The Golden Ratio
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From a GOLDEN RECTANGLE to GOLDEN QUADRILATERALS And
An example of constructive defining: TechSpace From a GOLDEN TechSpace RECTANGLE to GOLDEN QUADRILATERALS and Beyond Part 1 MICHAEL DE VILLIERS here appears to be a persistent belief in mathematical textbooks and mathematics teaching that good practice (mostly; see footnote1) involves first Tproviding students with a concise definition of a concept before examples of the concept and its properties are further explored (mostly deductively, but sometimes experimentally as well). Typically, a definition is first provided as follows: Parallelogram: A parallelogram is a quadrilateral with half • turn symmetry. (Please see endnotes for some comments on this definition.) 1 n The number e = limn 1 + = 2.71828 ... • →∞ ( n) Function: A function f from a set A to a set B is a relation • from A to B that satisfies the following conditions: (1) for each element a in A, there is an element b in B such that <a, b> is in the relation; (2) if <a, b> and <a, c> are in the relation, then b = c. 1It is not being claimed here that all textbooks and teaching practices follow the approach outlined here as there are some school textbooks such as Serra (2008) that seriously attempt to actively involve students in defining and classifying triangles and quadrilaterals themselves. Also in most introductory calculus courses nowadays, for example, some graphical and numerical approaches are used before introducing a formal limit definition of differentiation as a tangent to the curve of a function or for determining its instantaneous rate of change at a particular point. Keywords: constructive defining; golden rectangle; golden rhombus; golden parallelogram 64 At Right Angles | Vol. -
Fibonacci Number
Fibonacci number From Wikipedia, the free encyclopedia • Have questions? Find out how to ask questions and get answers. • • Learn more about citing Wikipedia • Jump to: navigation, search A tiling with squares whose sides are successive Fibonacci numbers in length A Fibonacci spiral, created by drawing arcs connecting the opposite corners of squares in the Fibonacci tiling shown above – see golden spiral In mathematics, the Fibonacci numbers form a sequence defined by the following recurrence relation: That is, after two starting values, each number is the sum of the two preceding numbers. The first Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for n = 0, 1, … , are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, ... (Sometimes this sequence is considered to start at F1 = 1, but in this article it is regarded as beginning with F0=0.) The Fibonacci numbers are named after Leonardo of Pisa, known as Fibonacci, although they had been described earlier in India. [1] [2] • [edit] Origins The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well-known text on these. -
The Astronomers Tycho Brahe and Johannes Kepler
Ice Core Records – From Volcanoes to Supernovas The Astronomers Tycho Brahe and Johannes Kepler Tycho Brahe (1546-1601, shown at left) was a nobleman from Denmark who made astronomy his life's work because he was so impressed when, as a boy, he saw an eclipse of the Sun take place at exactly the time it was predicted. Tycho's life's work in astronomy consisted of measuring the positions of the stars, planets, Moon, and Sun, every night and day possible, and carefully recording these measurements, year after year. Johannes Kepler (1571-1630, below right) came from a poor German family. He did not have it easy growing Tycho Brahe up. His father was a soldier, who was killed in a war, and his mother (who was once accused of witchcraft) did not treat him well. Kepler was taken out of school when he was a boy so that he could make money for the family by working as a waiter in an inn. As a young man Kepler studied theology and science, and discovered that he liked science better. He became an accomplished mathematician and a persistent and determined calculator. He was driven to find an explanation for order in the universe. He was convinced that the order of the planets and their movement through the sky could be explained through mathematical calculation and careful thinking. Johannes Kepler Tycho wanted to study science so that he could learn how to predict eclipses. He studied mathematics and astronomy in Germany. Then, in 1571, when he was 25, Tycho built his own observatory on an island (the King of Denmark gave him the island and some additional money just for that purpose). -
The Origins of the Ptolemaic Tradition and Its Adoption and Replacement in Colonial America
