Learning Geometry, Number and Design with the Spiral of Theodorus
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Plato As "Architectof Science"
Plato as "Architectof Science" LEONID ZHMUD ABSTRACT The figureof the cordialhost of the Academy,who invitedthe mostgifted math- ematiciansand cultivatedpure research, whose keen intellectwas able if not to solve the particularproblem then at least to show the methodfor its solution: this figureis quite familiarto studentsof Greekscience. But was the Academy as such a centerof scientificresearch, and did Plato really set for mathemati- cians and astronomersthe problemsthey shouldstudy and methodsthey should use? Oursources tell aboutPlato's friendship or at leastacquaintance with many brilliantmathematicians of his day (Theodorus,Archytas, Theaetetus), but they were neverhis pupils,rather vice versa- he learnedmuch from them and actively used this knowledgein developinghis philosophy.There is no reliableevidence that Eudoxus,Menaechmus, Dinostratus, Theudius, and others, whom many scholarsunite into the groupof so-called"Academic mathematicians," ever were his pupilsor close associates.Our analysis of therelevant passages (Eratosthenes' Platonicus, Sosigenes ap. Simplicius, Proclus' Catalogue of geometers, and Philodemus'History of the Academy,etc.) shows thatthe very tendencyof por- trayingPlato as the architectof sciencegoes back to the earlyAcademy and is bornout of interpretationsof his dialogues. I Plato's relationship to the exact sciences used to be one of the traditional problems in the history of ancient Greek science and philosophy.' From the nineteenth century on it was examined in various aspects, the most significant of which were the historical, philosophical and methodological. In the last century and at the beginning of this century attention was paid peredominantly, although not exclusively, to the first of these aspects, especially to the questions how great Plato's contribution to specific math- ematical research really was, and how reliable our sources are in ascrib- ing to him particular scientific discoveries. -
The Ordered Distribution of Natural Numbers on the Square Root Spiral
The Ordered Distribution of Natural Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 Et Germanytlingen, Germany ------------------------------ mathematical analysis by - Kay Schoenberger - Humboldt-University Berlin ----------------------------- 20. June 2007 Abstract : Natural numbers divisible by the same prime factor lie on defined spiral graphs which are running through the “Square Root Spiral“ ( also named as “Spiral of Theodorus” or “Wurzel Spirale“ or “Einstein Spiral” ). Prime Numbers also clearly accumulate on such spiral graphs. And the square numbers 4, 9, 16, 25, 36 … form a highly three-symmetrical system of three spiral graphs, which divide the square-root-spiral into three equal areas. A mathematical analysis shows that these spiral graphs are defined by quadratic polynomials. The Square Root Spiral is a geometrical structure which is based on the three basic constants: 1, sqrt2 and π (pi) , and the continuous application of the Pythagorean Theorem of the right angled triangle. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. Fibonacci Numbers divide the Square Root Spiral into areas and angle sectors with constant proportions. These proportions are linked to the “golden mean” ( golden section ), which behaves as a self-avoiding-walk- constant in the lattice-like structure of the square root spiral. Contents of the general section Page 1 Introduction to the Square Root Spiral 2 2 Mathematical description of the Square Root Spiral 4 3 The distribution -
The Spiral of Theodorus and Sums of Zeta-Values at the Half-Integers
The spiral of Theodorus and sums of zeta-values at the half-integers David Brink July 2012 Abstract. The total angular distance traversed by the spiral of Theodorus is governed by the Schneckenkonstante K introduced by Hlawka. The only published estimate of K is the bound K ≤ 0:75. We express K as a sum of Riemann zeta-values at the half-integers and compute it to 100 deci- mal places. We find similar formulas involving the Hurwitz zeta-function for the analytic Theodorus spiral and the Theodorus constant introduced by Davis. 1 Introduction Theodorus of Cyrene (ca. 460{399 B.C.) taught Plato mathematics and was himself a pupil of Protagoras. Plato's dialogue Theaetetus tells that Theodorus was distinguished in the subjects of the quadrivium and also contains the following intriguing passage on irrational square-roots, quoted here from [12]: [Theodorus] was proving to us a certain thing about square roots, I mean of three square feet and of five square feet, namely that these roots are not commensurable in length with the foot-length, and he went on in this way, taking all the separate cases up to the root of 17 square feet, at which point, for some reason, he stopped. It was discussed already in antiquity why Theodorus stopped at seventeen and what his method of proof was. There are at least four fundamentally different theories|not including the suggestion of Hardy and Wright that Theodorus simply became tired!|cf. [11, 12, 16]. One of these theories is due to the German amateur mathematician J. -
