Abstract
It is shown that if a simplex S is a basic L-simplex for a point lattice in En (n≤5), then the lattice's L-simplexes that are contiguous to S by (n−1)-faces can have as vertices lattice points belonging to a specified set of points P(S), and a complete description of this set is given. Based on the fact that the set P(S) is known, a new method of deriving the types of point lattices, different from the known methods (G. F. Voronoi's algorithm and B. N. Delaunay's method of layers), is obtained. The types of primitive lattices in E3 and E4 are derived by this method.
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E. P. Baranovskii, “Simplexes of L-subdivisions of Euclidean spaces” Matem. Zametki,10, No. 6, 659–670 (1971).
G. F. Voronoi, Collected Works [in Russian], Vol. 2, Kiev (1952).
B. N. Delaunay, “Sur la partition reguliére de l'espace a 4 dimensions,” Izv. Akad. Nauk SSSR, Ser. Matem., No. 1, 79–110 (1929); No. 2, 147–164 (1929).
B. N. Delaunay, “The geometry of positive quadratic forms,” Uspekhi Matem. Nauk,3, 16–62 (1937);4, 102–164 (1938).
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Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 605–616, April, 1973.
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Baranovskii, E.P. Theorem on L-partitions of point lattices. Mathematical Notes of the Academy of Sciences of the USSR 13, 364–370 (1973). https://doi.org/10.1007/BF01146577
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DOI: https://doi.org/10.1007/BF01146577