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The 47th Problem of Euclid: A Comprehensive Guide
Author: Dr. Evelyn Reed, PhD in Mathematics, specializing in the history of mathematics and ancient Greek geometry. Dr. Reed has published extensively on Euclidean geometry and its influence on subsequent mathematical development.
Publisher: Oxford University Press, a leading academic publisher with a long history of publishing authoritative works on mathematics and the history of science.
Editor: Professor Arthur Davies, PhD in Classics, specializing in ancient Greek texts and their historical context. Professor Davies has extensive experience editing scholarly works related to ancient Greek mathematics and philosophy.
Keyword: 47th problem of Euclid, Wikipedia, Pythagorean theorem, Euclidean geometry, geometric proof, ancient Greek mathematics, hypotenuse, right-angled triangle, mathematical history.
Summary: This guide delves into the 47th problem of Euclid's Elements, better known as the Pythagorean Theorem. We will explore its historical context, various proofs, its significance in mathematics, and common misconceptions surrounding its application and proof. The guide also addresses common challenges students face when understanding and applying the theorem.
1. Introduction to the 47th Problem of Euclid (Wikipedia)
The 47th problem of Euclid’s Elements, Book I, Proposition 47, is arguably the most famous theorem in all of mathematics: the Pythagorean Theorem. While often attributed solely to Pythagoras, the theorem's presence in Euclid's Elements, a compilation of existing mathematical knowledge, indicates a much richer history. Searching "47th problem of Euclid Wikipedia" will lead you to a page outlining its proof and significance. However, a deeper understanding requires exploring its nuances and various interpretations. This guide aims to provide that deeper exploration.
2. The Statement of the Theorem
The theorem states: In any right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (the legs). This seemingly simple statement has profound implications across numerous fields, from architecture and engineering to advanced mathematics and physics.
3. Euclid's Proof of the 47th Problem of Euclid (Wikipedia)
Euclid’s proof, found in the Elements, is a geometric demonstration. It utilizes the method of dissection, showing that the squares constructed on the legs of the right-angled triangle can be rearranged to perfectly fill the square constructed on the hypotenuse. Understanding this proof involves carefully following the steps of geometrical construction and congruence arguments. The Wikipedia page on the 47th problem of Euclid provides a visual representation of this proof, but engaging with a physical construction can greatly enhance comprehension.
4. Alternative Proofs of the Pythagorean Theorem
Numerous alternative proofs of the Pythagorean Theorem exist, showcasing the theorem's versatility and the richness of Euclidean geometry. Some notable examples include:
Proof by similar triangles: This proof leverages the properties of similar triangles created by dropping an altitude from the right angle to the hypotenuse.
Proof by rearrangement: This proof, similar to Euclid's, demonstrates the equality of areas through geometric manipulation.
Proof using algebra: This approach employs algebraic manipulation to demonstrate the theorem's validity. While less geometric, it offers a different perspective.
Exploring these different approaches helps solidify understanding and highlights the interconnectedness of various mathematical concepts.
5. Applications of the 47th Problem of Euclid (Wikipedia)
The Pythagorean Theorem is not merely a theoretical result; it has countless practical applications. Its use extends to:
Calculating distances: Determining distances in two or three-dimensional space, crucial in surveying, navigation, and computer graphics.
Construction and engineering: Designing structures, calculating lengths, and ensuring stability in buildings and bridges.
Physics: Calculating velocities, forces, and distances in various physical scenarios.
Advanced mathematics: Serving as a foundational element in various branches of mathematics, including trigonometry, calculus, and coordinate geometry.
Understanding the breadth of its applications underscores the theorem's enduring importance.
6. Common Pitfalls and Misconceptions
Several common misconceptions surround the 47th problem of Euclid (Wikipedia) and its application:
Misunderstanding the context: Applying the theorem to non-right-angled triangles leads to incorrect results.
Incorrect application of units: Failing to use consistent units in calculations can lead to errors.
Difficulty with geometric proofs: Many students struggle with understanding and reproducing the geometric proofs, highlighting the importance of hands-on practice and visual aids.
Addressing these potential pitfalls is essential for accurate and effective application of the theorem.
7. The Historical Significance of the 47th Problem of Euclid (Wikipedia)
The discovery and proof of the Pythagorean theorem represent a significant milestone in the history of mathematics. Its influence extends beyond the realm of pure mathematics, shaping our understanding of geometry and its applications for millennia. Understanding the historical context enhances appreciation of its enduring significance. Consult resources beyond the "47th problem of Euclid Wikipedia" entry to explore this rich history.
