Archimedes and the Mathematics of Spirals Transcript
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Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Squaring the Circle in Elliptic Geometry
Rose-Hulman Undergraduate Mathematics Journal Volume 18 Issue 2 Article 1 Squaring the Circle in Elliptic Geometry Noah Davis Aquinas College Kyle Jansens Aquinas College, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/rhumj Recommended Citation Davis, Noah and Jansens, Kyle (2017) "Squaring the Circle in Elliptic Geometry," Rose-Hulman Undergraduate Mathematics Journal: Vol. 18 : Iss. 2 , Article 1. Available at: https://scholar.rose-hulman.edu/rhumj/vol18/iss2/1 Rose- Hulman Undergraduate Mathematics Journal squaring the circle in elliptic geometry Noah Davis a Kyle Jansensb Volume 18, No. 2, Fall 2017 Sponsored by Rose-Hulman Institute of Technology Department of Mathematics Terre Haute, IN 47803 a [email protected] Aquinas College b scholar.rose-hulman.edu/rhumj Aquinas College Rose-Hulman Undergraduate Mathematics Journal Volume 18, No. 2, Fall 2017 squaring the circle in elliptic geometry Noah Davis Kyle Jansens Abstract. Constructing a regular quadrilateral (square) and circle of equal area was proved impossible in Euclidean geometry in 1882. Hyperbolic geometry, however, allows this construction. In this article, we complete the story, providing and proving a construction for squaring the circle in elliptic geometry. We also find the same additional requirements as the hyperbolic case: only certain angle sizes work for the squares and only certain radius sizes work for the circles; and the square and circle constructions do not rely on each other. Acknowledgements: We thank the Mohler-Thompson Program for supporting our work in summer 2014. Page 2 RHIT Undergrad. Math. J., Vol. 18, No. 2 1 Introduction In the Rose-Hulman Undergraduate Math Journal, 15 1 2014, Noah Davis demonstrated the construction of a hyperbolic circle and hyperbolic square in the Poincar´edisk [1]. -
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 -
He One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’S Elements?
ARCHIWUM HISTORII FILOZOFII I MYŚLI SPOŁECZNEJ • ARCHIVE OF THE HISTORY OF PHILOSOPHY AND SOCIAL THOUGHT VOL. 59/2014 • ISSN 0066–6874 Zbigniew Król he One and the Dyad: the Foundations of Ancient Mathematics1 What Exists Instead of Ininite Space in Euclid’s Elements? ABSTRACT: his paper contains a new interpretation of Euclidean geometry. It is argued that ancient Euclidean geometry was created in a quite diferent intuitive model (or frame), without ininite space, ininite lines and surfaces. his ancient intuitive model of Euclidean geometry is reconstructed in connection with Plato’s unwritten doctrine. he model cre- ates a kind of “hermeneutical horizon” determining the explicit content and mathematical methods used. In the irst section of the paper, it is argued that there are no actually ininite concepts in Euclid’s Elements. In the second section, it is argued that ancient mathematics is based on Plato’s highest principles: the One and the Dyad and the role of agrapha dogmata is unveiled. KEYWORDS: Euclidean geometry • Euclid’s Elements • ancient mathematics • Plato’s unwrit- ten doctrine • philosophical hermeneutics • history of science and mathematics • philosophy of mathematics • philosophy of science he possibility to imagine all of the theorems from Euclid’s Elements in a Tquite diferent intuitive framework is interesting from the philosophical point of view. I can give one example showing how it was possible to lose the genuine image of Euclidean geometry. In many translations into Latin (Heiberg) and English (Heath) of the theorems from Book X, the term “area” (spatium) is used. For instance, the translations of X. 26 are as follows: ,,Spatium medium non excedit medium spatio rationali” and “A medial area does not exceed a medial area by a rational area”2. -
Greece: Archimedes and Apollonius
