Computing the Continuous Discretely

1. Front Matter

   Pages i-xviii

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2. The Essentials of Discrete Volume Computations

   1. The Coin-Exchange Problem of Frobenius

      Pages 3-24

   2. A Gallery of Discrete Volumes

      Pages 25-55

   3. Counting Lattice Points in Polytopes:The Ehrhart Theory

      Pages 57-82

   4. Reciprocity

      Pages 83-93

   5. Face Numbers and the Dehn—Sommerville Relations in Ehrhartian Terms

      Pages 95-104

   6. Magic Squares

      Pages 105-120

3. Beyond the Basics

   1. Finite Fourier Analysis

      Pages 123-137

   2. Dedekind Sums, the Building Blocks of Lattice-point Enumeration

      Pages 139-153

   3. The Decomposition of a Polytope into Its Cones

      Pages 155-165

   4. Euler—Maclaurin Summation in ℝ d

      Pages 167-178

   5. Solid Angles

      Pages 179-190

   6. A Discrete Version of Green’s Theorem Using Elliptic Functions

      Pages 191-197

4. Back Matter

   Pages 199-234

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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.

• Combinatorics
• Lattice
• calculus
• combinatorial geometry
• computational geometry
• discrete geometry
• number theory

From the reviews:

"You owe it to yourself to pick up a copy … to read about a number of interesting problems in geometry, number theory, and combinatorics … . Even people who are familiar with the material would almost certainly learn something from the clear and engaging exposition … . It contains a large number of exercises … . Each chapter also ends with a series of relevant open problems … . it is also full of mathematics that is self-contained and worth reading on its own." (Darren Glass, MathDL, February, 2007)

"This beautiful book presents, at a level suitable for advanced undergraduates, a fairly complete introduction to the problem of counting lattice points inside a convex polyhedron. … Most importantly the book gives a complete presentation of the use of generating functions of various kinds to enumerate lattice points, as well as an introduction to the theory of Erhart quasipolynomials. … This book provides many well-crafted exercises, and even a list of open problems in each chapter." (Jesús A. De Loera, Mathematical Reviews, Issue 2007 h)

"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. … Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (2), 2007)

"This book is concerned with the mathematics of that connection between the discrete and the continuous, with significance for geometry, number theory and combinatorics. … The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the material, exercises, open problems and an extensive bibliography." (Margaret M. Bayer, Zentralblatt MATH, Vol. 1114 (16), 2007)

"The main topic of the book is initiated by a theorem of Ehrhart … . This is a wonderful book for various readerships. Students, researchers, lecturers in enumeration, geometry and number theory all find it very pleasing and useful. The presentation is accessible for mature undergraduates. … it is a clear introduction to graduate students and researchers with many exercises and with a list of open problems at the end of each chapter." (Péter Hajnal, Acta Scientiarum Mathematicarum, Vol. 75, 2009)

“The theme of this textbook … is the relation between the continuous and the discrete, namely, the interplay between the volume of a polytope and the number of grid points contained in it. … The text contains many exercises, some of which present material needed later (for these exercises hints are provided), and lists also many open research problems. – The book can be recommended both for its style and for its interesting … content.” (P. Schmitt, Monatshefte für Mathematik, Vol. 155 (2), October, 2008)

• Mathematics Department, San Francisco State University, San Francisco, USA

  Matthias Beck

• Department of Mathematics, Temple University, Philadelphia, USA

  Sinai Robins