Computing the Continuous Discretely

Computing the Continuous Discretely is a mathematics textbook that connects discrete and continuous geometry through the study of polytopes, lattice points, and their relationships. The book builds from fundamental concepts to advanced mathematical ideas, using the concrete example of coin denominations as an entry point. The text covers essential topics in geometric combinatorics, including Pick's theorem, Ehrhart polynomials, and generating functions. It bridges multiple mathematical domains - from elementary number theory to complex analysis - while maintaining accessibility for undergraduate readers through clear explanations and numerous exercises. Beck and Robins incorporate historical context and open research problems throughout the work. The material progresses from basic geometric concepts to sophisticated mathematical tools like Fourier transforms and Dedekind sums, with applications to specialized polytopes and magic square enumeration. The book represents a significant contribution to mathematical pedagogy, demonstrating how discrete methods can illuminate continuous phenomena. Its approach reveals the deep connections between seemingly disparate areas of mathematics while remaining grounded in concrete examples and applications.

Readers describe this as a clear and well-structured introduction to Ehrhart theory and polytopes. Multiple reviews mention it works well for both self-study and teaching, with detailed examples and exercises building systematically. Liked: - Progressive difficulty of exercises - Clear explanations of complex concepts - Inclusion of historical context and motivation - Quality illustrations and diagrams - Useful appendices reviewing prerequisites Disliked: - Some sections require more mathematical maturity than advertised - A few typographical errors in early printings - Limited coverage of advanced topics One reader noted: "The authors take care to develop intuition before diving into technicalities" while another mentioned "exercises that genuinely help build understanding rather than just busy work." Ratings: Goodreads: 4.4/5 (17 ratings) Amazon: 4.7/5 (11 ratings) Mathematics Stack Exchange: Frequently recommended in discrete math discussions Most critical reviews focus on wanting more advanced material rather than fundamental flaws.

A Course in Enumeration by George E. Martin This text connects discrete mathematics to continuous problems through generating functions and combinatorial methods.

Lattice Point Geometry by Martin Henk, Jürgen Richter-Gebert, and Günter M. Ziegler The book presents geometric approaches to counting lattice points in polytopes and explores connections to discrete optimization.

Lattice Points by Alexander Barvinok The text develops fundamental techniques for counting integer points in polyhedra with applications to optimization and algebraic geometry.

Integer Points in Polyhedra by Matthias Beck and Raman Sanyal This work builds on similar themes of lattice point counting while expanding into geometric and algebraic combinatorics.

Geometry of Numbers by Carl Ludwig Siegel The book establishes foundations for understanding discrete point sets in continuous geometric spaces through number theoretic methods.

🔢 The concept of counting lattice points in polygons has practical applications in cryptography and computer graphics, where discrete points need to be mapped to continuous spaces. 🎓 Author Matthias Beck is a professor at San Francisco State University known for making complex mathematical concepts accessible through innovative teaching methods. 📐 Ehrhart polynomials, a key topic in the book, were first discovered in the 1960s by Eugène Ehrhart while studying volume calculations of polyhedra. 🏛️ The mathematical principles discussed in the book have roots in ancient Greek mathematics, particularly in Pythagorean explorations of discrete versus continuous quantities. 💡 The book's coverage of Pick's theorem, which relates a polygon's area to its boundary and interior points, was originally discovered by Georg Pick in 1899 but remained relatively unknown until the 1950s.