Greece: Archimedes and Apollonius Chapter 4 Archimedes • “What we are told about Archimedes is a mix of a few hard facts and many legends. Hard facts – the primary sources –are the axioms of history. Unfortunately, a scarcity of fact creates a vacuum that legends happily fill, and eventually fact and legend blur into each other. The legends resemble a computer virus that leaps from book to book, but are harder, even impossible, to eradicate.” – Sherman Stein, Archimedes: What Did He Do Besides Cry Eureka?, p. 1. Archimedes • Facts: – Lived in Syracuse – Applied mathematics to practical problems as well as more theoretical problems – Died in 212 BCE at the hands of a Roman soldier during the attack on Syracuse by the forces of general Marcellus. Plutarch, in the first century A.D., gave three different stories told about the details of his death. Archimedes • From sources written much later: – Died at the age of 75, which would put his birth at about 287 BCE (from The Book of Histories by Tzetzes, 12th century CE). – The “Eureka” story came from the Roman architect Vitruvius, about a century after Archimedes’ death. – Plutarch claimed Archimedes requested that a cylinder enclosing a sphere be put on his gravestone. Cicero claims to have found that gravestrone in about 75 CE. Archimedes • From sources written much later: – From about a century after his death come tales of his prowess as a military engineer, creating catapults and grappling hooks connected to levers that lifted boats from the sea. – Another legend has it that he invented parabolic mirrors that set ships on fire. -
Some Curves and the Lengths of Their Arcs Amelia Carolina Sparavigna
Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna To cite this version: Amelia Carolina Sparavigna. Some Curves and the Lengths of their Arcs. 2021. hal-03236909 HAL Id: hal-03236909 https://hal.archives-ouvertes.fr/hal-03236909 Preprint submitted on 26 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Some Curves and the Lengths of their Arcs Amelia Carolina Sparavigna Department of Applied Science and Technology Politecnico di Torino Here we consider some problems from the Finkel's solution book, concerning the length of curves. The curves are Cissoid of Diocles, Conchoid of Nicomedes, Lemniscate of Bernoulli, Versiera of Agnesi, Limaçon, Quadratrix, Spiral of Archimedes, Reciprocal or Hyperbolic spiral, the Lituus, Logarithmic spiral, Curve of Pursuit, a curve on the cone and the Loxodrome. The Versiera will be discussed in detail and the link of its name to the Versine function. Torino, 2 May 2021, DOI: 10.5281/zenodo.4732881 Here we consider some of the problems propose in the Finkel's solution book, having the full title: A mathematical solution book containing systematic solutions of many of the most difficult problems, Taken from the Leading Authors on Arithmetic and Algebra, Many Problems and Solutions from Geometry, Trigonometry and Calculus, Many Problems and Solutions from the Leading Mathematical Journals of the United States, and Many Original Problems and Solutions. -
Archimedean Spirals ∗
Archimedean Spirals ∗ An Archimedean Spiral is a curve defined by a polar equation of the form r = θa, with special names being given for certain values of a. For example if a = 1, so r = θ, then it is called Archimedes’ Spiral. Archimede’s Spiral For a = −1, so r = 1/θ, we get the reciprocal (or hyperbolic) spiral. Reciprocal Spiral ∗This file is from the 3D-XploreMath project. You can find it on the web by searching the name. 1 √ The case a = 1/2, so r = θ, is called the Fermat (or hyperbolic) spiral. Fermat’s Spiral √ While a = −1/2, or r = 1/ θ, it is called the Lituus. Lituus In 3D-XplorMath, you can change the parameter a by going to the menu Settings → Set Parameters, and change the value of aa. You can see an animation of Archimedean spirals where the exponent a varies gradually, from the menu Animate → Morph. 2 The reason that the parabolic spiral and the hyperbolic spiral are so named is that their equations in polar coordinates, rθ = 1 and r2 = θ, respectively resembles the equations for a hyperbola (xy = 1) and parabola (x2 = y) in rectangular coordinates. The hyperbolic spiral is also called reciprocal spiral because it is the inverse curve of Archimedes’ spiral, with inversion center at the origin. The inversion curve of any Archimedean spirals with respect to a circle as center is another Archimedean spiral, scaled by the square of the radius of the circle. This is easily seen as follows. If a point P in the plane has polar coordinates (r, θ), then under inversion in the circle of radius b centered at the origin, it gets mapped to the point P 0 with polar coordinates (b2/r, θ), so that points having polar coordinates (ta, θ) are mapped to points having polar coordinates (b2t−a, θ). -
