Spirals: From Theodorus To Chaos In his introduction to Spirals From Theodorus to Chaos, Phil Davis writes, "To me around the study of a difference equation that Davis dubs theodorus of cyrene, the book takes http://www.ibuki-trading-post.com/dir_akp/akp_spifro.html
Extractions: Spirals: From Theodorus to Chaos Philip J. Davis ISBN 1-56881-010-5 AK Peters - 1993 Hardcover 248pp In his introduction to Spirals: From Theodorus to Chaos, Phil Davis writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past twenty-five hundred years. info@ibuki.com
A Lesson On Spirals exercise below has been attributed to theodorus of cyrene (~465399 BC). Theodorus was Platos tutor Socrates makes reference to Theodorus proving that the square roots of 3, 5 http://courses.wcupa.edu/jkerriga/Lessons/A Lesson on Spirals.html
Extractions: A Lesson on The Root Spiral Kate Long The Shipley School Klong@shipleyschool.org Objectives Practice with compass and straight edge Explore a geometric representation of square roots, deepening understanding Introduce students to spirals, curves that are seldom studied in traditional textbooks Develop an awareness of the historical context for the study of irrational numbers and spirals Recognize spirals in nature and appreciate the mathematics inherent in them Historical Perspectives Theaetus , Socrates makes reference to Theodorus proving that the square roots of 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, and 17 were irrational. Plato raises the question "Why did Theodorus stop at 17?". One possible answer is that the 17 hypotenuse belongs to the last triangle that does not overlap the figure. In 1958 E. Teuffel proved several more interesting facts about the root spiral. If the procedure for generating the spiral is continued indefinitely so that the figure overlaps, no two hypotenuses will coincide. In other words, they will never lie directly on top of each other. Also, if the sides of unit "one" length are extended forever, they will not pass through any of the other vertices of the total figure. The Spiral of Theodorus approximates the Logarithmic Spiral. By the early 1600s the logarithmic spiral was being studied in depth. See Historical Perspectives II in Appendix A for additional information.
History Of Mathematics: Greece 430) *SB. Hippias of Elis (c. 425) theodorus of cyrene (c. 425 c. 250) Nicoteles of Cyrene (c. 250) Strato (c http://aleph0.clarku.edu/~djoyce/mathhist/greece.html
Theodorus theodorus of cyrene. theodorus of cyrene was a pupil of Protagoras and himself thetutor of Plato, teaching him mathematics, and also the tutor of Theaetetus. http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html
Extractions: Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.
References For Theodorus 7 (1969), 359379. L Giacardi, On theodorus of cyrene s problem, Arch.Internat. Hist. Sci. 27 (101) (1977), 231-236. TL Heath, A http://www-gap.dcs.st-and.ac.uk/~history/References/Theodorus.html
Extractions: B Artmann, A proof for Theodorus' theorem by drawing diagrams, J. Geom. M S Brown, Theaetetus : Knowledge as Continued Learning, Journal of the History of Philosophy L Giacardi, On Theodorus of Cyrene's problem, Arch. Internat. Hist. Sci. T L Heath, A History of Greek Mathematics I (Oxford, 1921), 203-204, 209-212. R L McCabe, Theodorus' irrationality proofs, Math. Mag. A Wasserstein, Theaetetus and the History of the Theory of Numbers, Classical Quarterly Main index Birthplace Maps Biographies Index
Spirals: From Theodorus To Chaos, By P. Davis around the study of a difference equation that Davis dubs theodorus of cyrene, the book takes the conjectural, Davis examines Theodorus in light of mathematical concerns that have http://www.webbooks.net/books/_peters/Davis.html
Extractions: In his introduction, the author writes, "To me, mathematics has always been more than its form, or its content, its logic, its strategies, or its applications. Mathematics is one of the greatest of human intellectual experiences, and as such merits and requires a rather liberal approach." He takes just such an approach in this book inspired by the Hedrick Lectures of the Seventy-Fifth Anniversary of the Mathematical Association of America. Although loosely organized around the study of a difference equation that Davis dubs Theodorus of Cyrene, the book takes us on an eclectic whirlwind tour of history, philosophy, anecdote and, of course, mathematics. Incorporating the old and the new, the proved and the conjectural, Davis examines Theodorus in light of mathematical concerns that have grown and changed over the past 2,500 years. A. K. Peters, Ltd., 1993
GO.HRW.COM Lesson 12.4 The Distance Formula. theodorus of cyrene, theodorus of cyrene wasa mathematician, astronomer, musician, philosopher, and the tutor of Plato. http://go.hrw.com/hrw.nd/gohrw_rls1/pKeywordResults?MA1 Theodorus
Theodorus Biography of Theodorus (465BC398BC) theodorus of cyrene. Born 465 BC in Cyrene (now Shahhat, Libya) theodorus of cyrene was a pupil of Protagoras and himself the tutor of Plato, teaching him mathematics http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html
Extractions: Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato , teaching him mathematics, and also the tutor of Theaetetus Plato travelled to and from Egypt and on such occasions he spent time with Theodorus in Cyrene. Theodorus, however, did not spend his whole life in Cyrene for he was certainly in Athens at a time when Socrates was alive. Theodorus, in addition to his work in mathematics, was [5]:- ... distinguished ... in astronomy, arithmetic, music and all educational subjects. A member of the society of Pythagoras , Theodorus was one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. Our knowledge of Theodorus comes through Plato who wrote about him in his work Theaetetus.
