DOCSLIB.ORG

FYSE 1423 - the Story of Geometry Fall 2014

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
FYSE 1423 - the Story of Geometry Fall 2014 FYSE 1423 - The Story of Geometry Fall 2014 Emily Proctor [email protected] 310 Warner Hall Office Hours: Tu 11-12, W 11-12, Th 2:30-4:00, and by appointment Peer Mentor: Kate Eiseman [email protected] Librarian: Bryan Carson [email protected] Course Description In this course we will look at one storyline in the development of modern geometry. We will begin the semester with an examination of Euclid's Elements. Written over 2000 years ago, Elements is one of the most influential textbooks ever, and has long been held up as a shining example of deductive reasoning. While this book deserves great praise, we will see that it is not without flaws. After familiarizing ourselves with Elements, we will investigate its problems, and see how modern mathematicians, Hilbert in particular, shored up Euclid's work years later. Euclid's work is based on five postulates, the last of which is often referred to as the parallel postulate. For years, mathematicians believed that the parallel postulate could be deduced from the first four postulates, and gave many erroneous proofs that it was. Finally, in the 19th century, three mathematicians, Gauss, Bolyai, and Lobachevsky, came to the outstanding revelation that there are consistent geometric systems for which the first four postulates hold but for which the parallel postulate does not. This discovery shook the foundations of geometry and opened the door to modern geometry. We will examine one such geometric system, and at the end of the semester, we will learn about how modern geometers view the subject. Modern geometry does not look much like Euclid's Elements, but we will see how its roots come from Euclid's ancient work. Course Website The website for this course can be found at: http://f14.middlebury.edu/fyse1423a Look here for information about the course as well as homework and writing assignments. Texts • Euclid's Elements, edited by Dana Densmore, translated by T. L. Heath. • Geometry: Euclid and Beyond, by Robin Hartshorne. • The Shape of Space, second edition, by Jeffrey Weeks. 1 Homework We will have homework for each class meeting. Some of the assignments will be \traditional" math homework, in the form of problems to work out, and some of our homework will be reading and writing assignments. Please check the website for our daily assignments. Although you will not always have something to turn in, it is essential that you do the assignment each night as our activities in class will depend heavily on the work that you do between classes. Major Assignments Along with smaller assignments, there will be three main papers for this course. For each paper, you will need to produce both a polished draft, along with a substantial revision of that draft. For your planning purposes, the tentative due dates for the polished drafts are Monday, September 29 Monday, November 3 Monday, December 1 and the revisions would be due roughly two weeks later. I will pass out more specific information about the assignments and due dates during the semester. Please note that I have a very strict policy of not accepting late work without penalty, so make sure to plan accordingly. Support We have both a peer mentor, Kate Eiseman, and a librarian, Bryan Carson, assigned to our class to help support you in your transition to Middlebury. Our peer mentor is available to help with your writing process, as well as with your oral presentation and time-management skills. At some points in the semester, I will require that you meet with Kate to discuss your writing, but she is available and ready to help even if I do not require that you speak with her. She may also hold a workshop on oral presentation skills for you, depending on the interest of the class. Our librarian is available to help you learn to navigate the Middlebury College library. He will help acquaint you with the resources available through the library and can also help with specific questions you have about finding library materials. Both Kate and Bryan's contact information is listed at the top of the syllabus, so please do not hesitate to contact either of them with questions. Honor Code The Middlebury College Honor Code is central to the academic community that we foster at Middlebury. To make sure that you get started on the right foot with your college career, in class we will discuss the Honor Code as it applies to your life at Middlebury College in general. For this particular course, the Honor Code applies as follows: your written work and any homework problems you turn in must be entirely your own 2 work, and may