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Summary
Summary
The world is continuous, but the mind is discrete. David Mumford We seek to bridge some critical gaps between various ?elds of mathematics by studying the interplay between the continuous volume and the discrete v- ume of polytopes. Examples of polytopes in three dimensions include crystals, boxes, tetrahedra, and any convex object whose faces are all ?at. It is amusing to see how many problems in combinatorics, number theory, and many other mathematical areas can be recast in the language of polytopes that exist in some Euclidean space. Conversely, the versatile structure of polytopes gives us number-theoretic and combinatorial information that ?ows naturally from their geometry. Fig. 0. 1. Continuous and discrete volume. The discrete volume of a body P can be described intuitively as the number of grid points that lie inside P, given a ?xed grid in Euclidean space. The continuous volume of P has the usual intuitive meaning of volume that we attach to everyday objects we see in the real world. VIII Preface Indeed, the di?erence between the two realizations of volume can be thought of in physical terms as follows. On the one hand, the quant- level grid imposed by the molecular structure of reality gives us a discrete notion of space and hence discrete volume. On the other hand, the N- tonian notion of continuous space gives us the continuous volume.
Reviews 1
Choice Review
All mathematics majors study the topics they will need to know, should they want to go to graduate school. But most will not, and many departments recognize the need for capstone courses in which graduating students can see the tools they have acquired come together in some satisfying way. Beck (San Francisco State Univ.) and Robins (Temple Univ.) have written the perfect text for such a course. This material reinforces and extends ideas from calculus (partial fractions), advanced calculus (Green's theorem), numerical analysis (Euler-Maclauren summation), analysis (Fourier transform, but in the finite setting), complex analysis (elliptic functions), combinatorics (generating functions), and even recreational mathematics (magic squares). Students will also likely meet for the first time some basic topics (Dedekind sums). But no hodgepodge; the book develops a consistent theme--the relation between the (continuous) volume of a polytope (a notion that generalizes polygon and polyhedron) and its discrete volume, namely, the number of integer points that lie inside it. With these familiar tools, the authors lead students to a host of results that will seem simultaneously concrete and magical and almost certainly unfamiliar. Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty. D. V. Feldman University of New Hampshire
