2017;
119 pp;
Softcover

MSC: Primary 51;

Print ISBN: 978-1-4704-3714-5

Product Code: MBK/108

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

**Electronic ISBN: 978-1-4704-4303-0
Product Code: MBK/108.E**

List Price: $49.00

AMS Member Price: $39.20

MAA Member Price: $44.10

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#### Supplemental Materials

# Geometry of Lengths, Areas, and Volumes: Two-Dimensional Spaces, Volume 1

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*James W. Cannon*

This is the first of a three volume collection
devoted to the geometry, topology, and curvature of 2-dimensional
spaces. The collection provides a guided tour through a wide range of
topics by one of the twentieth century's masters of geometric
topology. The books are accessible to college and graduate students
and provide perspective and insight to mathematicians at all levels
who are interested in geometry and topology.

The first volume begins with length measurement as dominated by the
Pythagorean Theorem (three proofs) with application to number theory;
areas measured by slicing and scaling, where Archimedes uses the
physical weights and balances to calculate spherical volume and is led
to the invention of calculus; areas by cut and paste, leading to the
Bolyai-Gerwien theorem on squaring polygons; areas by counting,
leading to the theory of continued fractions, the efficient rational
approximation of real numbers, and Minkowski's theorem on convex
bodies; straight-edge and compass constructions, giving complete
proofs, including the transcendence of \(e\) and
\(\pi\), of the impossibility of squaring the circle,
duplicating the cube, and trisecting the angle; and finally to a
construction of the Hausdorff-Banach-Tarski paradox that shows some
spherical sets are too complicated and cloudy to admit a well-defined
notion of area.

#### Readership

Undergraduate and graduate students and researchers interested in topology.

#### Reviews & Endorsements

Many readers will be hooked by Cannon's aesthetics and proof exposition, where geometric intuition and topological arguments play leading roles...Cannon's books are worth every cent. I have in the past gifted Hilbert & Cohn-Voseen and Rademacher and Toeplitz to my students. Now I have Cannon's trio to add to my list of giftables.

-- Tushar Das, MAA Reviews

The presentation is accessible, generously illustrated, and supported by exercises.

-- Viktor Blasjö, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Geometry of Lengths, Areas, and Volumes: Two-Dimensional Spaces, Volume 1

- Cover Cover11
- Title page iii4
- Contents v6
- Preface to the Three Volume Set vii8
- Preface to Volume 1 xi12
- Chapter 1. Lengths—The Pythagorean Theorem 114
- 1.1. Proof 1. Proof by Algebra 215
- 1.2. Proof 2. Proof by Slicing 215
- 1.3. Proof 3. Proof by Similarity 316
- 1.4. The Sharp Version of the Pythagorean Theorem—The Law of Cosines 619
- 1.5. The Pythagorean Theorem in High Dimensions 821
- 1.6. Perpendicularity and Inner Products 1023
- 1.7. The Length of a Curve 1023
- 1.8. Riemannian Metrics: Exotic Distance Formulas 1124
- 1.9. Exercises 1427
- 1.10. Selected Solutions to the Exercises. 1528

- Chapter 2. Consequences of the Pythagorean Theorem 1730
- Areas 2538
- Chapter 3. Areas by Slicing and Scaling 2740
- Chapter 4. Areas by Cut and Paste 4356
- Chapter 5. Areas by Counting 5164
- Chapter 6. Unsolvable Problems in Euclidean Geometry 7992
- Chapter 7. Does Every Set Have a Size? 99112
- Bibliography 113126
- Back Cover Back Cover1133