proved that a lattice is extreme if and only if it is both perfect and eutactic.
{{harvtxt|Conway|Sloane|1988}} summarize the properties of eutactic lattices of dimension up to 7.
References
- {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | title=Low-dimensional lattices. III. Perfect forms | jstor=2398316 | mr=953277 | year=1988 | journal=Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences | issn=0962-8444 | volume=418 | issue=1854 | pages=43–80 | doi=10.1098/rspa.1988.0073}}
- {{Citation |jstor=2398351 |pages=441 |last1=Conway |first1=J. H. |last2=Sloane |first2=N. J. A. |title=Errata: Low-Dimensional Lattices. III. Perfect Forms |volume=426 |issue=1871 |journal=Proceedings of the Royal Society of London |year=1989 |doi=10.1098/rspa.1989.0134 |postscript=.}}
- {{Citation | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | title=Extreme forms | url=http://www.math.ca/10.4153/CJM-1951-045-8 | doi=10.4153/CJM-1951-045-8 | mr=0044580 | year=1951 | journal=Canadian Journal of Mathematics | issn=0008-414X | volume=3 | pages=391–441| doi-access=free }}
- {{Citation | last1=Korkine | last2=Zolotareff | title=Sur les formes quadratique positives | doi=10.1007/BF01442667 | year=1877 | journal=Mathematische Annalen | issn=0025-5831 | volume=11 | pages=242–292 | first1=A. | first2=G. | issue=2| url=https://zenodo.org/record/1896288/files/article.pdf }}
- {{Citation | last1=Martinet | first1=Jacques | title=Perfect lattices in Euclidean spaces | url=https://books.google.com/books?id=gd9CcFclBRIC | publisher=Springer-Verlag | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-44236-3 | mr=1957723 | year=2003 | volume=327}}
- {{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 | issn=0075-4102 | volume=133 | pages=97–178 | issue=133}}