论文信息 - Planar lattices and equilateral polygons

Planar lattices and equilateral polygons

Abstract If a planar lattice contains the vertices of a convex equilateral n -gon for some n ≠ 4 , then the lattice is similar to a sublattice of Z 5 in R 5 , and it contains the vertices of a convex equilateral 2 n -gon for every n ≥ 2 . If a planar lattice contains the vertices of an equilateral triangle, then the lattice is similar to a sublattice of Z 3 in R 3 , and it contains the vertices of a convex equilateral n -gon for every n ≥ 3 .

Hiroshi Maehara

[1] Jiri Matousek,et al. Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[2] N. Sloane,et al. Low-dimensional lattices V. Integral coordinates for integral lattices , 1989, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3] Bôcher's theorem , 1992 .

[4] Hiroshi Maehara. Circle lattice point problem, revisited , 2015, Discret. Math..

[5] The Fascination of the Elementary , 1987 .

[6] W. Plesken. Solving XXtr = A over the integers , 1995 .

[7] W. Scherrer. Die Einlagerung eines regulären Vielecks in ein Gitter. , 1946 .

[8] M. Beeson. Triangles with vertices on lattice points , 1992 .

[9] W. Narkiewicz. Classical problems in number theory , 1986 .

[10] Hiroshi Maehara. On a sphere that passes through n lattice points , 2010, Eur. J. Comb..

[11] I. J. Schoenberg. Mathematical Time Exposures , 1982 .

[12] I. Niven,et al. An introduction to the theory of numbers , 1961 .

[13] Hiroshi Maehara,et al. Embedding a polytope in a lattice , 1995, Discret. Comput. Geom..

[14] E. A. Maxwell,et al. Mathematical Gems II , 1976, The Mathematical Gazette.

[15] L. Mordell. The Representation of a Definite Quadratic Form as a Sum of Two Others , 1937 .

[16] Hiroshi Maehara,et al. Is There a Circle that Passes Through a Given Number of Lattice Points? , 1998, Eur. J. Comb..

[17] E. Bannai,et al. On a property of 2-dimensional integral Euclidean lattices , 2009, 0912.1659.