DOCSLIB.ORG

Spirals, Spots and Continued Fractions Robert Dixon Explains

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spirals, Spots and Continued Fractions Robert Dixon Explains © ATM 2012 • No reproduction (including Internet) except for legitimate academic purposes • [email protected] for permissions. SpiRalS, SpotS and continUed FRactionS Robert dixon explains ... there is a wealth of number theory to be found in a famous spatial pattern approximated by plants. 22 n his book On Spirals, archimedes Here the area swept out is ka 4 demonstrates from first principles what Because the area swept by a moving radius we now call integral calculus in polar i overlaps itself once it exceeds one revolution, coordinates. He finds the area under a curve archimedes performed subtractions to find swept out by a radius rotating about a fixed that the areas of successive coils are in the pole. His one example is the curve r = kθ, the proportions 1:6:12:18:24:... the radius of the archimedes spiral. it is the spiral whose radius archimedes spiral increases by the same amount increases uniformly with turning. divide the sector each turn, and so the area enclosed each turn of the spiral θ = 0 to θ = α into n small sectors of a increases with radius. Following in archimedes’ . angle n each small sector has an area between footsteps, Fermat applied the same calculations an inner and an outer circular sector. in radian to his spiral , and found that the areas measure, the ith outer circular sector area is of its successive coils are in the proportions ki22a 3 k23a 1:2:2:2:2:... an equal area is enclosed each turn, 3 .the sum of all these is Σi² times 3 . 2n 2n and the same may be said for any part of a turn. Knowing a number series, the sum of squares, it is the spiral whose area increases uniformly here solves a geometric problem, the area under with turning. to put this equal area property in an nnn32 a curve. Σi² = ++. So the approximating equivalent rule, the Fermat spiral traverses equal area is k2a3 1 1 1 . this has the limit annuli in equal angles. a + + 2 k ka23 6 4n 12n Whereas the archimedes spiral is a simple coil 6 as n → ∞. and the equiangular spiral is the curve of mollusc a) b) c) Figures 1(a)-(c): polar integration, the archimedes spiral and the Fermat spiral. archimedes shows that the sum of inner circular shells, we can look for the Fermat spiral in botany: sectors has the same limit, thus ‘exhausting’ the for example, in the whorls of florets on a mature difference to prove it is the spiral area. of course, daisy head. the reason for this becomes clear he did not have algebraic notation. nearly two if we learn to draw ‘Vogel daisies’, after Helmut thousand years later, a decade before the birth of newton, pierre de Fermat was reading this Vogel, who first used computers to draw them. work by archimedes and practising the first steps (See Figure 2.)the idea is to, a) draw a single in algebraic notation. He must have become Fermat spiral of small equal circles at constant intrigued by the use of number series to find angular interval, α; and to b) explore what values areas, and wondered what spiral would yield to of α gives the best distribution of the circles. the simplest non-trivial series of all - not Σi², but Without any circles overlapping, how much of the Σi. the answer is , the Fermat spiral. plane can be covered? What is the pattern of gaps? Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY 28 Mathematics Teaching 229 Journal of the Association of Teachers of Mathematics © ATM 2012 • No reproduction (including Internet) except for legitimate academic purposesSpiRalS, • [email protected] SpotS and continUed for permissions. FRactionS can we achieve a close-packing? With unit every irrational can be written uniquely as an circular measure, α can be any real value between infinite simple continued fraction: 1 0 and 1. in botany, α is called divergence, the α = [a ; a , a ,....