There is a nineteen\-year recurrence in the apparent position of the sun and moon against the background of the stars, a pattern observed long ago by the Babylonians. In the course of those nineteen years the Earth experiences 235 lunar cycles. Suppose we calculate the ratio of Earth\x27s period about the sun to the moon\x27s period about Earth. That ratio has 235\/19 as one of its early continued fraction convergents, which explains the apparent periodicity.\n\nExploring Continued Fractions explains this and other recurrent phenomena―astronomical transits and conjunctions, lifecycles of cicadas, eclipses―by way of continued fraction expansions. The deeper purpose is to find patterns, solve puzzles, and discover some appealing number theory. The reader will explore several algorithms for computing continued fractions, including some new to the literature. He or she will also explore the surprisingly large portion of number theory connected to continued fractions: Pythagorean triples, Diophantine equations, the Stern\-Brocot tree, and a number of combinatorial sequences.\n\nThe book features a pleasantly discursive style with excursions into music (The Well\-Tempered Clavier), history (the Ishango bone and Plimpton 322), classics (the shape of More\x27s Utopia) and whimsy (dropping a black hole on Earth\x27s surface). Andy Simoson has won both the Chauvenet Prize and Pólya Award for expository writing from the MAA and his Voltaire\x27s Riddle was a Choice magazine Outstanding Academic Title. This book is an enjoyable ramble through some beautiful mathematics. For most of the journey the only necessary prerequisites are a minimal familiarity with mathematical reasoning and a sense of fun.
Author(s): Andrew J. Simoson
Edition: 1
Publisher: American Mathematical Society
Year: 2021
Language: English
Pages: 480
Tags: Continued Fractions; Number Theory; Pythagorean Triples; Diophantine Equations; Stern-Brocot Tree
Cover
Title page
Copyright
Contents
Introduction
Strand I: Patterns
Tips on problem-solving and spotting patterns
A look ahead at three patterns
Chapter I: Tally Bones to the Integers
Tally bones
A table of primes?
The solution to a puzzle?
A base twelve or base sixty system?
Base ten, base twenty, base eight, base two
A binary digit interlude
Solving the shepherd’s puzzle and beyond
Three parting puzzles
Exercises
Strand II: Leibniz and the Binary Revolution
A continued fraction connection
Chapter II: Mathematical Induction
Set notation and the well-ordering principle
The principle of mathematical induction
The fundamental theorem of arithmetic
Equivalence classes
Nim*
Case Study: Mancala*
Mancala nim*
Exercises
Strand III: Al-Maghribî meets Sudoku
Chapter III: GCDs and Diophantine Equations
The greatest common divisor
An ancient algorithm for the greatest common divisor
The Diophantine solution
A litmus test for Euclid’s solution
Clock arithmetic
Systems of Diophantine equations
The totient is multiplicative
A problem from Diophantus’s Arithmetica
Exercises
Strand IV: Fractions in the Pythagorean Scale
A note-naming interlude
How Pythagoras generated his scale
Chapter IV: A Tree of Fractions
Unitary fractions in ancient Egypt
A continued fraction tradition
Farey sequences
A mediant interlude*
The Stern-Brocot tree
A grand finale*
Exercises
Strand V: Bach and The Well-Tempered Clavier
A well-tempered innovation
A musical interlude
An equal-tempered revolution
A continued fraction connection
Chapter V: The Harmonic Series
Case Study: Jeeps in the Desert
A look behind and a look ahead
A generating function finale*
Exercises
Strand VI: A Clay Tablet
The Babylonian number system
The accepted transliteration of Plimpton 322
Reciprocal pairs generate normalized Pythagorean triples
Finding the realm of potential generators
How the scribe may have screened for generators
The purpose of the tablet
Chapter VI: Families of Numbers
Primitive Pythagorean triples
Binomial coefficients
Fibonacci numbers
The continued fraction recursion for ?
The Catalan numbers*
Ben-Hur numbers*
Pogo-stick hikes along continued fractions
Exercises
Strand VII: Planetary Conjunctions
A few conjunction stories
A rough guess
A numerical approach
A continued fraction approach
Chapter VII: Simple and Strange Harmonic Motion
A heavenly approach to circular motion
An earthly approach to circular motion*
Strange harmonic motion
A where, what, and why interlude
The harmonic algorithm
A blue moon application
Exercises
Strand VIII: The Size and Shape of Utopia Island
Chapter VIII: Classic Elliptical Fractions
The prehistory of the ellipse
The trammel of Archimedes
An old elliptical puzzle
A model for the heavens
Newton’s case for a flattened Earth*
The French expeditions to Peru and Lapland
A final riddle
Exercises
Strand IX: The Cantor Set
A lotus-flower introduction
Ternary notation
A reality check*
Chapter IX: Continued Fractions
A local approach to continued fractions
A global approach to continued fractions
A plethora of continued fractions
Why the ugly duckling ? is really a swan
An interlude delineating Algorithm ?*
Dominance domains
The harmonic algorithm is a chameleon
Applying continued fractions to factoring integers
The first infinite continued fraction
Black holes and the receding Moon
Exercises
Strand X: The Longevity of the 17-year Cicada
Chapter X: Transits of Venus
A historical interlude
A Venus-Earth-Sun model
Conditions for a transit to occur
Recognizing the pattern
A reality check
An easier way to determine when transits occur
A final thought
Exercises
Strand XI: Meton of Athens
Chapter XI: Lunar Rhythms
Predicting the time lapse between successive new moons
Checking the expected length of short and long spans
Expected value of the variation in spans of years*
Final thoughts
Exercises
Strand XII: Eclipse Lore and Legends
Chapter XII: Diophantine Eclipses
Adapting the Earth-Moon-Sun model
Eclipse duration
A sufficient condition for eclipses
Finding ? at any lunation
Using Condition 1 to find the lapse between successive eclipses
Continued fraction insight
Some Diophantine magic
Lunar eclipses
A reality check
A final note
Exercises
Appendix I: List of Symbols Used in the Text
Appendix II: An Introduction to Vectors and Matrices
Appendix III: Computer Algebra System Codes
Appendix IV: Comments on Selected Exercises
Bibliography
Index
Back Cover
