Geometry of Lengths, Areas, and Volumes Two-Dimensional Spaces

133 Pages·2017·1.24 MB·English

by  Cannon

Two-Dimensional Spaces, Volume 1
GEOMETRY OF LENGTHS,
AREAS, AND VOLUMES
James W. Cannon
Two-Dimensional Spaces, Volume 1
GEOMETRY OF LENGTHS,
AREAS, AND VOLUMES

Two-Dimensional Spaces, Volume 1
GEOMETRY OF LENGTHS,
AREAS, AND VOLUMES
James W. Cannon

A M E R I C A N M AT H E M AT I C A L S O C I E T Y
Providence, Rhode Island
2010 Mathematics Subject Classification. Primary 51-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/mbk-108
Library of Congress Cataloging-in-Publication Data
Names: Cannon,JamesW.,author.
Title: Two-dimensionalspaces/JamesW.Cannon.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2017]|Includesbibli-
ographicalreferences.
Identifiers: LCCN2017024690|ISBN9781470437145(v. 1)|ISBN9781470437152(v. 2)|ISBN
9781470437169(v. 3)
Subjects: LCSH: Geometry. | Geometry,Plane. |Non-Euclidean geometry. | AMS:Geometry–
Instructionalexposition(textbooks,tutorialpapers,etc.). msc
Classification: LCCQA445.C272017|DDC516–dc23
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Contents
Preface to the Three Volume Set vii
Preface to Volume 1 xi
Chapter 1. Lengths—The Pythagorean Theorem 1
1.1. Proof 1. Proof by Algebra 2
1.2. Proof 2. Proof by Slicing 2
1.3. Proof 3. Proof by Similarity 3
1.4. The Sharp Version of the Pythagorean Theorem—The Law of Cosines 6
1.5. The Pythagorean Theorem in High Dimensions 8
1.6. Perpendicularity and Inner Products 10
1.7. The Length of a Curve 10
1.8. Riemannian Metrics: Exotic Distance Formulas 11
1.9. Exercises 14
1.10. Selected Solutions to the Exercises. 15
Chapter 2. Consequences of the Pythagorean Theorem 17
2.1. The Square Root of 2 Is Irrational 17
2.2. Pythagorean Triples 19
2.3. The Euclidean Algorithm 21
2.4. Proof of the Rational-Root Theorem, Theorem 2.5 22
2.5. Exercises 23
Areas 25
Chapter 3. Areas by Slicing and Scaling 27
3.1. Slicing and Scaling 27
3.2. What Is π? 29
3.3. Archimedes Discovers the Volume of a Sphere 31
3.4. Wallis Discovers a Product Formula for π 35
3.5. Fourier Discovers Fourier Series 38
3.6. Exercises 40
Chapter 4. Areas by Cut and Paste 43
4.1. Euclidean Constructions 43
4.2. Cut and Paste Constructions 45
4.3. Exercises 49
Chapter 5. Areas by Counting 51
5.1. The Area Formula 51
5.2. Three Basic Geometric Consequences 54
v
vi CONTENTS
5.3. Farey Sequences 56
5.4. Lattice Generators 59
5.5. Efficient Rational Approximation 60
5.6. Continued Fractions 63
5.7. The Complement of the Lattice Set 69
5.8. Exercises 77
Chapter 6. Unsolvable Problems in Euclidean Geometry 79
6.1. The Basic Constructions in Euclidean Geometry 80
6.2. Translation into Algebra 81
6.3. Algebraic Lemmas 86
6.4. Impossibilities 88
6.5. e and π Are Transcendental 89
6.6. Exercises 96
Chapter 7. Does Every Set Have a Size? 99
7.1. A Subset of [0,1] that Is Not Measurable 100
7.2. The Free Group F 101
7.3. The Hausdorff-Banach-Tarski Paradox 103
7.4. Exercises 109
Bibliography 113
Preface to the Three Volume Set
Geometry measures space (geo = earth, metry = measurement). Einstein’s
theory of relativity measures space-time and might be called geochronometry (geo
= space, chrono= time, metry = measurement). The arc of mathematical history
that has led us from the geometry of the plane of Euclid and the Greeks after 2500
years to the physics of space-time of Einstein is an attractive mathematical story.
Geometricalreasoninghasprovedinstrumentalinourunderstandingoftherealand
complexnumbers, algebraandnumber theory,thedevelopmentofcalculuswithits
elaboration in analysis and differential equations, our notions of length, area, and
volume, motion, symmetry, topology, and curvature.
These threevolumes form averypersonal excursion throughthose parts of the
mathematics of 1- and 2-dimensional geometry that I have found magical. In all
cases,thispointofviewistheonemostmeaningfultome. Everysectionisdesigned
around results that, as a student, I found interesting in themselves and not just
as preparation for something to come later. Where is the magic? Why are these
things true?
Where is the tension? Every good theorem should have tension
between hypothesis and conclusion. — Dennis Sullivan
Where is the Sullivan tension in the statement and proofs of the theorems?
What are the key ideas? Why is the given proof natural? Are the theorems almost
false? Is there a nice picture? I am not interestedin quotingresults without proof.
I am not afraid of a little algebra, or calculus, or linear algebra. I do not care
about complete rigor. I want to understand. If every formula in a book cuts the
readership in half, my audience is a small, elite audience. This book is for the
student who likes the magic and wants to understand.
A scientist is someone who is always a child, asking ‘Why?
why? why?’. — Isidor Isaac Rabi, Nobel Prize in Physics 1944
Wir mu¨ssen wissen, wir werden wissen. [We must know, we
will know.] — David Hilbert