= a + 0 1 2 0 1 angle between successive like parts - leaves, fruits, a1 + branches - about the axis of a stem. By choosing 1 a2 + different values for α in a computer graphic daisy, a3 + ... a) b) c) d) ( 15- 3) ( 5- 1) Figures 2a to d: computer daisies, with a 22, π, , and = 7 3 2 . different patterns of scattering of the circles will all ai are positive integers, except that ao may p be zero. the th principal convergent is the finite result. For rational α = /q, the pattern will be q radial i pi spokes. only q directions will be occupied. this is continued fraction, [a0; a1, a2,…, ai] = /qi, given by remedied by choosing irrational values for α. With recursion formulae: irrational α, no two circles in the spiral can occupy p0 = a0, p1 = a1a0 + 1, pi = ai pi-1 + pi-2 the same direction. the pattern will be a mesh q0 = 1, q1 = a1, qi = aiqi-1 + qi-2 which at a succession of radial distances resolves into sets of symmetrically spaced spirals, some sets For example, π = [3; 7, 15, 1, 292, ...] So, the spiralling one way, some the other. the numbers principal convergents are: of spirals in each set are the denominators of the closest rational approximations, the convergents of α. these can be found as follows. Academic copyright permission does NOT extend to publishing on Internet or similar system. Provide link ONLY July 2012 www.atm.org.uk 29 SpiRalS,© SpotSATM 2012 and • No continUed reproduction (including FRactionS Internet) except for legitimate academic purposes • [email protected] for permissions. principal convergents approach their limit - π in this case - alternately from below and from above. Between two consecutive principal convergents pi-2 pi on the same side of the limit, qi-2 and qi there are ai − 1 intermediate convergents on that side, (npi-1 + pi-2) given by /(nqi-1 + qi-2), n = 1, 2, 3,..., ai – 1. For example, with π, because a2 = 15, there are fourteen improving underestimates between 3 333 25 47 69 /1 to /106, namely /8, /15, /22, and so on to 311 /99, but because a3 = 1, there are no improving 22 355 overestimates from /7 to /113. this theory can be visually represented in a c) Figure 3: the Klein lattice diagram, for α = √2 diagram named after Felix Klein. on a unit square lattice, let any line drawn from lattice point (0, 0) 1 x the answer is α = [0; 1, 1, 1, ...]. Why? Well, it to infinity represent α = /gradient, or / . thus, the y can be shown that the size of gaps in each of the line for any rational α = p/ goes from (0, 0) to q successive sets of spirals showing up in the mesh (p, q), and from there to (2p, 2q), and so on forever is a function of the a’s in the continued fraction through the infinity of lattice points at all multiples i of α, and that size is minimised by minimising of the distance √(p2 + q2). the line for an irrational each corresponding a. Uniquely among all α, on the other hand, goes from (0, 0) to infinity i irrationals, with all a = 1, there are no intermediate without ever passing through another lattice i convergents. the principal convergents are point. We next imagine such a line as a taut string 0, 1, ½, , , , , ... these fractions are anchored at infinity, and all the lattice points as composed of consecutive Fibonacci numbers, upright pins. taking the end at (0, 0), we imagine Fi F . a computer graphic daisy drawn with this the taut string swept first one way then the other, / i+1 value of α will show concentric sets of spirals in in the plane. in each case, the string coming into Fibonacci numbers, as found in botanical daisies. contact with the pins - lattice points - forms a polygon. the polygon vertices are at each of the in fact, [0; 1, 1, 1, ...] = ½(√5 – 1), none other than the golden section, τ. continued fraction theory (pi, qi) corresponding to the principal convergents p i/ . elsewhere the string may touch other lattice formulates the degree of approximation to a given qi points along the polygon sides, between polygon irrational for each of its principal convergents, vertices: these correspond to the intermediate and this degree is poorest with smallest ai’s. the convergents. golden ratio is the ‘slowest converger’. it is the most remote from a fraction of numbers up to Figure 3 takes the case of α = √2. any given size of denominator. You might say it is √2 = 1 + (√2 – 1) = 1 + 1/(√2 + 1) the most irrational. as a point on the open string = 1 + 1/(2 + (√2 – 1) = 1 + 1/(2 + 1/(√2 + 1)), etc., in music, it must be the least harmonious. But in which gives √2 = [1; 2, 2, 2, ...]. the mod 1 arithmetic of serial packing or spacing all quadratic irrationals are periodic continued round an axis, it is the best divergence. fractions, and vice versa. the principal the discovery of Fibonacci numbers in botanical convergents for α = √2 are , , , ,... phyllotaxis - ‘leaf arrangement’ - dates back to the the intermediate convergents are , , ,... 18th century. this is seen in the numbers of spirals or helices in compound flowers or fruits forming So, when drawing a Fermat spiral of equal circles a disc or cone or cylinder - for example, daisy, at constant divergence α, what irrational value of sunflower, pine cone, pineapple - or is seen in the α gives the best spread? numbers of leaves and revolutions in a spiral Academic copyright permission does NOT extend to publishing on Internet or similar system.