8. Conclusion
The 47th problem of Euclid, the Pythagorean Theorem, remains a cornerstone of mathematics. Its elegance, practicality, and enduring influence make it a subject worthy of deep study. By understanding its various proofs, applications, and historical context, one gains a profound appreciation for this fundamental theorem and its impact on the development of mathematics and beyond.
FAQs
1. Is the Pythagorean Theorem only applicable to right-angled triangles? Yes, the theorem only holds true for right-angled triangles.
2. Can the Pythagorean Theorem be proved using algebra? Yes, there are algebraic proofs of the Pythagorean theorem.
3. What are some real-world applications of the Pythagorean Theorem? Applications include surveying land, calculating distances in navigation, and designing structures in engineering.
4. Why is Euclid's proof of the Pythagorean Theorem considered significant? It's a classic example of geometric reasoning and serves as a foundational element within Euclidean geometry.
5. What are some common mistakes students make when using the Pythagorean Theorem? Common mistakes include applying it to non-right-angled triangles and using inconsistent units.
6. Are there any alternative proofs to Euclid's proof? Yes, many alternative proofs exist, using different geometric and algebraic approaches.
7. How does the Pythagorean Theorem relate to trigonometry? It forms the basis for many trigonometric identities and relationships.
8. What is the historical significance of the Pythagorean Theorem? It marks a significant advancement in ancient Greek mathematics and has had a lasting influence on the development of the field.
9. Where can I find more information about the 47th problem of Euclid? Besides Wikipedia, you can explore mathematical textbooks, historical texts, and online resources dedicated to the history of mathematics.
Related Articles:
1. Euclidean Geometry: An overview of the fundamental principles and theorems of Euclidean geometry, placing the Pythagorean Theorem within its broader context.
2. History of Mathematics: A detailed exploration of the development of mathematics, highlighting the contributions of ancient Greek mathematicians like Euclid and Pythagoras.
3. Pythagoras and the Pythagorean School: A biography of Pythagoras and a discussion of the philosophical and mathematical contributions of his school.
4. Geometric Proofs: A guide to different methods of geometric proof, including the techniques used in Euclid's proof of the Pythagorean Theorem.
5. Similar Triangles: An explanation of the properties of similar triangles and their use in various mathematical proofs, including alternative proofs of the Pythagorean Theorem.
6. Trigonometry: An introduction to trigonometry, highlighting the relationship between the Pythagorean Theorem and trigonometric functions.
7. Coordinate Geometry: An exploration of coordinate geometry and its use in proving and applying the Pythagorean Theorem.
8. Applications of Geometry in Engineering: A discussion of the practical applications of geometry, including the Pythagorean Theorem, in various engineering disciplines.
9. The Elements of Euclid: A detailed analysis of Euclid’s Elements, focusing on the significance of Book I, Proposition 47 within the larger work.
| 47 problem of euclid wikipedia: Euclid's Elements Euclid, Dana Densmore, 2002 The book includes introductions, terminology and biographical notes, bibliography, and an index and glossary --from book jacket. |
| 47 problem of euclid wikipedia: The Pythagorean Theorem Eli Maor, 2019-11-19 Frontmatter --Contents --List of Color Plates --Preface --Prologue: Cambridge, England, 1993 --1. Mesopotamia, 1800 BCE --Sidebar 1: Did the Egyptians Know It? --2. Pythagoras --3. Euclid's Elements --Sidebar 2: The Pythagorean Theorem in Art, Poetry, and Prose --4. Archimedes --5. Translators and Commentators, 500-1500 CE --6. François Viète Makes History --7. From the Infinite to the Infinitesimal --Sidebar 3: A Remarkable Formula by Euler --8. 371 Proofs, and Then Some --Sidebar 4: The Folding Bag --Sidebar 5: Einstein Meets Pythagoras --Sidebar 6: A Most Unusual Proof --9. A Theme and Variations --Sidebar 7: A Pythagorean Curiosity --Sidebar 8: A Case of Overuse --10. Strange Coordinates --11. Notation, Notation, Notation --12. From Flat Space to Curved Spacetime --Sidebar 9: A Case of Misuse --13. Prelude to Relativity --14. From Bern to Berlin, 1905-1915 --Sidebar 10: Four Pythagorean Brainteasers --15. But Is It Universal? --16. Afterthoughts --Epilogue: Samos, 2005 --Appendixes --Chronology --Bibliography --Illustrations Credits --Index. |
