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

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 Read more... 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 s, 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. -- This is the second 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 second volume deals with the topology of 2-dimensional spaces. The attempts encountered in Volume 1 to understand length and area in the plane lead to examples most easily described by the methods of topology (fluid geometry): finite curves of infinite length, 1-dimensional curves of positive area, space-filling curves (Peano curves), 0-dimensional subsets of the plane through which no straight path can pass (Cantor sets), etc. Volume 2 describes such sets. All of the standard topological results about 2-dimensional spaces are then proved, such as the Fundamental Theorem of Algebra (two proofs), the No Retraction Theorem, the Brouwer Fixed Point Theorem, the Jordan Curve Theorem, the Open Mapping Theorem, the Riemann-Hurwitz Theorem, and the Classification Theorem for Compact 2-manifolds. Volume 2 also includes a number of theorems usually assumed without proof since their proofs are not readily available, for example, the Zippin Characterization Theorem for 2-dimensional spaces that are locally Euclidean, the Schoenflies Theorem characterizing the disk, the Triangulation Theorem for 2-manifolds, and the R. L. Moore's Decomposition Theorem so useful in understanding fractal sets. -- This is the final volume 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. Einstein showed how to interpret gravity as the dynamic response to the curvature of space-time. Bill Thurston showed us that non-Euclidean geometries and curvature are essential to the understanding of low-dimensional spaces. This third and final volume aims to give the reader a firm intuitive understanding of these concepts in dimension 2. The volume first demonstrates a number of the most important properties of non-Euclidean geometry by means of simple infinite graphs that approximate that geometry. This is followed by a long chapter taken from lectures the author gave at MSRI, which explains a more classical view of hyperbolic non-Euclidean geometry in all dimensions. Finally, the author explains a natural intrinsic obstruction to flattening a triangulated polyhedral surface into the plane without distorting the constituent triangles. That obstruction extends intrinsically to smooth surfaces by approximation and is called curvature. Gauss's original definition of curvature is extrinsic rather than intrinsic. The final two chapters show that the book's intrinsic definition is equivalent to Gauss's extrinsic definition (Gauss's 2Theorema Egregium3 (2Great Theorem3)) Cover 1 Title page 4 Contents 6 Preface to the Three Volume Set 8 Preface to Volume 1 12 Chapter 1. Lengths—The Pythagorean Theorem 14 1.1. Proof 1. Proof by Algebra 15 1.2. Proof 2. Proof by Slicing 15 1.3. Proof 3. Proof by Similarity 16 1.4. The Sharp Version of the Pythagorean Theorem—The Law of Cosines 19 1.5. The Pythagorean Theorem in High Dimensions 21 1.6. Perpendicularity and Inner Products 23 1.7. The Length of a Curve 23 1.8. Riemannian Metrics: Exotic Distance Formulas 24 1.9. Exercises 27 1.10. Selected Solutions to the Exercises. 28 Chapter 2. Consequences of the Pythagorean Theorem 30 2.1. The Square Root of 2 Is Irrational 30 2.2. Pythagorean Triples 32 2.3. The Euclidean Algorithm 34 2.4. Proof of the Rational-Root Theorem, Theorem 2.5 35 2.5. Exercises 36 Areas 38 Chapter 3. Areas by Slicing and Scaling 40 3.1. Slicing and Scaling 40 3.2. What Is π? 42 3.3. Archimedes Discovers the Volume of a Sphere 44 3.4. Wallis Discovers a Product Formula for π 48 3.5. Fourier Discovers Fourier Series 51 3.6. Exercises 53 Chapter 4. Areas by Cut and Paste 56 4.1. Euclidean Constructions 56 4.2. Cut and Paste Constructions 58 4.3. Exercises 62 Chapter 5. Areas by Counting 64 5.1. The Area Formula 64 5.2. Three Basic Geometric Consequences 67 5.3. Farey Sequences 69 5.4. Lattice Generators 72 5.5. Efficient Rational Approximation 73 5.6. Continued Fractions 76 5.7. The Complement of the Lattice Set 82 5.8. Exercises 90 Chapter 6. Unsolvable Problems in Euclidean Geometry 92 6.1. The Basic Constructions in Euclidean Geometry 93 6.2. Translation into Algebra 94 6.3. Algebraic Lemmas 99 6.4. Impossibilities 101 6.5. e and π Are Transcendental 102 6.6. Exercises 109 Chapter 7. Does Every Set Have a Size? 112 7.1. A Subset of [0,1] that Is Not Measurable 113 7.2. The Free Group F 114 7.3. The Hausdorff-Banach-Tarski Paradox 116 7.4. Exercises 122 Bibliography 126 Back Cover 133