The Origins of the Ptolemaic Tradition and its Adoption and Replacement in Colonial America Master Thesis By Ben Baumann 11592583 June 2018 University of Amsterdam Graduate School of Humanities Master’s Program in Classics and Ancient Civilizations: Ancient Studies First Supervisor: Dr. Jan Willem van Henten Second Supervisor: Dr. Fred Spier Baumann 1 I. Introduction The mapping of the universe and the attempt to understand the cosmos through scholarly investigation has been a constant endeavor of the human race since ancient times. This investigation is known as cosmography. In this thesis, I will analyze how early Colonial American scholars made sense of ancient Greek understandings of cosmography. In particular, I will focus on the way these Greek ideas shaped American thinking not only about the cosmos itself, but also about the way cosmographic understanding became intertwined with views about God and theology. When they first arrived in North America, Colonial Americans generally had a cosmography that was based on the Ptolemaic tradition. But, once they became established in the new world, and especially after the founding of Harvard University, a cosmographic revolution taking place in Europe would begin to resonate in the so-called New World. Thus, some Colonial American scholars willingly engaged in contemplation of the new outlook and proved receptive to the ideas of Nicolas Copernicus. Not surprisingly, in a land where scientific and religious thought overlapped so extensively, this caused heated scholarly debate over the topic and even resulted in student protests at Harvard University. The record of these debates can be traced back to the 17th-century writings of Colonial America, such as several astronomical almanacs, that since have been preserved as vitally significant artifacts of intellectual life in the colonies. -
University of California, San Diego
UNIVERSITY OF CALIFORNIA, SAN DIEGO THE SCIENCE OF THE STARS IN DANZIG FROM RHETICUS TO HEVELIUS A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in History (Science Studies) by Derek Jensen Committee in charge: Professor Robert S. Westman, Chair Professor Luce Giard Professor John Marino Professor Naomi Oreskes Professor Donald Rutherford 2006 The dissertation of Derek Jensen is approved, and it is acceptable in quality and form for publication on microfilm: _________________________________________ _________________________________________ _________________________________________ _________________________________________ _________________________________________ Chair University of California, San Diego 2006 iii FOR SARA iv TABLE OF CONTENTS Signature Page........................................................................................................... iii Dedication ................................................................................................................. iv Table of Contents ...................................................................................................... v List of Figures ........................................................................................................... vi Acknowledgments..................................................................................................... vii Vita, Publications and Fields of Study...................................................................... x A Note on Dating -
Page 1 Golden Ratio Project When to Use This Project
Golden Ratio Project When to use this project: Golden Ratio artwork can be used with the study of Ratios, Patterns, Fibonacci, or Second degree equation solutions and with pattern practice, notions of approaching a limit, marketing, review of long division, review of rational and irrational numbers, introduction to ϕ . Appropriate for students in 6th through 12th grades. Vocabulary and concepts Fibonacci pattern Leonardo Da Pisa = Fibonacci = son of Bonacci Golden ratio Golden spiral Golden triangle Phi, ϕ Motivation Through the investigation of the Fibonacci sequence students will delve into ratio, the notion of irrational numbers, long division review, rational numbers, pleasing proportions, solutions to second degree equations, the fascinating mathematics of ϕ , and more. Introductory concepts In the 13th century, an Italian mathematician named Leonardo Da Pisa (also known as Fibonacci -- son of Bonacci) described an interesting pattern of numbers. The sequence was this; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... Notice that given the first two numbers, the remaining sequence is the sum of the two previous elements. This pattern has been found to be in growth structures, plant branchings, musical chords, and many other surprising realms. As the Fibonacci sequence progresses, the ratio of one number to its proceeding number is about 1.6. Actually, the further along the sequence that one continues, this ratio approaches 1.618033988749895 and more. This is a very interesting number called by the Greek letter phi ϕ . Early Greek artists and philosophers judged that a page 1 desirable proportion in Greek buildings should be width = ϕ times height. The Parthenon is one example of buildings that exhibit this proportion. -
Golden Ratio: a Subtle Regulator in Our Body and Cardiovascular System?