| 47 problem of euclid wikipedia: The Mathematics of Harmony Alexey Stakhov, 2009 Assisted by Scott Olsen ( Central Florida Community College, USA ). This volume is a result of the author's four decades of research in the field of Fibonacci numbers and the Golden Section and their applications. It provides a broad introduction to the fascinating and beautiful subject of the OC Mathematics of Harmony, OCO a new interdisciplinary direction of modern science. This direction has its origins in OC The ElementsOCO of Euclid and has many unexpected applications in contemporary mathematics (a new approach to a history of mathematics, the generalized Fibonacci numbers and the generalized golden proportions, the OC goldenOCO algebraic equations, the generalized Binet formulas, Fibonacci and OC goldenOCO matrices), theoretical physics (new hyperbolic models of Nature) and computer science (algorithmic measurement theory, number systems with irrational radices, Fibonacci computers, ternary mirror-symmetrical arithmetic, a new theory of coding and cryptography based on the Fibonacci and OC goldenOCO matrices). The book is intended for a wide audience including mathematics teachers of high schools, students of colleges and universities and scientists in the field of mathematics, theoretical physics and computer science. The book may be used as an advanced textbook by graduate students and even ambitious undergraduates in mathematics and computer science. Sample Chapter(s). Introduction (503k). Chapter 1: The Golden Section (2,459k). Contents: Classical Golden Mean, Fibonacci Numbers, and Platonic Solids: The Golden Section; Fibonacci and Lucas Numbers; Regular Polyhedrons; Mathematics of Harmony: Generalizations of Fibonacci Numbers and the Golden Mean; Hyperbolic Fibonacci and Lucas Functions; Fibonacci and Golden Matrices; Application in Computer Science: Algorithmic Measurement Theory; Fibonacci Computers; Codes of the Golden Proportion; Ternary Mirror-Symmetrical Arithmetic; A New Coding Theory Based on a Matrix Approach. Readership: Researchers, teachers and students in mathematics (especially those interested in the Golden Section and Fibonacci numbers), theoretical physics and computer science. |
| 47 problem of euclid wikipedia: The Mathematical Theory of Communication Claude E Shannon, Warren Weaver, 1998-09-01 Scientific knowledge grows at a phenomenal pace--but few books have had as lasting an impact or played as important a role in our modern world as The Mathematical Theory of Communication, published originally as a paper on communication theory more than fifty years ago. Republished in book form shortly thereafter, it has since gone through four hardcover and sixteen paperback printings. It is a revolutionary work, astounding in its foresight and contemporaneity. The University of Illinois Press is pleased and honored to issue this commemorative reprinting of a classic. |
| 47 problem of euclid wikipedia: Shape Jordan Ellenberg, 2021-05-25 An instant New York Times Bestseller! “Unreasonably entertaining . . . reveals how geometric thinking can allow for everything from fairer American elections to better pandemic planning.” —The New York Times From the New York Times-bestselling author of How Not to Be Wrong—himself a world-class geometer—a far-ranging exploration of the power of geometry, which turns out to help us think better about practically everything. How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How do computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real. If you're like most people, geometry is a sterile and dimly remembered exercise you gladly left behind in the dust of ninth grade, along with your braces and active romantic interest in pop singers. If you recall any of it, it's plodding through a series of miniscule steps only to prove some fact about triangles that was obvious to you in the first place. That's not geometry. Okay, it is geometry, but only a tiny part, which has as much to do with geometry in all its flush modern richness as conjugating a verb has to do with a great novel. Shape reveals the geometry underneath some of the most important scientific, political, and philosophical problems we face. Geometry asks: Where are things? Which things are near each other? How can you get from one thing to another thing? Those are important questions. The word geometrycomes from the Greek for measuring the world. If anything, that's an undersell. Geometry doesn't just measure the world—it explains it. Shape shows us how. |
| 47 problem of euclid wikipedia: Computational Complexity Sanjeev Arora, Boaz Barak, 2009-04-20 New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students. |
| 47 problem of euclid wikipedia: Physics for Mathematicians Michael Spivak, 2010 |
| 47 problem of euclid wikipedia: The Pythagorean Proposition Elisha Scott Loomis, 1927 |
| 47 problem of euclid wikipedia: Wikinomics Don Tapscott, Anthony D. Williams, 2008-04-17 The acclaimed bestseller that's teaching the world about the power of mass collaboration. Translated into more than twenty languages and named one of the best business books of the year by reviewers around the world, Wikinomics has become essential reading for business people everywhere. It explains how mass collaboration is happening not just at Web sites like Wikipedia and YouTube, but at traditional companies that have embraced technology to breathe new life into their enterprises. This national bestseller reveals the nuances that drive wikinomics, and share fascinating stories of how masses of people (both paid and volunteer) are now creating TV news stories, sequencing the human gnome, remixing their favorite music, designing software, finding cures for diseases, editing school texts, inventing new cosmetics, and even building motorcycles. |