not be work that you have turned in for another class prior to ours. Please properly cite any sources that you reference in your papers. For any homework problems, please cite any help that you have received from a classmate. Online sources are not allowed for solving homework problems. If you have any questions at any point, please ask me! Attendance I expect to attend every class during the term and to participate fully during class. Please on time, and be ready to start working at the time that class starts. I will not take formal attendance but I will notice if you are gone. Depending on the circumstances, missing class could have a negative impact on your final grade. If you are seriously ill for a number of days, contact me and your dean as soon as possible so that we can work with you to help you keep up with the rest of the class. Grading I will determine final grades according to the following percentages: First Major Paper 30% Second Major Paper 30% Third Major Paper 30% Active Class Participation and Smaller Assignments 10% 3 FYSE 1423: The Story of Geometry Fall 2014 First Paper Assignment: Book IV of Euclid’s Elements. Your task for this assignment is to explore Book IV of Euclid’s Elements.Iwouldlike you to read and work to understand all of the definitions, all of the statements of the sixteen propositions in Book IV, and as many proofs of the propositions as you would like. Once you have done this, please write a 4-6 page paper that summarizes what you have found. In your paper, you should include a summary and analysis of the definitions and the propositions. You aim here should be to try to explain to an intelligent lay person your observations about the contents of Book IV. In other words, you should read Euclid’s work, internalize it, and explain it in your own language. In order to give some depth to your work, I would like you to read at least one proof carefully and incorporate the proof, again in your own words, into your paper. For this, please choose one of the proofs of Proposition 10, Proposition 11, or Proposition 12. When you write about the proof, first summarize the main steps of the proof and then work through the reasoning in finer detail. Whenever Euclid makes reference to an earlier proposition in his proof, please include in your exposition a statement, in your own words, of what the earlier proposition says. If Euclid leaves out any references, please supply them. Again, you should think of your audience here as an intelligent lay person: someone who has basic background in mathematics but who is not necessarily an expert in geometry. Note that although you do not need to read all of the proofs in Book IV, the more proofs you read, the better your paper is likely to be. Even if you don’t incorporate all of the proofs into your paper, knowing them will give you a fuller and more nuanced understanding of the work. Dates: Your mostly polished draft is due in class on Monday, September 29. Please also • email me an electronic copy by 5pm on that day. IwillreadyourdraftsandgetthembacktoyouwithcommentsbyMonday, • October 6. Your revised paper is due back to me in class on Wednesday, October 15. Please also email me an electronic copy by 5pm on that day. Each of you will meet with our peer mentor, Kate Eiseman, in the process of • writing your initial draft of your paper. She will email the class a list of available meeting times during September 17-28. Please set up a meeting with her and send her a copy of your current draft 24 hours before your meeting so that she has time to read it before talking with you. A note about sources: As you know, Euclid’s Elements is an ancient text and many, many people have written about it over the years. Although there may be other sources out there that describe the work in Book IV of Elements,forthispaperIaminterestedinyour interpretation and explanation of Book IV. Thus, the only resources you may use for this paper are your copy of Elements; your copy of Hartshorne (Chapter 1, Section 4 in particular might be worth a read); and conversations with each other, Kate Eiseman, and me. FYSE 1423: The Story of Geometry Fall 2014 Second Paper Assignment: Geometry in Context Your task for this assignment is to research either the life and work of a mathematician who was influential in the development of geometry or a period of major development in the history of geometry. Write about your findings in a 4-6 page (double spaced, 12pt font) paper. The goal of this assignment is to give some mathematical and historical context as well as life and a human side to the material we are studying in class.
Load more

Recommended publications
  • 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.
