Hardcover ISBN:  9780821843161 
Product Code:  MBK/46 
List Price:  $69.00 
MAA Member Price:  $62.10 
AMS Member Price:  $55.20 
Electronic ISBN:  9781470418120 
Product Code:  MBK/46.E 
List Price:  $65.00 
MAA Member Price:  $58.50 
AMS Member Price:  $52.00 

Book Details2007; 463 ppMSC: Primary 00;
The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an awardwinning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.
ReadershipUndergraduates, graduate students, and research mathematicians interested in mathematics.

Table of Contents

Algebra and arithmetics

Chapter 1. Arithmetic and combinatorics

Lecture 1. Can a number be approximately rational?

Lecture 2. Arithmetical properties of binomial coefficients

Lecture 3. On collecting like terms, on Euler, Gauss, and MacDonald, and on missed opportunities

Chapter 2. Equations

Lecture 4. Equations of degree three and four

Lecture 5. Equations of degree five

Lecture 6. How many roots does a polynomial have?

Lecture 7. Chebyshev polynomials

Lecture 8. Geometry of equations

Geometry and topology

Chapter 3. Envelopes and singularities

Lecture 9. Cusps

Lecture 10. Around four vertices

Lecture 11. Segments of equal areas

Lecture 12. On plane curves

Chapter 4. Developable surfaces

Lecture 13. Paper sheet geometry

Lecture 14. Paper Möbius band

Lecture 15. More on paper folding

Chapter 5. Straight lines

Lecture 16. Straight lines on curved surfaces

Lecture 17. Twentyseven lines

Lecture 18. Web geometry

Lecture 19. The Crofton formula

Chapter 6. Polyhedra

Lecture 20. Curvature and polyhedra

Lecture 21. Noninscribable polyhedra

Lecture 22. Can one make a tetrahedron out of a cube?

Lecture 23. Impossible tilings

Lecture 24. Rigidity of polyhedra

Lecture 25. Flexible polyhedra

Chapter 7. Two surprising topological constructions

Lecture 26. Alexander’s horned sphere

Lecture 27. Cone eversion

Chapter 8. On ellipses and ellipsoids

Lecture 28. Billiards in ellipses and geodesics on ellipsoids

Lecture 29. The Poncelet porism and other closure theorems

Lecture 30. Gravitational attraction of ellipsoids

Lecture 31. Solutions to selected exercises


Additional Material

Reviews

The authors manage to breathe new life into topics that at first glance appear to be old hat.
Springer Science & Business 
This is an enjoyable book with suggested uses ranging from a text for a undergraduate Honors Mathematics Seminar to a coffee table book. It is appropriate for either It could also be used as a starting point for undergraduate research topics or a place to find a short undergraduate seminar talk. This is a wonderful book that is not only fun to read, but gives the reader new ideas to think about.
MAA Reviews 
Dmitry Fuchs and Serge Tabachnikov display impeccable taste in their choice of the material, level of exposition, and the balance between concrete and more conceptual mathematical themes. Each of the thirty lectures tells a unique mathematical story, each with a display of mathematical narrative art, with great care for the details, revealing masters of their craft at work. Both novice and more experienced readers will find many pleasant surprises at all levels of exposition. ...[A] book suitable for such a noble and demanding goal to serve as an introduction to the world of 'serious mathematics' for new generations of mathematicians. ...[E]very page has one or more diagrams, graphs, and pictures illustrating the material. ...[T]he special artistic spirit and atmosphere the book owes to numerous, witty, humorous, and mysterious illustrations of the artist Sergey Ivanov. Without much exaggeration, one may say that these provocative, yet mathematically correct drawings can alone serve as a layman's guide to the beauty and mystery of mathematics. Summarizing, we can say that Mathematical Omnibus is a 'desert island book,' a 'coffee table book,' a book to share with friends, colleagues, and students, a gift for a beginner and an expert alike. In short, it is a wonderful addition to our personal, school, and university libraries.
Rade T. Zivaljevic, The American Mathematical Monthly


RequestsReview Copy – for reviewers who would like to review an AMS bookPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an awardwinning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.
Undergraduates, graduate students, and research mathematicians interested in mathematics.

Algebra and arithmetics

Chapter 1. Arithmetic and combinatorics

Lecture 1. Can a number be approximately rational?

Lecture 2. Arithmetical properties of binomial coefficients

Lecture 3. On collecting like terms, on Euler, Gauss, and MacDonald, and on missed opportunities

Chapter 2. Equations

Lecture 4. Equations of degree three and four

Lecture 5. Equations of degree five

Lecture 6. How many roots does a polynomial have?

Lecture 7. Chebyshev polynomials

Lecture 8. Geometry of equations

Geometry and topology

Chapter 3. Envelopes and singularities

Lecture 9. Cusps

Lecture 10. Around four vertices

Lecture 11. Segments of equal areas

Lecture 12. On plane curves

Chapter 4. Developable surfaces

Lecture 13. Paper sheet geometry

Lecture 14. Paper Möbius band

Lecture 15. More on paper folding

Chapter 5. Straight lines

Lecture 16. Straight lines on curved surfaces

Lecture 17. Twentyseven lines

Lecture 18. Web geometry

Lecture 19. The Crofton formula

Chapter 6. Polyhedra

Lecture 20. Curvature and polyhedra

Lecture 21. Noninscribable polyhedra

Lecture 22. Can one make a tetrahedron out of a cube?

Lecture 23. Impossible tilings

Lecture 24. Rigidity of polyhedra

Lecture 25. Flexible polyhedra

Chapter 7. Two surprising topological constructions

Lecture 26. Alexander’s horned sphere

Lecture 27. Cone eversion

Chapter 8. On ellipses and ellipsoids

Lecture 28. Billiards in ellipses and geodesics on ellipsoids

Lecture 29. The Poncelet porism and other closure theorems

Lecture 30. Gravitational attraction of ellipsoids

Lecture 31. Solutions to selected exercises

The authors manage to breathe new life into topics that at first glance appear to be old hat.
Springer Science & Business 
This is an enjoyable book with suggested uses ranging from a text for a undergraduate Honors Mathematics Seminar to a coffee table book. It is appropriate for either It could also be used as a starting point for undergraduate research topics or a place to find a short undergraduate seminar talk. This is a wonderful book that is not only fun to read, but gives the reader new ideas to think about.
MAA Reviews 
Dmitry Fuchs and Serge Tabachnikov display impeccable taste in their choice of the material, level of exposition, and the balance between concrete and more conceptual mathematical themes. Each of the thirty lectures tells a unique mathematical story, each with a display of mathematical narrative art, with great care for the details, revealing masters of their craft at work. Both novice and more experienced readers will find many pleasant surprises at all levels of exposition. ...[A] book suitable for such a noble and demanding goal to serve as an introduction to the world of 'serious mathematics' for new generations of mathematicians. ...[E]very page has one or more diagrams, graphs, and pictures illustrating the material. ...[T]he special artistic spirit and atmosphere the book owes to numerous, witty, humorous, and mysterious illustrations of the artist Sergey Ivanov. Without much exaggeration, one may say that these provocative, yet mathematically correct drawings can alone serve as a layman's guide to the beauty and mystery of mathematics. Summarizing, we can say that Mathematical Omnibus is a 'desert island book,' a 'coffee table book,' a book to share with friends, colleagues, and students, a gift for a beginner and an expert alike. In short, it is a wonderful addition to our personal, school, and university libraries.
Rade T. Zivaljevic, The American Mathematical Monthly