Chapter 1. Classical Greek Mathematics
Chapter 1. Classical Greek Mathematics Greek science and mathematics is distinguished from that of earlier cultures by its desire to know, in contrast to a need to make purely utilitarian advances or improvements. Greek geometry displays abstract and deductive elements which were largely lost during the Dark Ages, following the collapse of the Roman Empire, and only gradually recovered in the 16th and 17th centuries. It must be understood that many of the great discoveries in geometry were made about two and half thousand years ago. Given the difficulty of preserving fragile manuscripts, written on parchment or papyrus, over centuries when warfare could wipe out civilizations, it is not too surprising to find that we do not have many reliable records about the origin of Greek geometry or of its practitioners. We may count ourselves lucky that a few commentaries on Greek geometry, written in the fourth or fifth centuries of the present era, have survived to provide us with what details we have. Greeks from Ionia had settled in Asia Minor and there they had contact with two ancient civilizations, those of Babylon and Egypt. Although knowledge of science was elementary among the Babylonians and Egyptians, nonetheless they supplied the initial impetus that directed the Greeks towards the pursuit of systematic science. All authorities trace the beginnings of Greek geometry to Egypt. The rudimentary study of geometry in Egypt arose out of practical needs. Revenue was raised by taxation of landed prop- erty, and its assessment depended on the accurate fixing of the boundaries of fields in the possession of the landowners. -
A Short History of Greek Mathematics
Cambridge Library Co ll e C t i o n Books of enduring scholarly value Classics From the Renaissance to the nineteenth century, Latin and Greek were compulsory subjects in almost all European universities, and most early modern scholars published their research and conducted international correspondence in Latin. Latin had continued in use in Western Europe long after the fall of the Roman empire as the lingua franca of the educated classes and of law, diplomacy, religion and university teaching. The flight of Greek scholars to the West after the fall of Constantinople in 1453 gave impetus to the study of ancient Greek literature and the Greek New Testament. Eventually, just as nineteenth-century reforms of university curricula were beginning to erode this ascendancy, developments in textual criticism and linguistic analysis, and new ways of studying ancient societies, especially archaeology, led to renewed enthusiasm for the Classics. This collection offers works of criticism, interpretation and synthesis by the outstanding scholars of the nineteenth century. A Short History of Greek Mathematics James Gow’s Short History of Greek Mathematics (1884) provided the first full account of the subject available in English, and it today remains a clear and thorough guide to early arithmetic and geometry. Beginning with the origins of the numerical system and proceeding through the theorems of Pythagoras, Euclid, Archimedes and many others, the Short History offers in-depth analysis and useful translations of individual texts as well as a broad historical overview of the development of mathematics. Parts I and II concern Greek arithmetic, including the origin of alphabetic numerals and the nomenclature for operations; Part III constitutes a complete history of Greek geometry, from its earliest precursors in Egypt and Babylon through to the innovations of the Ionic, Sophistic, and Academic schools and their followers. -
Welcome to the Complete Pythagoras
Welcome to The Complete Pythagoras A full-text, public domain edition for the generalist & specialist Edited by Patrick Rousell for the World Wide Web. I first came across Kenneth Sylvan Guthrie’s edition of the Complete Pythagoras while researching a book on Leonardo. I had been surfing these deep waters for a while and so the value of Guthrie’s publication was immediately apparent. As Guthrie explains in his own introduction, which is at the beginning of the second book (p 168), he was initially prompted to publish these writings in the 1920’s for fear that this information would become lost. As it is, much of this information has since been published in fairly good modern editions. However, these are still hard to access and there is no current complete collection as presented by Guthrie. The advantage here is that we have a fairly comprehensive collection of works on Pythagoras and the Pythagoreans, translated from the origin- al Greek into English, and presented as a unified, albeit electronic edition. The Complete Pythagoras is a compilation of two books. The first is entitled The Life Of Py- thagoras and contains the four biographies of Pythagoras that have survived from antiquity: that of Iamblichus (280-333 A.D.), Porphry (233-306 A.D.), Photius (ca 820- ca 891 A.D.) and Diogenes Laertius (180 A.D.). The second is entitled Pythagorean Library and is a complete collection of the surviving fragments from the Pythagoreans. The first book was published in 1920, the second a year later, and released together as a bound edition. -