Post-Euclid Greek Mathematics
Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Some high points of Greek mathematics after Euclid Algebra Through History October 2019 Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Outline 1 Archimedes 2 Apollonius and the Conics 3 How Apollonius described and classified the conic sections Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Who was Archimedes? Lived ca. 287 - 212 BCE, mostly in Greek city of Syracuse in Sicily Studied many topics in what we would call mathematics, physics, engineering (less distinction between them at the time) We don’t know much about his actual life; much of his later reputation was based on somewhat dubious anecdotes, e.g. the “eureka moment,” inventions he was said to have produced to aid in defense of Syracuse during Roman siege in which he was killed, etc. Perhaps most telling: we do know he designed a tombstone for himself illustrating the discovery he wanted most to be remembered for (discussed by Plutarch, Cicero) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Figure: Sphere inscribed in cylinder of equal radius 3Vsphere = 2Vcyl and Asphere = Acyl (lateral area) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works On the Equilibrium of Planes (2 books) On Floating Bodies (2 books) Measurement of a Circle On Conoids and Spheroids On Spirals On the Sphere and Cylinder (2 books) Algebra Through History Greek Math Post Euclid Archimedes Apollonius and the Conics How Apollonius described and classified the conic sections Surviving works, cont. -
Pappus of Alexandria: Book 4 of the Collection
Pappus of Alexandria: Book 4 of the Collection For other titles published in this series, go to http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences Managing Editor J.Z. Buchwald Associate Editors J.L. Berggren and J. Lützen Advisory Board C. Fraser, T. Sauer, A. Shapiro Pappus of Alexandria: Book 4 of the Collection Edited With Translation and Commentary by Heike Sefrin-Weis Heike Sefrin-Weis Department of Philosophy University of South Carolina Columbia SC USA [email protected] Sources Managing Editor: Jed Z. Buchwald California Institute of Technology Division of the Humanities and Social Sciences MC 101–40 Pasadena, CA 91125 USA Associate Editors: J.L. Berggren Jesper Lützen Simon Fraser University University of Copenhagen Department of Mathematics Institute of Mathematics University Drive 8888 Universitetsparken 5 V5A 1S6 Burnaby, BC 2100 Koebenhaven Canada Denmark ISBN 978-1-84996-004-5 e-ISBN 978-1-84996-005-2 DOI 10.1007/978-1-84996-005-2 Springer London Dordrecht Heidelberg New York British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Control Number: 2009942260 Mathematics Classification Number (2010) 00A05, 00A30, 03A05, 01A05, 01A20, 01A85, 03-03, 51-03 and 97-03 © Springer-Verlag London Limited 2010 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licenses issued by the Copyright Licensing Agency. -
Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica
Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Greek Mathematics Recovered in Books 6 and 7 of Clavius’ Geometria Practica John B. Little Department of Mathematics and CS College of the Holy Cross June 29, 2018 Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions I’ve always been interested in the history of mathematics (in addition to my nominal specialty in algebraic geometry/computational methods/coding theory, etc.) Want to be able to engage with original texts on their own terms – you might recall the talks on Apollonius’s Conics I gave at the last Clavius Group meeting at Holy Cross (two years ago) So, I’ve been taking Greek and Latin language courses in HC’s Classics department The subject for today