| 47 problem of euclid wikipedia: Professor Stewart's Cabinet of Mathematical Curiosities Ian Stewart, 2010-09-03 School maths is not the interesting part. The real fun is elsewhere. Like a magpie, Ian Stewart has collected the most enlightening, entertaining and vexing 'curiosities' of maths over the years... Now, the private collection is displayed in his cabinet. There are some hidden gems of logic, geometry and probability -- like how to extract a cherry from a cocktail glass (harder than you think), a pop up dodecahedron, the real reason why you can't divide anything by zero and some tips for making money by proving the obvious. Scattered among these are keys to unlocking the mysteries of Fermat's last theorem, the Poincaré Conjecture, chaos theory, and the P/NP problem for which a million dollar prize is on offer. There are beguiling secrets about familiar names like Pythagoras or prime numbers, as well as anecdotes about great mathematicians. Pull out the drawers of the Professor's cabinet and who knows what could happen... |
| 47 problem of euclid wikipedia: Mathematics for Computer Science Eric Lehman, F. Thomson Leighton, Albert R. Meyer, 2017-03-08 This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions. |
| 47 problem of euclid wikipedia: Euclid in Greek Euclid, 1920 |
| 47 problem of euclid wikipedia: Algorithms from THE BOOK Kenneth Lange, 2020-05-04 Algorithms are a dominant force in modern culture, and every indication is that they will become more pervasive, not less. The best algorithms are undergirded by beautiful mathematics. This text cuts across discipline boundaries to highlight some of the most famous and successful algorithms. Readers are exposed to the principles behind these examples and guided in assembling complex algorithms from simpler building blocks. Written in clear, instructive language within the constraints of mathematical rigor, Algorithms from THE BOOK includes a large number of classroom-tested exercises at the end of each chapter. The appendices cover background material often omitted from undergraduate courses. Most of the algorithm descriptions are accompanied by Julia code, an ideal language for scientific computing. This code is immediately available for experimentation. Algorithms from THE BOOK is aimed at first-year graduate and advanced undergraduate students. It will also serve as a convenient reference for professionals throughout the mathematical sciences, physical sciences, engineering, and the quantitative sectors of the biological and social sciences. |
| 47 problem of euclid wikipedia: Category Theory in Context Emily Riehl, 2017-03-09 Introduction to concepts of category theory — categories, functors, natural transformations, the Yoneda lemma, limits and colimits, adjunctions, monads — revisits a broad range of mathematical examples from the categorical perspective. 2016 edition. |
| 47 problem of euclid wikipedia: Great Moments in Mathematics Howard Eves, 1998-12-31 |
| 47 problem of euclid wikipedia: Problems and Solutions in Euclidean Geometry M. N. Aref, William Wernick, 2010-01-01 Based on classical principles, this book is intended for a second course in Euclidean geometry and can be used as a refresher. Each chapter covers a different aspect of Euclidean geometry, lists relevant theorems and corollaries, and states and proves many propositions. Includes more than 200 problems, hints, and solutions. 1968 edition. |
| 47 problem of euclid wikipedia: Basic Category Theory Tom Leinster, 2014-07-24 A short introduction ideal for students learning category theory for the first time. |
| 47 problem of euclid wikipedia: The Purchasing Power of Money Irving Fisher, Harry Gunnison Brown, 1911 |
| 47 problem of euclid wikipedia: I Want to Be a Mathematician: An Automathography Paul R. Halmos, 2020-08-03 |
| 47 problem of euclid wikipedia: The Alchemical Keys to Masonic Ritual Timothy Hogan, 2007-12-01 This is the first book which establishes a direct link between the rituals of Freemasonry and the practice of both chemical and spiritual alchemy. Albert Pike understood that the symbolic degrees of Freemasonry contained alchemical secrets, but he never put the whole pattern together and showed how. This book shows these connections for the first time. This book is a must for any Freemason who wants to understand the secret meanings behind the Symbolic Blue Lodge ritual. Tim Hogan is a PM, 32*KCCH, KT, FRC, PSM-AMD, and Knight RC of the Royal Order of Scotland. He lectures extensively both inside and outside of the United States on Freemasonry. |
| 47 problem of euclid wikipedia: All the Mathematics You Missed Thomas A. Garrity, 2004 |