Euclidean Geometry
Euclidean Geometry Rich Cochrane Andrew McGettigan Fine Art Maths Centre Central Saint Martins http://maths.myblog.arts.ac.uk/ Reviewer: Prof Jeremy Gray, Open University October 2015 www.mathcentre.co.uk This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 2 Contents Introduction 5 What's in this Booklet? 5 To the Student 6 To the Teacher 7 Toolkit 7 On Geogebra 8 Acknowledgements 10 Background Material 11 The Importance of Method 12 First Session: Tools, Methods, Attitudes & Goals 15 What is a Construction? 15 A Note on Lines 16 Copy a Line segment & Draw a Circle 17 Equilateral Triangle 23 Perpendicular Bisector 24 Angle Bisector 25 Angle Made by Lines 26 The Regular Hexagon 27 © Rich Cochrane & Andrew McGettigan Reviewer: Jeremy Gray www.mathcentre.co.uk Central Saint Martins, UAL Open University 3 Second Session: Parallel and Perpendicular 30 Addition & Subtraction of Lengths 30 Addition & Subtraction of Angles 33 Perpendicular Lines 35 Parallel Lines 39 Parallel Lines & Angles 42 Constructing Parallel Lines 44 Squares & Other Parallelograms 44 Division of a Line Segment into Several Parts 50 Thales' Theorem 52 Third Session: Making Sense of Area 53 Congruence, Measurement & Area 53 Zero, One & Two Dimensions 54 Congruent Triangles 54 Triangles & Parallelograms 56 Quadrature 58 Pythagoras' Theorem 58 A Quadrature Construction 64 Summing the Areas of Squares 67 Fourth Session: Tilings 69 The Idea of a Tiling 69 Euclidean & Related Tilings 69 Islamic Tilings 73 Further Tilings -
Galileo, Ignoramus: Mathematics Versus Philosophy in the Scientific Revolution
Galileo, Ignoramus: Mathematics versus Philosophy in the Scientific Revolution Viktor Blåsjö Abstract I offer a revisionist interpretation of Galileo’s role in the history of science. My overarching thesis is that Galileo lacked technical ability in mathematics, and that this can be seen as directly explaining numerous aspects of his life’s work. I suggest that it is precisely because he was bad at mathematics that Galileo was keen on experiment and empiricism, and eagerly adopted the telescope. His reliance on these hands-on modes of research was not a pioneering contribution to scientific method, but a last resort of a mind ill equipped to make a contribution on mathematical grounds. Likewise, it is precisely because he was bad at mathematics that Galileo expounded at length about basic principles of scientific method. “Those who can’t do, teach.” The vision of science articulated by Galileo was less original than is commonly assumed. It had long been taken for granted by mathematicians, who, however, did not stop to pontificate about such things in philosophical prose because they were too busy doing advanced scientific work. Contents 4 Astronomy 38 4.1 Adoption of Copernicanism . 38 1 Introduction 2 4.2 Pre-telescopic heliocentrism . 40 4.3 Tycho Brahe’s system . 42 2 Mathematics 2 4.4 Against Tycho . 45 2.1 Cycloid . .2 4.5 The telescope . 46 2.2 Mathematicians versus philosophers . .4 4.6 Optics . 48 2.3 Professor . .7 4.7 Mountains on the moon . 49 2.4 Sector . .8 4.8 Double-star parallax . 50 2.5 Book of nature . -
Irrationals Or Incommensurables III: the Greek Solution
Irrationals or Incommensurables III: The Greek Solution PH I L L I P S. J 0 N E S CBoTH THE PYTHAGOREAN number-theoretic with 17; but the fact that he treated only these proof of the irrationality of ;f2 (or of the incom cases indicates that he still did not have a general mensurability of the side and diagonal of a square) process.2 Theaetetus, on the other hand, is supposed and the successive subtraction process which we to have been the original source for such general applied to the square and the pentagon in earlier theorems as Euclid, Book X, Proposition 8: If two articles' required generalization before one could magnitudes have not to one another the ratio which a claim that mathematicians had a theory of number has to a number, the magnitudes will be in irrationals. Some generalizations developed quite commensurable; and Book X, 9: The squares on rapidly. We shall merely sketch in the steps in this straight lines commensurable in length have to one development and explain the finished product as it another the ratio which a square number has to a appeared in Euclid. square number; . .. ; and Squares which have not to The chief contributors to this development one another the ratio which a square number has to a were Theodorus of Cyrene (ca. 400 B.C.), Theae square number will not have their sides commensu tetus (ca. 375 B.C.), and Eudoxus of Cnidos (ca. rable in length either.3 40Q-347 B.C.). Theodorus showed the irrationality The use in these propositions of the words of the non-square integers from 3 to 17. -
