Proofs from THE BOOK

This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises.

From the Reviews


"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... "
Notices of the AMS, August 1999


"... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..."
LMS Newsletter, January 1999


"Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio
ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... "
SIGACT News, December 2011.

List of contents

Number Theory: 1. Six proofs of the infinity of primes.- 2. Bertrand's postulate.- 3. Binomial coefficients are (almost) never powers.- 4. Representing numbers as sums of two squares.- 5. The law of quadratic reciprocity.- 6. Every finite division ring is a field.- 7. The spectral theorem and Hadamard's determinant problem.- 8. Some irrational numbers.- 9. Three times pi2/6.- Geometry: 10. Hilbert's third problem: decomposing polyhedral.- 11. Lines in the plane and decompositions of graphs.- 12. The slope problem.- 13. Three applications of Euler's formula.- 14. Cauchy's rigidity theorem.- 15. The Borromean rings don't exist.- 16. Touching simplices.- 17. Every large point set has an obtuse angle.- 18. Borsuk's conjecture.- Analysis: 19. Sets, functions, and the continuum hypothesis.- 20. In praise of inequalities.- 21. The fundamental theorem of algebra.- 22. One square and an odd number of triangles.- 23. A theorem of Pólya on polynomials.- 24. On a lemma of Littlewood and Offord.- 25. Cotangent and the Herglotz trick.- 26. Buffon's needle problem.- Combinatorics: 27. Pigeon-hole and double counting.- 28. Tiling rectangles.- 29. Three famous theorems on finite sets.- 30. Shuffling cards.- 31. Lattice paths and determinants.- 32. Cayley's formula for the number of trees.- 33. Identities versus bijections.- 34. The finite Kakeya problem.- 35. Completing Latin squares.- Graph Theory: 36. The Dinitz problem.- 37. Permanents and the po
wer of entropy.- 38. Five-coloring plane graphs.- 39. How to guard a museum.- 40. Turán's graph theorem.- 41. Communicating without errors.- 42. The chromatic number of Kneser graphs.- 43. Of friends and politicians.- 44. Probability makes counting (sometimes) easy.- About the Illustrations.- Index.

About the author

Martin Aigner received his Ph.D. from the University of Vienna and has been professor of mathematics at the Freie Universität Berlin since 1974. He has published in various fields of combinatorics and graph theory and is the author of several monographs on discrete mathematics, among them the Springer books Combinatorial Theory and A Course on Enumeration. Martin Aigner is a recipient of the 1996 Lester R. Ford Award for mathematical exposition of the Mathematical Association of America MAA.
 
Günter M. Ziegler received his Ph.D. from M.I.T. and has been professor of mathematics in Berlin - first at TU Berlin, now at Freie Universität - since 1995. He has published in discrete mathematics, geometry, topology, and optimization, including the Lectures on Polytopes with Springer, as well as "Do I Count? Stories from Mathematics". Günter M. Ziegler is a recipient of the 2006 Chauvenet Prize of the MAA for his expository writing and the 2008 Communicatoraward of the German Science Foundation.
 
Martin Aigner and Günter M. Ziegler have started their work on Proofs from THE BOOK in 1995 together with Paul Erdös. The first edition of this book appeared in 1998 - it has since been translated into 13 languages: Brazilian, Chinese, German, Farsi, French, Hungarian, Italian, Japanese, Korean, Polish, Russian, Spanish, and Turkish.

Summary

This revised and enlarged fifth edition features four new chapters, which contain highly original and delightful proofs for classics such as the spectral theorem from linear algebra, some more recent jewels like the non-existence of the Borromean rings and other surprises.

From the Reviews

"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... "
Notices of the AMS, August 1999

"... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautifuldrawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..."
LMS Newsletter, January 1999

"Martin Aigner and Günter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erdös. The theorems are so fundamental, their proofs so elegant and the remaining open questio
ns so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... "

SIGACT News, December 2011.

Additional text

“This book by Aigner and Ziegler, now in its fifth edition, seeks to pay homage to the late Paul Erdõs by attempting to provide an approximation of ‘The Book.’ … Throughout, illustrations and figures are used to support the arguments in the main text; these can greatly help the readability of the proofs, especially for novices like me. … the book is a marvelous project and this new edition provides a good amount of fresh material.” (Harry Strange, Computing Reviews, March, 2015)

Report

"This book by Aigner and Ziegler, now in its fifth edition, seeks to pay homage to the late Paul Erdõs by attempting to provide an approximation of 'The Book.' ... Throughout, illustrations and figures are used to support the arguments in the main text; these can greatly help the readability of the proofs, especially for novices like me. ... the book is a marvelous project and this new edition provides a good amount of fresh material." (Harry Strange, Computing Reviews, March, 2015)

Product details

• Authors: Marti Aigner, Martin Aigner, Günter M Ziegler, Günter M. Ziegler
• Assisted by: Karl H. Hofmann (Illustration)
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2014
• EAN: 9783662495926
• ISBN: 978-3-662-49592-6
• No. of pages: 308
• Dimensions: 190 mm x 18 mm x 246 mm
• Weight: 646 g
• Illustrations: VIII, 308 p. 255 illus., 9 illus. in color.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics; Analysis, Informatik, B, Diskrete Mathematik, Mathematische Analysis, allgemein, Mathematics, computer science, Computer Science, general, geometry, Combinatorics, Mathematics and Statistics, Mathematics, general, Discrete Mathematics, Number Theory, Combinatorics & graph theory, Calculus & mathematical analysis, Analysis (Mathematics), Mathematical analysis