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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. 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 e 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 G is really a swan An interlude delineating Algorithm O* 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 H 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 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's period about the sun to the moon's period about Earth. That ratio has 235/19 as one of its early continued fraction convergents, which explains the apparent periodicity. Exploring 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. The 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's Utopia) and whimsy (dropping a black hole on Earth's surface). Andy Simoson has won both the Chauvenet Prize and Pólya Award for expository writing from the MAA and his Voltaire's 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