The Geometry of Numbers available in Paperback
- Pub. Date:
- Mathematical Association of America
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1.1 The Fundamental Lattice The theme of this book is the geometry of numbers, a branch of the theory of numbers that was discovered by Hermann Minkowski (1864-1909). Where other mathematicians had attacked problems of certain types algebraically, Minkowski's genius was to approach them from a geometrical point of view. Through the visible order of geometrical constructs, he was able to reveal and explore many numerical relationships. We shall trace Minkowski's explorations in the second part of this book. Here, we begin at the beginning, by defining the two elementary concepts on which the entire geometry of numbers rests: the fundamental lattice Land the fundamental point-lattice A that L determines. When we speak of lattice systems, we are imagining grids of points in space connected like monkey bars on a playground. We can construct a simple planar lattice on an ordinary rectangular coordinate system just by drawing straight lines. First we draw lines parallel to the y-axis through the points ..., ((-2,0), (-1,0), (0,0), (1,0), (2,0), (3,0),..., and then we draw lines parallel to the x-axis through the points ..., (0,-2), (0,-1),(0,0),(0,1), (0,2),..., These lines form the fundamental lattice L. The points where these lines intersect, called lattice points, are the vertices of squares. Lattice points are points with integer coordinates (x,y), as Figure 1.1 indicates. For clarity, to distinguish between a lattice point and any point (x,y), we shall designate a lattice point by (p,q), wher ex=p, y=q are integers. It is the lattice points that determine the fundamental point-lattice A. For now, we shall consider only lattices in the plane, saving n-dimensional figures for Part II. Evenly spaced along lines running parallel to the x- and y- axes, at first glance lattice points seem to be of little interest in themselves. In 1837, Carl Friedrich Gauss (1777-1855)  published a seminal paper on the number of lattice points contained within and on a circle of radius r. Since then, mathematicians have proposed a great many intriguing related problems. Many of these problems have now been answered; some are only partially solved; and others-including the optimal form of Gauss's original questions- still defy solutions. To prepare ourselves for examining such problems, we must first discuss some of the relationships between straight lines and lattice points.
Table of Contents1. Geometry and algebra; 2. Trigonometry, calculus and analytic geometry; 3. Inequalities; 4. Integer sums; 5. Sequences and series.