    [Show full text]
  • A CONCISE MINI HISTORY of GEOMETRY 1. Origin And
    Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 5{21. A CONCISE MINI HISTORY OF GEOMETRY LEOPOLD VERSTRAELEN 1. Origin and development in Old Greece Mathematics was the crowning and lasting achievement of the ancient Greek cul- ture. To more or less extent, arithmetical and geometrical problems had been ex- plored already before, in several previous civilisations at various parts of the world, within a kind of practical mathematical scientific context. The knowledge which in particular as such first had been acquired in Mesopotamia and later on in Egypt, and the philosophical reflections on its meaning and its nature by \the Old Greeks", resulted in the sublime creation of mathematics as a characteristically abstract and deductive science. The name for this science, \mathematics", stems from the Greek language, and basically means \knowledge and understanding", and became of use in most other languages as well; realising however that, as a matter of fact, it is really an art to reach new knowledge and better understanding, the Dutch term for mathematics, \wiskunde", in translation: \the art to achieve wisdom", might be even more appropriate. For specimens of the human kind, \nature" essentially stands for their organised thoughts about sensations and perceptions of \their worlds outside and inside" and \doing mathematics" basically stands for their thoughtful living in \the universe" of their idealisations and abstractions of these sensations and perceptions. Or, as Stewart stated in the revised book \What is Mathematics?" of Courant and Robbins: \Mathematics links the abstract world of mental concepts to the real world of physical things without being located completely in either".
    [Show full text]
  • A Survey of the Development of Geometry up to 1870
    A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.
    [Show full text]
  • The Project Gutenberg Ebook #31061: a History of Mathematics
    The Project Gutenberg EBook of A History of Mathematics, by Florian Cajori This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: A History of Mathematics Author: Florian Cajori Release Date: January 24, 2010 [EBook #31061] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK A HISTORY OF MATHEMATICS *** Produced by Andrew D. Hwang, Peter Vachuska, Carl Hudkins and the Online Distributed Proofreading Team at http://www.pgdp.net transcriber's note Figures may have been moved with respect to the surrounding text. Minor typographical corrections and presentational changes have been made without comment. This PDF file is formatted for screen viewing, but may be easily formatted for printing. Please consult the preamble of the LATEX source file for instructions. A HISTORY OF MATHEMATICS A HISTORY OF MATHEMATICS BY FLORIAN CAJORI, Ph.D. Formerly Professor of Applied Mathematics in the Tulane University of Louisiana; now Professor of Physics in Colorado College \I am sure that no subject loses more than mathematics by any attempt to dissociate it from its history."|J. W. L. Glaisher New York THE MACMILLAN COMPANY LONDON: MACMILLAN & CO., Ltd. 1909 All rights reserved Copyright, 1893, By MACMILLAN AND CO. Set up and electrotyped January, 1894. Reprinted March, 1895; October, 1897; November, 1901; January, 1906; July, 1909. Norwood Pre&: J. S. Cushing & Co.|Berwick & Smith.
    [Show full text]
  • Timeline for Euclidean Geometry and the “Fifth Postulate”
    Timeline For Euclidean Geometry and the “Fifth Postulate” There is a lot of history of geometry before Euclid can onto the scene. The Babylonians, Egyptians, Greeks, and Pythagoreans did a great deal to advance the subject of geometry. The area of geometry that is taught most widely in high school mathematics classes is Euclidean Geometry. The focus for my project will be Euclidean Geometry. ~2000-500 BC- The Babylonians and Egyptians created many rules that became the basis for Geometry. They created rules, from experimentation, that were used only by the engineers within the civilizations. They had little previous knowledge on the subject of Geometry and both civilizations spent much time assuming rules and then confirming or refuting what they had declared to be true. ~500 BC- Pythagoras of Samos began to use logical thinking to make geometrical conclusions. He also used basic geometrical knowledge to make conclusions regarding advanced geometrical properties. ~300 BC (Before Euclid and The Elements)- Plato developed the precise idea of what a proof should entail. He also pushed for all conclusions, such as definitions and hypothesis, to be clear and able to be understood by most educated individuals. ~300 BC- Euclid wrote what is know as The Elements. This work is a collection of thirteen different books that are collection of geometric knowledge that was decided even before Euclid’s existence. This was knowledge expressed by Pythagoras, Plato, Hippocrates, and other mathematicians that were the pioneers in expressing geometrical conclusions. The first of these books was composed of 23 definitions, 5 postulates, 5 common notions, and 48 propositions.