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/306051060 Golden Ratio: A subtle regulator in our body and cardiovascular system? Article in International journal of cardiology · August 2016 DOI: 10.1016/j.ijcard.2016.08.147 CITATIONS READS 8 266 3 authors, including: Selcuk Ozturk Ertan Yetkin Ankara University Istinye University, LIV Hospital 56 PUBLICATIONS 121 CITATIONS 227 PUBLICATIONS 3,259 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: microbiology View project golden ratio View project All content following this page was uploaded by Ertan Yetkin on 23 August 2019. The user has requested enhancement of the downloaded file. International Journal of Cardiology 223 (2016) 143–145 Contents lists available at ScienceDirect International Journal of Cardiology journal homepage: www.elsevier.com/locate/ijcard Review Golden ratio: A subtle regulator in our body and cardiovascular system? Selcuk Ozturk a, Kenan Yalta b, Ertan Yetkin c,⁎ a Abant Izzet Baysal University, Faculty of Medicine, Department of Cardiology, Bolu, Turkey b Trakya University, Faculty of Medicine, Department of Cardiology, Edirne, Turkey c Yenisehir Hospital, Division of Cardiology, Mersin, Turkey article info abstract Article history: Golden ratio, which is an irrational number and also named as the Greek letter Phi (φ), is defined as the ratio be- Received 13 July 2016 tween two lines of unequal length, where the ratio of the lengths of the shorter to the longer is the same as the Accepted 7 August 2016 ratio between the lengths of the longer and the sum of the lengths. -
Thinking Outside the Sphere Views of the Stars from Aristotle to Herschel Thinking Outside the Sphere
Thinking Outside the Sphere Views of the Stars from Aristotle to Herschel Thinking Outside the Sphere A Constellation of Rare Books from the History of Science Collection The exhibition was made possible by generous support from Mr. & Mrs. James B. Hebenstreit and Mrs. Lathrop M. Gates. CATALOG OF THE EXHIBITION Linda Hall Library Linda Hall Library of Science, Engineering and Technology Cynthia J. Rogers, Curator 5109 Cherry Street Kansas City MO 64110 1 Thinking Outside the Sphere is held in copyright by the Linda Hall Library, 2010, and any reproduction of text or images requires permission. The Linda Hall Library is an independently funded library devoted to science, engineering and technology which is used extensively by The exhibition opened at the Linda Hall Library April 22 and closed companies, academic institutions and individuals throughout the world. September 18, 2010. The Library was established by the wills of Herbert and Linda Hall and opened in 1946. It is located on a 14 acre arboretum in Kansas City, Missouri, the site of the former home of Herbert and Linda Hall. Sources of images on preliminary pages: Page 1, cover left: Peter Apian. Cosmographia, 1550. We invite you to visit the Library or our website at www.lindahlll.org. Page 1, right: Camille Flammarion. L'atmosphère météorologie populaire, 1888. Page 3, Table of contents: Leonhard Euler. Theoria motuum planetarum et cometarum, 1744. 2 Table of Contents Introduction Section1 The Ancient Universe Section2 The Enduring Earth-Centered System Section3 The Sun Takes -
Leonhard Euler: His Life, the Man, and His Works∗
SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St. -
The Golden Ratio and the Diagonal of the Square
Bridges Finland Conference Proceedings The Golden Ratio and the Diagonal of the Square Gabriele Gelatti Genoa, Italy [email protected] www.mosaicidiciottoli.it Abstract An elegant geometric 4-step construction of the Golden Ratio from the diagonals of the square has inspired the pattern for an artwork applying a general property of nested rotated squares to the Golden Ratio. A 4-step Construction of the Golden Ratio from the Diagonals of the Square For convenience, we work with the reciprocal of the Golden Ratio that we define as: φ = √(5/4) – (1/2). Let ABCD be a unit square, O being the intersection of its diagonals. We