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Notes on Greek Mathematics
    MATHEMATICS IN ANCIENT GREECE Periods in Greek history [AG]. Archaic Period: 750 BC to 490 BC (Emergence of city- states to Battle of Marathon) The Iliad and the Odyssey were composed c. 750-720 (based on the `Trojan war’ of 1250-1225 BC.) Between 670-500 BC, many city-states were ruled by tyrants (a word borrowed into the Greek language from Asia Minor, to signify a man who seizes control of the state by a coup and governs illegally.) Classical Period: 490 to 323 BC (Battle of Marathon to death of Alexander). The years from 480 (Persians driven from Greece) to 430 saw the rise of Athenian democracy (under Pericles). The Parthenon was built, 447-432. Playwrights Sophocles, Aristophanes, Euripides were active in Athens (c. 430-400). Following undeclared war between Athens and Sparta from 460-445, the Peloponnesian war (431-404; account by Thucydides) kept these two states (and their allies throughout the region) busy. This led to the ascendance of Sparta and the `rule of thirty tyrants’ in Athens (404-403). Socrates was tried and executed in 399. His follower Plato wrote his dialogues (399-347) and founded the Academy. The accession of Philip II of Macedon (359) signals the beginning of a period of Macedonian rule. Philip invaded Asia in 336, and was assassinated the same year. His successor Alexander III (`the Great’, 356-323 BC) continued the program of world conquest over the next 13 years, invading India in 325 BC. Hellenistic Period (323-30 BC). Science, mathematics (Euclid) and culture flourished in Egypt (Alexandria) under the Ptolemaic dynasty.
    [Show full text]
  • Ioannis M. VANDOULAKIS, « the Arbitrariness of the Sign in Greek Mathematics »
    Ioannis M. VANDOULAKIS, « The Arbitrariness of the sign in Greek Mathematics » Communication donnée dans l’atelier de Jean-Yves Beziau, The Arbitrariness of the Sign, au colloque Le Cours de Linguistique Générale, 1916-2016. L’émergence, Genève, 9-13 janvier 2017. CERCLE FERDINAND DE SAUSSURE N° D’ISBN : 978-2-8399-2282-1 Pour consulter le programme complet de l’atelier de Jean-Yves Beziau, The Arbitrariness of the Sign : https://www.clg2016.org/geneve/programme/ateliers-libres/the- arbitrariness-of-the-sign/ 1 Ioannis M. Vandoulakis The Arbitrariness of the Sign in Greek Mathematics Ioannis M. Vandoulakis Abstract. In this paper, we will examine some modes of signification of mathematical entities used in Greek mathematical texts in the light of Saussure’s conceptualization of sign. In particular, we examine certain mathematical texts from Early Greek period, the Euclidean and Neo- Pythagorean traditions, and Diophantus. 0. Introduction Greek mathematicians have used a wide variety of modes of signification to express mathematical entities. By signification here, we understand the relation between the form of the sign (the signifier) and its meaning (the signified), as used in Ferdinand de Saussure’s semiology. According to Saussure, this relation is essentially arbitrary, motivated only by social convention. Moreover, for Saussure, signifier and signified are inseparable. One does not exist without the other, and conversely, one always implicates the other. Each one of them is the other’s condition of possibility. In this paper, we will examine some modes of signification of mathematical entities used in Greek mathematical texts in the light of Saussure’s conceptualization of sign.
    [Show full text]
  • Copyrighted Material
    11194_Darling_c01_f.qxd 6/10/04 9:56 AM Page 3 A abacus The Chinese suan pan differs from the European aba- A counting frame that started out, several thousand years cus in that the board is split into two decks, with two ago, as rows of pebbles in the desert sands of the Middle beads on each rod in the upper deck and five beads, rep- East. The word appears to come from the Hebrew âbâq resenting the digits 0 through 4, on each rod in the bot- (dust) or the Phoenician abak (sand) via the Greek abax, tom. When all five beads on a rod in the lower deck are which refers to a small tray covered with sand to hold the moved up, they’re reset to the original position, and one pebbles steady. The familiar frame-supporting rods or bead in the top deck is moved down as a carry. When wires, threaded with smoothly running beads, gradually both beads in the upper deck are moved down, they’re emerged in a variety of places and mathematical forms. reset and a bead on the adjacent rod on the left is moved In Europe, there was a strange state of affairs for more up as a carry. The result of the computation is read off than 1,500 years. The Greeks and the Romans, and then from the beads clustered near the separator beam the medieval Europeans, calculated on devices with a between the upper and lower decks. In a sense, the aba- place-value system in which zero was represented by an cus works as a 5-2-5-2-5-2...–based number system in empty line or wire.