relates to a Latin-to-English translation project I have recently begun – working with the Geometria Practica of Christopher Clavius, S.J. (1538 - 1612, CE) Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Overview 1 Introduction – Clavius and Geometria Practica 2 Book 6 and Greek approaches to duplication of the cube 3 Book 7 and squaring the circle via the quadratrix 4 Conclusions Greek Mathematics in Clavius Introduction – Clavius and Geometria Practica Book 6 and Greek approaches to duplication of the cube Book 7 and squaring the circle via the quadratrix Conclusions Clavius’ Principal Mathematical Textbooks Euclidis Elementorum, Libri XV (first ed. -
Analysis of Spiral Curves in Traditional Cultures
Forum _______________________________________________________________ Forma, 22, 133–139, 2007 Analysis of Spiral Curves in Traditional Cultures Ryuji TAKAKI1 and Nobutaka UEDA2 1Kobe Design University, Nishi-ku, Kobe, Hyogo 651-2196, Japan 2Hiroshima-Gakuin, Nishi-ku, Hiroshima, Hiroshima 733-0875, Japan *E-mail address: [email protected] *E-mail address: [email protected] (Received November 3, 2006; Accepted August 10, 2007) Keywords: Spiral, Curvature, Logarithmic Spiral, Archimedean Spiral, Vortex Abstract. A method is proposed to characterize and classify shapes of plane spirals, and it is applied to some spiral patterns from historical monuments and vortices observed in an experiment. The method is based on a relation between the length variable along a curve and the radius of its local curvature. Examples treated in this paper seem to be classified into four types, i.e. those of the Archimedean spiral, the logarithmic spiral, the elliptic vortex and the hyperbolic spiral. 1. Introduction Spiral patterns are seen in many cultures in the world from ancient ages. They give us a strong impression and remind us of energy of the nature. Human beings have been familiar to natural phenomena and natural objects with spiral motions or spiral shapes, such as swirling water flows, swirling winds, winding stems of vines and winding snakes. It is easy to imagine that powerfulness of these phenomena and objects gave people a motivation to design spiral shapes in monuments, patterns of cloths and crafts after spiral shapes observed in their daily lives. Therefore, it would be reasonable to expect that spiral patterns in different cultures have the same geometrical properties, or at least are classified into a small number of common types. -
Math 109: Mathematics for Design
MATH 109: MATHEMATICS FOR DESIGN SUMMARY OF GOALS FOR THE COURSE In our contemporary culture the dialogue between math and art, while sometimes strained by Course details misunderstandings, is a dynamic and living one. Art Time: MWF 10:20 – 11:30am continues to inspire and inform mathematical Place: Mon, Wed PPHAC 235 thinking, and mathematics helps artists develop Fri PPHAC 112 additional insight when reasoning about the content and structure of their work. The tools of mathematics Instructor: Kevin Hartshorn also aid in the construction of conceptual Office: PPHAC 215 frameworks that are useful in all aspects of life. Hours: Mon/Wed 2:30-3:30pm, Tue/Thu 8:30-9:30am, This course will introduce students to ideas in or by appointment mathematical thinking that are related to artistic considerations. Students will need to show e-mail: [email protected] proficiency with some mathematical ideas and then Web: https://sites.google.com/a/moravian.edu/ apply those ideas in creating their own works of art. math-109-spring-2015 In the process, students will also be called to analyze Text: Squaring the Circle: Geometry in Art existing artwork with a mathematical eye. In this and Architecture, by Paul Calter way, students will be provided a new tool to use in their approach to art and aesthetics. KEY IDEAS FOR THIS COURSE Each assignment and class discussion will be aimed at expanding on these key notions: • Mathematics and mathematical thinking involve a creative effort, not just rote memorization. • There is a rich and complex connection between mathematics and art. • Very basic mathematical concepts can be used to solve seemingly complex real-world problems.