Computing the Continuous Discretely

Integer-Point Enumeration in Polyhedra

  • Matthias Beck
  • Sinai Robins

Part of the Undergraduate Texts in Mathematics book series (UTM)

Table of contents

  1. Front Matter
    Pages i-xx
  2. The Essentials of Discrete Volume Computations

    1. Front Matter
      Pages 1-1
    2. Matthias Beck, Sinai Robins
      Pages 3-26
    3. Matthias Beck, Sinai Robins
      Pages 27-58
    4. Matthias Beck, Sinai Robins
      Pages 59-88
    5. Matthias Beck, Sinai Robins
      Pages 89-100
    6. Matthias Beck, Sinai Robins
      Pages 113-129
  3. Beyond the Basics

    1. Front Matter
      Pages 131-131
    2. Matthias Beck, Sinai Robins
      Pages 133-148
    3. Matthias Beck, Sinai Robins
      Pages 149-165
    4. Matthias Beck, Sinai Robins
      Pages 167-182
    5. Matthias Beck, Sinai Robins
      Pages 183-197
    6. Matthias Beck, Sinai Robins
      Pages 199-211
    7. Matthias Beck, Sinai Robins
      Pages 213-225
    8. Matthias Beck, Sinai Robins
      Pages 227-239
    9. Matthias Beck, Sinai Robins
      Pages 241-248
  4. Back Matter
    Pages 249-285

About this book


This richly illustrated textbook explores the amazing interaction between combinatorics, geometry, number theory, and analysis which arises in the interplay between polyhedra and lattices. Highly accessible to advanced undergraduates, as well as beginning graduate students, this second edition is perfect for a capstone course, and adds two new chapters, many new exercises, and updated open problems. For scientists, this text can be utilized as a self-contained tooling device.

The topics include a friendly invitation to Ehrhart’s theory of counting lattice points in polytopes, finite Fourier analysis, the Frobenius coin-exchange problem, Dedekind sums, solid angles, Euler–Maclaurin summation for polytopes, computational geometry, magic squares, zonotopes, and more.

With more than 300 exercises and open research problems, the reader is an active participant, carried through diverse but tightly woven mathematical fields that are inspired by an innocently elementary question: What are the relationships between the continuous volume of a polytope and its discrete volume?

Reviews of the first edition:

“You owe it to yourself to pick up a copy of Computing the Continuous Discretely to read about a number of interesting problems in geometry, number theory, and combinatorics.”

— MAA Reviews

“The book is written as an accessible and engaging textbook, with many examples, historical notes, pithy quotes, commentary integrating the mate

rial, exercises, open problems and an extensive bibliography.”

— Zentralblatt MATH

“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.”

— Mathematical Reviews

“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 and Robins have written the perfect text for such a course.”



Dehn-Sommerville relations Ehrhart theory Euler-Maclaurin summation combinatorics discrete geometry discrete mathematics discrete volume of a polytope finite Fourier analysis magic squares number theory

Authors and affiliations

  • Matthias Beck
    • 1
  • Sinai Robins
    • 2
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Department of Mathematics, Brown UniversityProvidenceUSA

Bibliographic information