Über die Autorin bzw. den Autor

Alfred S. Posamentier is dean of the School of Education and professor of mathematics education at Mercy College in Dobbs Ferry, New York. Previously, he had the same positions at the City College of the City University of New York for forty years.  He has published over fifty-five books in the area of mathematics and mathematics education, including The Fabulous Fibonacci Numbers (with Ingmar Lehmann). 
 

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The Pythagorean Theorem

Prometheus Books

Contents

Acknowledgments.................................................................................................9Introduction....................................................................................................11Chapter 1: Pythagoras and His Famous Theorem....................................................................17Chapter 2: Proving the Pythagorean Theorem without (Many) Words.................................................37Chapter 3: Applications of the Pythagorean Theorem..............................................................77Chapter 4: Pythagorean Triples and Their Properties.............................................................123Chapter 5: The Pythagorean Means................................................................................169Chapter 6: Tuning the Soul: Pythagoras and Music................................................................185Chapter 7: The Pythagorean Theorem in Fractal Art...............................................................211Final Thoughts..................................................................................................235Ultimately to Pythagoras by Dr. Herbert A. Hauptman.............................................................237Pictorial Depictions of Pythagoras and His Famous Theorem.......................................................241Appendix A: Some Selected Proofs................................................................................253Appendix B: Some More Proofs and Solutions......................................................................265Appendix C: List of Primitive Pythagorean Triples...............................................................273Appendix D: List of Pythagorean Triples-Primitive and Nonprimitive..............................................279Further References..............................................................................................303Index...........................................................................................................305

Chapter One

As we embark on our exploration of the Pythagorean Theorem, we are faced with some questions. Chief among them is why is the relationship that historically bears that name-the Pythagorean Theorem-so important? There are many reasons: perhaps because it is easy to remember; perhaps because it can be easily visualized; perhaps because it has fascinating applications in many fields of mathematics; or perhaps because it is the basis for much of mathematics that has been studied over the past millennia. We shall explore these aspects in the chapters that follow. But, perhaps it is best to begin at its roots, with the mathematician whom we credit as being the first to prove this theorem, and examine the man himself, his life, and his society.

The first biography of Pythagoras was written about eight hundred years after his death by Iamblichus, one of many Pythagoras enthusiasts, who tried to glorify him. And, although Pythagoras has been mentioned numerous other times throughout history, by well-known writers such as Plato, Aristotle, Eudoxus, Herodotus, Empedocles, and others, we still do not have reliable information about him. Some of his contemporary followers actually believed that he was a demigod, a son of Apollo-a conviction they supported by noting that his mother was said to be a very beautiful woman. Some reported that he even worked wonders.

But even though he was called the greatest mathematician and philosopher of antiquity by some, he was not without critics who tried to revile him. They say that he was merely the founder and chief of a sect-the Pythagoreans-and that the many scientific results that came from them were written by the members of the sect and dedicated to its leader, and thus were not the work of Pythagoras himself. The critics considered him a collector of facts without any deeper understanding of the related concepts, and therefore felt he did not really contribute to a deep understanding of mathematics. Similar criticism also was aimed at such luminaries as Plato, Aristotle, and Euclid. We must remain mindful of these uncertainties when we consider the "facts" about Pythagoras's life and work.

Pythagoras was born circa 575 BCE on the island Samos (located off the west coast of Asia Minor). His initial and perhaps most influential teacher was Pherecydes, who was primarily a theologist who taught him religion and mysticism along with mathematics. As a young man he traveled to Phoenicia, Egypt, and Mesopotamia, where he advanced his knowledge of mathematics and also pursued a variety of other interests, such as philosophy, religion, and mysticism. Some biographers believe that, while in his late teens, Pythagoras traveled first to Miletus, a coastal town in Asia Minor near Samos, where he continued his studies in mathematics under the tutelage of the famous philosopher and mathematician Thales of Miletus. It is very likely that he also attended lectures from another Miletic philosopher, Anaximander, who further inspired Pythagoras in geometry. When he returned to Samos, the tyrant Polycrates, who ruled Samos from 538 to 522 BCE, had come to power. It is not clear if Pythagoras disagreed with Polycrates' leadership. But soon after returning home, he moved to Croton (today, Crotone in southern Italy), about 530 BCE, a region that had had a considerable Greek population since the eighth century. There he founded a community-or society-whose main interests were religion, mathematics, astronomy, and music (acoustics). The Pythagoreans' conviction that all aspects of nature and the universe could be explained and expressed by means of the natural numbers and the ratios of numbers suffered a setback when they learned that the emblem of their community, the pentagram, contradicted their core numerical principles.

The Pythagoreans tried to explain the nature of the world and the universe with the help of numbers. In particular, they studied vibrating strings and found that two strings sound harmonious if their lengths can be expressed as the ratio of two small natural numbers such as 1:2, 2:3, 3:4, 4:5, and so on. They came to believe that the entire universe is ordered by such simple relations of natural numbers. This ties in with their study of the three most popular means: the arithmetic mean, the geometric mean, and the harmonic mean, which relate to each other. We will visit these means in chapter 5.

One of the core beliefs of the Pythagoreans is that there is a strong connection between religion and mathematics. They believed that the sun, the moon, the planets, and the stars were of a divine nature and therefore they could move only along circular paths. Furthermore, they believed that the movements of these bodies caused sounds of different frequencies because of their different velocities, which in turn depended on their radii. These sounds were said to generate a harmonic scale, which they called the "harmony of the spheres." Yet they believed that man cannot actually hear this sound, as it surrounds humans constantly from birth. Even the great scientist Johannes Kepler (1571-1630) was sometimes characterized as a late Pythagorean since he believed that the diameters of the orbits of the planets could be explained by inscribed and circumscribed...