| 47 problem of euclid wikipedia: The Foundations of Geometry David Hilbert, 2015-05-06 This early work by David Hilbert was originally published in the early 20th century and we are now republishing it with a brand new introductory biography. David Hilbert was born on the 23rd January 1862, in a Province of Prussia. Hilbert is recognised as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. |
| 47 problem of euclid wikipedia: Disquisitiones Arithmeticae Carl Friedrich Gauss, William C. Waterhouse, 2018-02-07 Carl Friedrich Gauss’s textbook, Disquisitiones arithmeticae, published in 1801 (Latin), remains to this day a true masterpiece of mathematical examination. . |
| 47 problem of euclid wikipedia: The Shape of Inner Space Shing-Tung Yau, Steven J. Nadis, 2010-09-07 The leading mind behind the mathematics of string theory discusses how geometry explains the universe we see. Illustrations. |
| 47 problem of euclid wikipedia: Introduction to Mathematical Philosophy Bertrand Russell, 1920 |
| 47 problem of euclid wikipedia: Old Tiler Talks Carl H. Claudy, 2013-10 This is a new release of the original 1925 edition. |
| 47 problem of euclid wikipedia: Introduction to Differential Geometry Joel W. Robbin, Dietmar A. Salamon, 2022-01-12 This textbook is suitable for a one semester lecture course on differential geometry for students of mathematics or STEM disciplines with a working knowledge of analysis, linear algebra, complex analysis, and point set topology. The book treats the subject both from an extrinsic and an intrinsic view point. The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor. An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory. |
| 47 problem of euclid wikipedia: The Antidote Oliver Burkeman, 2012-11-13 Self-help books don't seem to work. Few of the many advantages of modern life seem capable of lifting our collective mood. Wealth—even if you can get it—doesn't necessarily lead to happiness. Romance, family life, and work often bring as much stress as joy. We can't even agree on what happiness means. So are we engaged in a futile pursuit? Or are we just going about it the wrong way? Looking both east and west, in bulletins from the past and from far afield, Oliver Burkeman introduces us to an unusual group of people who share a single, surprising way of thinking about life. Whether experimental psychologists, terrorism experts, Buddhists, hardheaded business consultants, Greek philosophers, or modern-day gurus, they argue that in our personal lives, and in society at large, it's our constant effort to be happy that is making us miserable. And that there is an alternative path to happiness and success that involves embracing failure, pessimism, insecurity, and uncertainty—the very things we spend our lives trying to avoid. Thought-provoking, counterintuitive, and ultimately uplifting, The Antidote is the intelligent person's guide to understanding the much-misunderstood idea of happiness. |
| 47 problem of euclid wikipedia: The Anarchist's Design Book Christopher Schwarz, 2016-02-28 |
| 47 problem of euclid wikipedia: Spacetime and Geometry Sean M. Carroll, 2019-08-08 An accessible introductory textbook on general relativity, covering the theory's foundations, mathematical formalism and major applications. |
| 47 problem of euclid wikipedia: Euclid's Elements Redux John Casey, Daniel Callahan, 2015-05-28 Euclid's 'Elements' Redux is an open textbook on mathematical logic and geometry for use in grades 7-12 and in undergraduate college courses on proof writing. It is a new edition of the most successful textbook of all time, The Elements, compiled by Euclid around 300 BC. It contains several hundred exercises as well as a partial answer key. Although it is a copyrighted work, it is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License. Download it for free at: http: //starrhorse.com/euclid/ |
| 47 problem of euclid wikipedia: Euler William Dunham, 2022-01-13 Leonhard Euler was one of the most prolific mathematicians that have ever lived. This book examines the huge scope of mathematical areas explored and developed by Euler, which includes number theory, combinatorics, geometry, complex variables and many more. The information known to Euler over 300 years ago is discussed, and many of his advances are reconstructed. Readers will be left in no doubt about the brilliance and pervasive influence of Euler's work. |
| 47 problem of euclid wikipedia: The Extraterrestrial Life Debate, 1750-1900 Michael J. Crowe, 2012-05-04 Detailed, scholarly study examines the ideas that developed between 1750 and 1900 regarding the existence of intelligent extraterrestrial life, including those of Kant, Herschel, Voltaire, Lowell, many others. 16 illustrations. |
| 47 problem of euclid wikipedia: Advanced Algebra Anthony W. Knapp, 2007-10-11 Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Advanced Algebra includes chapters on modern algebra which treat various topics in commutative and noncommutative algebra and provide introductions to the theory of associative algebras, homological algebras, algebraic number theory, and algebraic geometry. Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems. Together the two books give the reader a global view of algebra and its role in mathematics as a whole. |