|||GET||| Geometry Theorems and Constructions 1St Edition
GEOMETRY THEOREMS AND CONSTRUCTIONS 1ST EDITION DOWNLOAD FREE Allan Berele | 9780130871213 | | | | | Geometry: Theorems and Constructions For readers pursuing a career in mathematics. Angle bisector theorem Exterior angle theorem Euclidean algorithm Euclid's theorem Geometric mean theorem Greek geometric algebra Hinge theorem Inscribed Geometry Theorems and Constructions 1st edition theorem Intercept theorem Pons asinorum Pythagorean theorem Thales's theorem Theorem of the gnomon. Then the 'construction' or 'machinery' follows. For example, propositions I. The Triangulation Lemma. Pythagoras c. More than editions of the Elements are known. Euclidean geometryelementary number theoryincommensurable lines. The Mathematical Intelligencer. A History of Mathematics Second ed. The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Arcs and Angles. Applies and reinforces the ideas in the geometric theory. Scholars believe that the Elements is largely a compilation of propositions based on books by earlier Greek mathematicians. Enscribed Circles. Return for free! Description College Geometry offers readers a deep understanding of the basic results in plane geometry and how they are used. The books cover plane and solid Euclidean geometryelementary number theoryand incommensurable lines. Cyrene Library of Alexandria Platonic Academy. You can find lots of answers to common customer questions in our FAQs View a detailed breakdown of our shipping prices Learn about our return policy Still need help? View a detailed breakdown of our shipping prices. Then, the 'proof' itself follows. Distance between Parallel Lines. By careful analysis of the translations and originals, hypotheses have been made about the contents of the original text copies of which are no longer available. -
Some Important Historical Names, Dates,* and Events
Some Important Historical Names, Dates,* and Events MATHEMATICAL GENERAL Early Beginnings (Before the Sixth Century B.C.) B.C. 30,000 Notched wolf bone B.C. 3300 Menes unites Egypt 8000 Ishango bone 2600 Great Pyramid at Gizeh 2500 Table tablets from Nippur 2100 Code of Hammurabi 1900 Plimpton 322 1500 Phoenician alphabet 1850 Moscow Papyrus 1200 Trojan War 1650 Rhind Papyrus 700 Homer: The Odyssey Classical Period (Sixth Century B.C. to Fifth Century) B.C. 622-547 Thales of Miletus B.C. 558-486 Darius the Great 585-500 Pythagoras of Samos 485-430 Herodotus ca. 470 Theodorus of Cyrene 480 Battle of Thermopylae 460-380 Hippocrates of Chios 469-399 Socrates ca. 420 Hippias of Elis 431 Peloponnesian War 408-355 Euxodus of Cnidos 388 Plato founds Academy 323-285 Euclid 356-323 Alexander the Great 287-212 Archimedes ca. 370 Eudemus of Rhodes 262-190 Apollonius of Perga 331 Foundation of Alexandria ca. 240 Nicomedes 213 Books burned in China ca. 230 Eratosthenes of Cyrene 212 Fall of Syracuse to Romans A.D. ca. 75 Heron of Alexandria 195 Rosetta Stone engraved ca. 100 Nicomachus of Gerasa 106-43 Cicero 85-160 Claudius Ptolemy 44 Assassination of Caesar ca. 250 Diophantus 27 Beginning of Roman Empire ca. 260 Liu Hui A.D. 100 Paper made in China ca. 300 Pappus of Alexandria 272-337 Constantine the Great 365-395 Theon of Alexandria 286 Division of the Empire / d. 415 Hypatia 324 Constantinople founded 410-485 Proclus 455 Vandals sack Rome *Most dates before 600 B.C. -
Pythagorean Theorem Project
Wheel of Theodorus Project Grading Rubric Grading rubric allows student to track their progress and calculate total possible points earned for the completion of this assigned project. The points listed represent the maximum total points available for satisfactory completion of each assigned task. Based on the student’s answers and quality of work on these assigned tasks; all, part or none of the points may be earned. Construction and Labeling of the Wheel: (25 points) Wheel contains a minimum of 17 triangles. (5 pts) _________ True center is marked clearly with minimal radius. (5 pts) _________ All triangles constructed properly with no gaps or overlays. (5 pts) _________ All right angles and legs are marked correctly. (5 pts) _________ Each hypotenuse length is marked correctly (radical form). (5 pts) _________ Calculation Chart: (40 points) Pythagorean Theorem is written at top of chart (10 pts) _________ All entries completed for the first 17 triangles. (10 pts) _________ Measure for hypotenuse, shown in radical form. (10 pts) _________ Measure for hypotenuse, shown in decimal form . (10 pts) _________ Project Questions: (20 points) Question #1 (4 pts) _________ Question #2 (4 pts) _________ Question #3 (4 pts) _________ Question #4 (4 pts) _________ Question #5 (4 pts) _________ Creativity: (15 points) Neatness and showing all work for each step. (3 pts) _________ Design is original. (3 pts) _________ Design is decorated with attention to detail (3 pts) _________ Effective incorporation of the wheel into the theme of the design. (3 pts) _________ Color is used effectively unless B&W contrast is part of intentional design.