    [Show full text]
  • MA 341 Fall 2011
    MA 341 Fall 2011 The Origins of Geometry 1.1: Introduction In the beginning geometry was a collection of rules for computing lengths, areas, and volumes. Many were crude approximations derived by trial and error. This body of knowledge, developed and used in construction, navigation, and surveying by the Babylonians and Egyptians, was passed to the Greeks. The Greek historian Herodotus (5th century BC) credits the Egyptians with having originated the subject, but there is much evidence that the Babylonians, the Hindu civilization, and the Chinese knew much of what was passed along to the Egyptians. The Babylonians of 2,000 to 1,600 BC knew much about navigation and astronomy, which required knowledge of geometry. Clay tablets from the Sumerian (2100 BC) and the Babylonian cultures (1600 BC) include tables for computing products, reciprocals, squares, square roots, and other mathematical functions useful in financial calculations. Babylonians were able to compute areas of rectangles, right and isosceles triangles, trapezoids and circles. They computed the area of a circle as the square of the circumference divided by twelve. The Babylonians were also responsible for dividing the circumference of a circle into 360 equal parts. They also used the Pythagorean Theorem (long before Pythagoras), performed calculations involving ratio and proportion, and studied the relationships between the elements of various triangles. See Appendices A and B for more about the mathematics of the Babylonians. 1.2: A History of the Value of π The Babylonians also considered the circumference of the circle to be three times the diameter. Of course, this would make 3 — a small problem.
    [Show full text]
  • Alan Hirshfeld, Eureka Man: the Life and Legacy of Archimedes Walker & Company, New York, 2009, ISBN: 978-0-8027-1618-7, 242 Pp, Price: U.S
    Sci & Educ (2011) 20:83–87 DOI 10.1007/s11191-010-9244-0 BOOK REVIEW Alan Hirshfeld, Eureka Man: The Life and Legacy of Archimedes Walker & Company, New York, 2009, ISBN: 978-0-8027-1618-7, 242 pp, price: U.S. $26.00 A. K. T. Assis Published online: 19 March 2010 Ó Springer Science+Business Media B.V. 2010 Alan Hirshfeld is a professor of physics at the University of Massachusetts Dartmouth. He conducts research in the history of physics and astronomy. He is an Associate of the Harvard College Observatory and is the director of the UMass Dartmouth Observatory. He obtained his B.A. in astrophysics from Princeton University in 1973, his M.S. in astronomy from Yale in 1975, and his Ph.D. in astronomy also from Yale in 1978. Hirshfeld previously published Parallax: The Race to Measure the Cosmos, (Henry Holt & Co., 2002); The Electric Life of Michael Faraday, (Walker & Co., 2006); and the Astronomy Activity and Laboratory Manual, (Jones & Bartlett Publishers, 2008). He is also the lead author of two reference books in astronomy: Sky Catalogue 2000.0, Vol. 1: Stars to Magnitude 8.0 (Cambridge U.P. and Sky Publishing Corp., 1982 and 1991); and Sky Catalogue 2000.0, Vol. 2: Double Stars, Variable Stars and Nonstellar Objects (Cam- bridge U.P. and Sky Publishing Corp., 1985). Apart from books, he has written articles on the history of astrophysics, on ancient Greek measurements of the solar system, and biographies of astronomers like James Bradley. This book is divided into two parts: Master of Thought (six Chapters), and A Palimpsest’s Tale (eight Chapters).