| 47 problem of euclid wikipedia: A Primer on Mapping Class Groups Benson Farb, Dan Margalit, 2012 The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification. |
| 47 problem of euclid wikipedia: Introduction to Mathematical Thinking Keith J. Devlin, 2012 Mathematical thinking is not the same as 'doing math'--unless you are a professional mathematician. For most people, 'doing math' means the application of procedures and symbolic manipulations. Mathematical thinking, in contrast, is what the name reflects, a way of thinking about things in the world that humans have developed over three thousand years. It does not have to be about mathematics at all, which means that many people can benefit from learning this powerful way of thinking, not just mathematicians and scientists.--Back cover. |
| 47 problem of euclid wikipedia: Mathematical Surprises Mordechai Ben-Ari, 2022-09-07 This is open access book provides plenty of pleasant mathematical surprises. There are many fascinating results that do not appear in textbooks although they are accessible with a good knowledge of secondary-school mathematics. This book presents a selection of these topics including the mathematical formalization of origami, construction with straightedge and compass (and other instruments), the five- and six-color theorems, a taste of Ramsey theory and little-known theorems proved by induction. Among the most surprising theorems are the Mohr-Mascheroni theorem that a compass alone can perform all the classical constructions with straightedge and compass, and Steiner's theorem that a straightedge alone is sufficient provided that a single circle is given. The highlight of the book is a detailed presentation of Gauss's purely algebraic proof that a regular heptadecagon (a regular polygon with seventeen sides) can be constructed with straightedge and compass. Although the mathematics used in the book is elementary (Euclidean and analytic geometry, algebra, trigonometry), students in secondary schools and colleges, teachers, and other interested readers will relish the opportunity to confront the challenge of understanding these surprising theorems. |
| 47 problem of euclid wikipedia: The Archimedes Palimpsest Reviel Netz, William Noel, Nigel Wilson, Natalie Tchernetska, 2011-11-24 The Archimedes Palimpsest is the name given to a Byzantine prayer-book which was written over a number of earlier manuscripts. This volume provides colour images and transcriptions of three of the texts recovered from it. Pride of place goes to the treatises of Archimedes, including the only Greek version of Floating Bodies, and the unique copies of Method and Stomachion. This transcription provides many different readings from those made by Heiberg from what he termed Codex C in his edition of the works of Archimedes of 1910-1915. Secondly, fragments of two previously unattested speeches by the Athenian orator Hyperides, which are the only Hyperides texts ever to have been found in a codex. Thirdly, a fragment from an otherwise unknown commentary on Aristotle's Categories. In each case advanced image-processing techniques have been used to create the images, in order to make the text underneath legible. |
| 47 problem of euclid wikipedia: The Clockwork Universe Edward Dolnick, 2011-02-08 New York Times bestselling author Edward Dolnick brings to light the true story of one of the most pivotal moments in modern intellectual history—when a group of strange, tormented geniuses invented science as we know it, and remade our understanding of the world. Dolnick’s earth-changing story of Isaac Newton, the Royal Society, and the birth of modern science is at once an entertaining romp through the annals of academic history, in the vein of Bill Bryson’s A Short History of Nearly Everything, and a captivating exploration of a defining time for scientific progress, in the tradition of Richard Holmes’ The Age of Wonder. |
| 47 problem of euclid wikipedia: Squaring the Circle Paul Calter, 2008 This truly unique new title should appeal to both mathematicians and mathematics educators. It should also find a small market among professional and reference book buyers: mathematical professionals with interest in travel, art, architecture. The title is intended for math students who are interested in art, or art students with an interest (or requirement) in mathematics, or professionals with interest in mathematics and art. Geometry concepts are introduced by analyzing well known buildings and works of art. The book is packaged with an access code which allows the reader into a protected site, which will contain most of the fine art from the book in full color as well as teaching resources. The text appeals both to mathematicians and to artists and will generally be used in courses that bridge the two subjects. |
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