GO.HRW.COM The activity also provides links to interesting Internet sites related to physics.MA1 Theodorus Learn about theodorus of cyrene and the golden spiral. http://go.hrw.com/ndNSAPI.nd/gohrw_rls1/pKeywordResults?MA1 CH12
Theodorus theodorus of cyrene. Born 465 BC in Cyrene (now Shahhat,Libya) Died 398 BC in Cyrene (now Shahhat, Libya). http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Thdrs.htm
Extractions: Previous (Alphabetically) Next Welcome page Theodorus was a tutor of Plato and Theaetetus and is best remembered by mathematicians for his contribution to the development of irrational numbers. Theodorus was also one of the main philosophers in the Cyrenaic school of moral philosophy. He believed that pleasures and pains are neither good nor bad. Cheerfulness and wisdom, he believed, were sufficient for happiness. References (7 books/articles) Previous (Chronologically) Next Biographies Index
Full Alphabetical Index Miletus (404*) Theaetetus of Athens (82) theodorus of cyrene (58) Theodosius http://intranet.woodvillehs.sa.edu.au/pages/resources/maths/History/Flllph.htm
Square Roots Chronology About 425BC. theodorus of cyrene shows that the square roots of 3, 5, 6, 7, 8, 10, 11 and 17 are irrational. Theodorus approximates R(3) as 7/4 since http://rpimath.topcities.com/irrationals/squareroots.html
Extractions: Note: This page uses R(n) to indicate the square root of n (this notation was commonly used before the modern day notation was adopted). This page also uses pi for the irrational number 3.14159..., phi for the golden mean (which is the irrational number 1.618...), and ^ to indicate exponents. About 1750BC
Extractions: Bernard SUZANNE Last updated December 24, 2001 Plato and his dialogues : Home Biography Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations - Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author This page is part of the "tools" section of a site, Plato and his dialogues , dedicated to developing a new interpretation of Plato's dialogues. The "tools" section provides historical and geographical context (chronology, maps, entries on characters and locations) for Socrates, Plato and their time. This page provides an index to the entries on persons (*) and locations of interest in the study of the historical context of Socrates and Plato that are available on other pages of this site ( names in italic are names for wich there is no specific entry, but which are delt with through another entry By clicking on a name in the index, you can go to individual entries on famous Greek leaders, writers, thinkers of the Vth and IVth centuries B. C., and also on characters staged in Plato's dialogues, or on the main cities and locations of Ancient Greece that are of interest in the study of Plato's dialogues, either as the location of noteworthy historical events of that time, or as the birthplace of famous writers or philosopher, or as locations mentioned in one or another dialogue. By clicking on the minimap below a city's name, you can go to a full size map for a better viewing of the city's location. You may also click on the area number at the beginning of the text to go to a director map that will show you where the specific portion of the full size map shown in the minimap is located in the larger map (this option is not available for locations in Attica, the relationship between the minimap and the full size map being obvious in that case).
Plato FAQ: The Allegory Of The Cave in a ditch while looking at the stars in the caricature of philosopher he putsin the mouth of Socrates for his friend theodorus of cyrene, the scientist, in http://plato-dialogues.org/faq/faq002.htm
Extractions: Bernard SUZANNE Last updated May 1st, 1999 Plato and his dialogues : Home Biography - Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author F requently A sked Q uestions about Plato "Could you tell me in what work of Plato I might find his "cave analogy". It is a story of men chained in a cave only able to see their own shadows and delude into thinking that this was all there was to reality." The story this letter refers to, usually called "the allegory of the cave", is found at the beginning of book VII of Plato's dialogue called The Republic . The Stephanus references (the universal way of quoting Plato, available in all editions of his works) for the section telling the allegory are Republic , VII, 514a-517a. It is followed by an interpretation of the allegory put by Plato in the mouth of Socrates, as is the allegory itself. The text of this section of the Republic is available in various English translations on the web, including :
Plato FAQ: The Allegory Of The Cave he puts in the mouth of Socrates for his friend theodorus of cyrene, the scientist, in the Theætetus caricature along the lines of what Theodorus sees as the proper behavior http://www.plato-dialogues.org/faq/faq002.htm
Extractions: Bernard SUZANNE Last updated May 1st, 1999 Plato and his dialogues : Home Biography - Works and links to them History of interpretation New hypotheses - Map of dialogues : table version or non tabular version . Tools : Index of persons and locations Detailed and synoptic chronologies - Maps of Ancient Greek World . Site information : About the author F requently A sked Q uestions about Plato "Could you tell me in what work of Plato I might find his "cave analogy". It is a story of men chained in a cave only able to see their own shadows and delude into thinking that this was all there was to reality." The story this letter refers to, usually called "the allegory of the cave", is found at the beginning of book VII of Plato's dialogue called The Republic . The Stephanus references (the universal way of quoting Plato, available in all editions of his works) for the section telling the allegory are Republic , VII, 514a-517a. It is followed by an interpretation of the allegory put by Plato in the mouth of Socrates, as is the allegory itself. The text of this section of the Republic is available in various English translations on the web, including :
Biography-center - Letter T Theodore II, www.knight.org/advent/cathen/14570b.htm; theodorus of cyrene,wwwhistory.mcs.st-and.ac.uk/~history/Mathematicians/Theodorus.html; http://www.biography-center.com/t.html
Extractions: random biography ! Any language Arabic Bulgarian Catalan Chinese (Simplified) Chinese (Traditional) Croatian Czech Danish Dutch English Estonian Finnish French German Greek Hebrew Hungarian Icelandic Indonesian Italian Japanese Korean Latvian Lithuanian Norwegian Polish Portuguese Romanian Russian Serbian Slovak Slovenian Spanish Swedish Turkish 361 biographies Tabern, Donalee L.