The three volumes indicate three natural parts into which the material on 2-
dimensional spaces may be divided:
Volume 1: The geometry of the plane, with various historical attempts to
understand lengths and areas: areas by similarity, by cut and paste, by counting,
byslicing. Applicationstotheunderstanding oftherealnumbers, algebra, number
theory, and the development of calculus. Limitations imposed on the measurement
of size given by nonmeasurable sets and the wonderful Hausdorff-Banach-Tarski
paradox.
Volume 2: The topology of the plane, with all of the standard theorems of
1- and 2-dimensional topology, the fundamental theorem of algebra, the Brouwer
vii
viii PREFACE TO THE THREE VOLUME SET
fixed-pointTheorem, space-fillingcurves, curvesofpositivearea, theJordanCurve
Theorem, the topological characterization of the plane, the Schoenflies Theorem,
the R. L. Moore Decomposition Theorem, the Open Mapping Theorem, the trian-
gulation of 2-manifolds, the classification of 2-manifolds via orientation and Euler
characteristic, dimension theory.
Volume 3: An introduction to non-Euclidean geometry and curvature. What
is the analogy between the standard trigonometric functions and the hyperbolic
trig functions? Why is non-Euclidean geometry called hyperbolic? What are the
gross intuitive differences between Euclidean and hyperbolic geometry?
The approach to curvature is backwards to that of Gauss, with definitions
that are obviously invariant under bending, with the intent that curvature should
obviouslymeasurethedegreetowhichasurfacecannotbeflattenedintotheplane.
Gauss’s Theorema Egregium then comes at the end of the discussion.
Prerequisites: An undergraduate student with a reasonable memory of cal-
culus and linear algebra, but with no fear of proofs, should be able to understand
almostallofthefirstvolume. Astudentwiththerudimentsoftopology—openand
closed sets, continuous functions, compact sets and uniform continuity—should be
able to understand almost all of the second volume with the exeption of a little
bit of algebraic topology used to prove results that are intuitively reasonable and
can be assumed if necessary. The final volume should be well within the reach of
someonewhoiscomfortablewithintegrationandchangeofvariables. Wewillmake
an attempt in many places to review the tools needed.
Comments on exercises: Mostexercisesareinterlacedwiththetextinthose
places where the development suggests them. They are an essential part of the
text, and the reader should at least make note of their content. Exercise sections
which appear at the end of most chapters refer back to these exercises, sometimes
with hints, occasionally with solutions, and sometimes add additional exercises.
Readersshouldtryasmanyexercisesasattractthem,firstwithoutlookingathints
or solutions.
Commentsondifficulty: Typically,sectionsandchaptersbecomemorediffi-
culttowardtheend. Don’tbeafraidtoquitachapterwhenitbecomestoodifficult.
Digest as much as interests you and move on to the next chapter or section.
Comments on the bibliography: The book was written with very little
direct reference to sources, and many of the proofs may therefore differ from the
standard ones. But there are many wonderful books and wonderful teachers that
we can learn from. I have therefore collected an annotated bibliography that you
maywanttoexplore. Iparticularlyrecommend[1,G.H.Hardy,AMathematician’s
Apology], [2, G. P´olya, How to Solve It], and [3, T. W. K¨orner, The Pleasure of
Counting], just for fun, light reading. For a bit of hero worship, I also recommend
the biographical references [21, E. T. Bell, Men of Mathematics], [22, C. Henrion,
Women of Mathematics], and [23, W. Dunham, Journey Through Genius]. And
I have to thank my particular heroes: my brother Larry, who taught me about
uncountable sets, space-filling curves, and mathematical induction; Georg Po´lya,
who invited me into his home and showed me his mathematical notebooks; my ad-
visorC.E.Burgess,whointroducedmetothewondersofTexas-stylemathematics;
R.H.Bing,whoseSling,DogboneSpace,HookedRug,BaseballMove,epslumsand
deltas, andCrumpledCubesaddedcolorandwonder tothestudyoftopology; and
W.P.Thurston,whooftenmademefeellikeGaryLarson’scharacteroflittlebrain
PREFACE TO THE THREE VOLUME SET ix
(“Stop, professor, my brain is full.”) They were all kind and encouraging to me.
And then there are those whom I only know from their writing: especially Euclid,
Archimedes, Gauss, Hilbert, and Poincar´e.
Finally, I must thankBill Floyd and WalterParry for more thanthree decades
of mathematical fun. When we would get together, we would work hard every
morning, thentalkmathematicsfortherestofthedayaswehikedthecities,coun-
trysides, mountains, and woods of Utah, Virginia, Michigan, Minnesota, England,
France, and any other place we could manage to get together. And special thanks
to Bill for cleaning up and improving almost all of those figures in these books
which he had not himself originally drawn.