obtain O' by symmetry, reflecting O on the line segment CD. Let C' be the point on BD such that |C'D| = |CD|. We now consider the circle centred at O' and having radius |C'O'|. Let C" denote the intersection of this circle with the line segment AD. We claim that C" cuts AD in the Golden Ratio. B C' C' O O' O' A C'' C'' E Figure 1: Construction of φ from the diagonals of the square and demonstration. Demonstration In Figure 1 since |CD| = 1, we have |C'D| = 1 and |O'D| = √(1/2). By the Pythagorean Theorem: |C'O'| = √(3/2) = |C''O'|, and |O'E| = 1/2 = |ED|, so that |DC''| = √(5/4) – (1/2) = φ. Golden Ratio Pattern from the Diagonals of Nested Squares The construction of the Golden Ratio from the diagonal of the square has inspired the research of a pattern of squares where the Golden Ratio is generated only by the diagonals. -
Chapter 1 Chapter 2 Chapter 3
Notes CHAPTER 1 1. Herbert Westren Turnbull, The Great Mathematicians in The World of Mathematics. James R. Newrnan, ed. New York: Sirnon & Schuster, 1956. 2. Will Durant, The Story of Philosophy. New York: Sirnon & Schuster, 1961, p. 41. 3. lbid., p. 44. 4. G. E. L. Owen, "Aristotle," Dictionary of Scientific Biography. New York: Char1es Scribner's Sons, Vol. 1, 1970, p. 250. 5. Durant, op. cit., p. 44. 6. Owen, op. cit., p. 251. 7. Durant, op. cit., p. 53. CHAPTER 2 1. Williarn H. Stahl, '' Aristarchus of Samos,'' Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 1, 1970, p. 246. 2. Jbid., p. 247. 3. G. J. Toorner, "Ptolerny," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 11, 1975, p. 187. CHAPTER 3 1. Stephen F. Mason, A History of the Sciences. New York: Abelard-Schurnan Ltd., 1962, p. 127. 2. Edward Rosen, "Nicolaus Copernicus," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 3, 1971, pp. 401-402. 3. Mason, op. cit., p. 128. 4. Rosen, op. cit., p. 403. 391 392 NOTES 5. David Pingree, "Tycho Brahe," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 2, 1970, p. 401. 6. lbid.. p. 402. 7. Jbid., pp. 402-403. 8. lbid., p. 413. 9. Owen Gingerich, "Johannes Kepler," Dictionary of Scientific Biography. New York: Charles Scribner's Sons, Vol. 7, 1970, p. 289. 10. lbid.• p. 290. 11. Mason, op. cit., p. 135. 12. Jbid .. p. 136. 13. Gingerich, op. cit., p. 305. CHAPTER 4 1. -
The Mathematics of Art: an Untold Story Project Script Maria Deamude Math 2590 Nov 9, 2010
The Mathematics of Art: An Untold Story Project Script Maria Deamude Math 2590 Nov 9, 2010 Art is a creative venue that can, and has, been used for many purposes throughout time. Religion, politics, propaganda and self-expression have all used art as a vehicle for conveying various messages, without many references to the technical aspects. Art and mathematics are two terms that are not always thought of as being interconnected. Yet they most definitely are; for art is a highly technical process. Following the histories of maths and sciences – if not instigating them – art practices and techniques have developed and evolved since man has been on earth. Many mathematical developments occurred during the Italian and Northern Renaissances. I will describe some of the math involved in art-making, most specifically in architectural and painting practices. Through the medieval era and into the Renaissance, 1100AD through to 1600AD, certain significant mathematical theories used to enhance aesthetics developed. Understandings of line, and placement and scale of shapes upon a flat surface developed to the point of creating illusions of reality and real, three dimensional space. One can look at medieval frescos and altarpieces and witness a very specific flatness which does not create an illusion of real space. Frescos are like murals where paint and plaster have been mixed upon a wall to create the image – Michelangelo’s work in the Sistine Chapel and Leonardo’s The Last Supper are both famous examples of frescos. The beginning of the creation of the appearance of real space can be seen in Giotto’s frescos in the late 1200s and early 1300s.