Math History - Pre-historic And Ancient Times About 425BC, theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://lahabra.seniorhigh.net/pages/teachers/pages/math/timeline/MpreAndAncient.
Extractions: Prehistory and Ancient Times Middle Ages Renaissance Reformation ... External Resources About 30000BC Palaeolithic peoples in central Europe and France record numbers on bones. About 25000BC Early geometric designs used. About 4000BC Babylonian and Egyptian calendars in use. About 3400BC The first symbols for numbers, simple straight lines, are used in Egypt. About 3000BC Babylonians begin to use a sexagesimal number system for recording financial transactions. It is a place-value system without a zero place value. About 3000BC Hieroglyphic numerals in use in Egypt. About 3000BC The abacus is developed in the Middle East and in areas around the Mediterranean. A somewhat different type of abacus is used in China. About 1950BC Babylonians solve quadratic equations.
A History Of Irrational Numbers Square roots. Pythagoras of Samos. Born about 569 BC in Samos, Ionia Someclaim he had proofed that the sqrt(2) is irrational. theodorus of cyrene. http://home.zonnet.nl/mathematics/Geschiedenis/Getallen/sub4.htm
Extractions: Theodorus of Cyrene was a pupil of Protagoras and himself the tutor of Plato, Our whole knowledge of Theodorus's mathematical achievements are given by this passage from Plato. Yet there are points of interest which immediately arise. The first point is that Plato does not credit Theodorus with a proof that the square root of two was irrational. This must be because 2 was proved irrational before Theodorus worked on the problem, as stated before, some claim this was proved by Pythagoras himself. Born: about 417 BC in Athens, Greece Born: about 325 BC Book ten deals with the theory of irrational numbers and is mainly the work of Theaetetus. Euclid changed the proofs of several theorems in this book so that they fitted the new definition of proportion given by Eudoxus. The number e was first studied by the Swiss mathematician Leonhard Euler in the 1720s, although its existence was more or less implied in the work of John Napier, the inventor of logarithms, in 1614. Euler was also the first to use the letter e for it in 1727 (the fact that it is the first letter of his surname is coincidental). As a result, sometimes e is called the Euler Number, the Eulerian Number, or Napier's Constant (but not Euler's Constant).
Ancient Greece Mathematics Timeline About 425 BC theodorus of cyrene shows that certain square roots are irrational.This had been shown earlier but it is not known by whom. http://www.mlahanas.de/Greeks/TLMathematics.htm
Extractions: the Cretan poet Epimenides is attributed to have invented the linguistic paradox with his phrase "Cretans are ever liars" - the Liar's Paradox. 2500 years later, the mathematician Kurt Gödel invents an adaptation of the Liar's Paradox that reveals serious axiomatic problems at the heart of modern mathematics. About 600 BC Thales of Miletus , He brings Babylonian mathematical knowledge to Greece and uses geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. About 530 BC Pythagoras no common rational measure is discoverable About 480 BC Parmenides of Elea founded the Eleatic School where he taught that 'all is one,' not an aggregation of units as Pythagoras had said, and that to arrive at a true statement, logical argument is necessary. Truth "is identical with the thought that recognizes it" (Lloyd 1963:327). Change or movement and non-being, he held, are impossibilities since everything is 'full' and 'nothing' is a contradiction which, as such, cannot exist. "Parmenides is said to have been the first to assert that the Earth is spherical in shape...; there was, however, an alternative tradition stating that it was Pythagoras" (Heath 1913:64). Corollary to Parmenides' rejection of the existence of 'nothing' is the Greek number system which, like the later Roman system, refused to use the Babylonian positional number system with its marker for 'nothing.' Making no clear distinction between nature and geometry, "mathematics, instead of being a science of possible relations, was to [the Greeks] the study of situations thought to subsist in nature" (Boyer 1949:25). Moreover, "almost everything in [Greek] philosophy became subordinated to the problem of change.... All temporal changes observed by the senses were mere permutations and combinations of 'eternal principles,' [and] the historical sequence of events (which formed part of the 'flux') lost all fundamental significance" (Toulmin and Goodfield 1965:40).