    [Show full text]
  • The Distribution of Prime Numbers on the Square Root Spiral
    The Distribution of Prime Numbers on the Square Root Spiral - Harry K. Hahn - Ludwig-Erhard-Str. 10 D-76275 EtGermanytlingen, Germany ------------------------------ contribution from “ The Number Spiral “ by - Robert Sachs - ------------------------------ 30. June 2007 Abstract : Prime Numbers accumulate on defined spiral graphs, which run through the Square Root Spiral. These spiral graphs can be assigned to different spiral-systems, in which all spiral-graphs have the same direction of rotation and the same “second difference” between the numbers, which lie on these spiral-graphs. A mathematical analysis shows, that these spiral-graphs are caused exclusively by quadratic polynomials. For example the well known Euler Polynomial x2+x+41 appears on the Square Root Spiral in the form of three spiral-graphs, which are defined by three different quadratic polynomials. All natural numbers, divisible by a certain prime factor, also lie on defined spiral graphs on the Square Root Spiral ( or “Spiral of Theodorus”, or “Wurzelspirale” ). And the Square Numbers 4, 9, 16, 25, 36 … even form a highly three-symmetrical system of three spiral graphs, which divides the square root spiral into three equal areas. Fibonacci number sequences also play a part in the structure of the Square Root Spiral. To learn more about these amazing facts, see my detailed introduction to the Square Root Spiral : “ The ordered distribution of natural numbers on the Square Root Spiral “ With the help of the “Number-Spiral” , described by Mr. Robert Sachs, a comparison can be drawn between the Square Root Spiral and the Ulam Spiral. With the kind permission of Mr Robert Sachs, I show some sections of his webside : www.numberspiral.com in this study.
    [Show full text]
  • Did Archimedes Do Calculus?
    Did Archimedes Do Calculus? Jeff Powers jeff[email protected] Grand Rapids Community College 143 Bostwick Ave NE Grand Rapids, MI 49503 March 30, 2020 1 Introduction The works of Archimedes (c. 287–212 BCE) read like modern mathemat- ics: fully-formed, to the point, and clever. Heath calls them “monuments of mathematical exposition” that are “so impressive in their perfection as to create a feeling akin to awe in the mind of the reader” [14]. Many histories of the calculus begin with Archimedes, and they highlight the parallels between his discoveries and results that are found today by limits, infinite series, and integration techniques, even going so far as to assert that the differences in approaches may be more in words than in ideas [4]. A student may wonder why, if Archimedes’ discoveries are so novel and modern in style, he is not credited with the discovery of the calculus. Historians award that distinc- tion unequivocally to Newton and Leibniz, who lived nearly two millennia after Archimedes. To investigate whether Archimedes should instead be ac- knowledged as the first mathematician to “do calculus,” we will explore three aspects of his work that are related to the calculus: his mechanical method, which anticipates integration techniques; his quadrature of the parabola, which demonstrates how to handle infinite series; and his investigations into spirals, which involve properties of tangent lines. As a whole, ancient Greek mathematicians banned infinity from their formal demonstrations, because at the time the concept was not grounded on any logical basis. Intuition often failed when considering questions of the infinite and how to divide the continuum, as exemplified by Zeno’s famous paradoxes which, to some, have since been resolved by modern formulations of the calculus [4].
    [Show full text]
  • Archimedes of Syracuse: Solving Problems and Applied Mathematics
    Archimedes of Syracuse: Solving Problems and Applied Mathematics Archimedes of Syracuse: Solving Problems and Applied Mathematics Waseda University, SILS, History of Mathematics Archimedes of Syracuse: Solving Problems and Applied Mathematics Domenico Fetti, “Archimedes,” 1620. Archimedes of Syracuse: Solving Problems and Applied Mathematics Outline Introduction Archimedes’ Works Measurement of the circle Archimedes of Syracuse: Solving Problems and Applied Mathematics Introduction What we know about Archimedes (c. 287-212) I He himself tells us, in various writings, that his father was an astronomer and one of his short works is addressed to the young prince of Syracuse (now in Sicily). He is reported to be related to king Hiero II (of Syracuse). I He apparently lived and worked in Syracuse and corresponded with other mathematicians, mostly in Alexandria. I A number of his mechanical writings are preserved in Doric Greek, the dialect of ancient Sicily. I There are a number of reports that Archimedes was killed, in his 70s, when the Romans conquered Syracuse, 212 BCE. I There are many legends about him, but it is difficult to sift the truth from the fantasy. Archimedes of Syracuse: Solving Problems and Applied Mathematics Introduction The legend of Archimedes, 1 I Already in antiquity, Archimedes was seen as the iconographic image of the brilliant scientist (like an ancient Einstein). I Archimedes is said to have been a gifted engineer as well as a mathematician, however, few of his mechanical works survive. I He is reported to have invented a planetarium that could model, or predict, the positions of the sun, moon and planets. I There is the famous story of him discovering the principle of density (specific weight) while taking a bath.
    [Show full text]