    [Show full text]
  • Mathematics History in the Geometry Classroom an Honor Thesis
    - Mathematics History in the Geometry Classroom An Honor Thesis (HONRS 499) by Lana D. smith Thesis Advisor Dr. Hubert Ludwig Ball State University Muncie, Indiana April, 1993 Graduation: May 8, 1993 ,- s~rl 1he.slS t...D ~HS9 ."Zlf I '1ct 3 - Abstract ,. SC:>51f This thesis is a collection of history lessons to be used in a mathematics classroom, specifically the geometry class, but many of the topics and persons discussed will lend themselves to discussion in other mathematics class. There are fourteen lessons to correlate with chapters of most geometry texts. The introduction contains several useful suggestions for integrating history in the classroom and use of the material. The goal of these lessons is to add interest in enthusiasm in an area that generally receives negative reactions from many students--math. - "In most sciences one generation tears down what another has - built, and what one has established another undoes. In mathematics alone each generation builds a new story to the old structure." -Hermann Hankel (Boyer and Merzbach, p. 619). Why use history? As Posamentier and Stepelman put it, in reference to the mathematics classroom fl ••• to breathe life into what otherwise might be a golem." using the lives, loves, successes, and failures of the people that created mathematics gives interest to a sometimes boring topic. History can be integrated into the classroom in a light and lively way (Posamentier and Stepelman, p. 156). The use of history can attract student interest and enthusiasm as it did with me. The following are ways to use history in the classroom: -mention past mathematicians anecdotally -mention historical introductions to concepts that are new to students -encourage pupils to understand the historical problems which the concepts they are learning are answers -give "history of mathematics" lessons -devise classroom/homework exercises using mathematical texts from the past -direct dramatic activity which reflects mathematical interaction -encourage creation of poster displays or other projects with a historical theme.
    [Show full text]
  • A Brief Historical Antecedents to the Evolution of Geometry Education
    A Brief Historical Antecedents to the Evolution of Geometry Education Mohammad A. Yazdani, Ph.D. † Abstract The contributions of geometry to the development and advancement of the other branches of mathematics, physics, astronomy, chemistry, biology, engineering, architecture, and arts are recognized by historians, mathematicians, scientists, artists, and educators. Perhaps, the need for learning geometric concepts and skills and their applications in the 21st century is more crucial than any other time in the history of mankind. We observe the implications of geometric skills and concepts in our every day lives, whether we are in need of a simple estimation such as the number of shingles to replace the roof in our house or a complicated subject such as defining the curvature of space and the path of moving bodies to learn the laws of universe. Utilizing geometric concepts and skills in learning and problem solving in other scientific disciplines require an efficient level of proficiency in a large number of geometric concepts or skills. Considering the importance of geometry in school curriculum, an effective method of teaching and learning geometry is desirable for mathematics educators. A brief review of geometry textbooks published in the United States between 1854 and 2001 helps us to explore and examine the instructional design and methods utilized for teaching and learning of this discipline. Background From 8000 B.C. (the beginning of the development of agriculture) when Sumerians used simple plain clay tokens in the shapes of disks, spheres, cylinders, and cones to keep track of their different kinds of goods (before the invention of writing) to the 20th century when Einstein's new geometry of space and time attempted to explain the natural motions of the objects in the universe, geometry has helped mankind to organize, classify, describe, represent, explain, interpret, define, and express in a logical way the world surrounding us.
    [Show full text]
  • Geometry and Empirical Science Author(S): C
    Geometry and Empirical Science Author(s): C. G. Hempel Source: The American Mathematical Monthly, Vol. 52, No. 1 (Jan., 1945), pp. 7-17 Published by: Mathematical Association of America Stable URL: http://www.jstor.org/stable/2304830 . Accessed: 03/02/2015 13:33 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Mathematical Association of America is collaborating with JSTOR to digitize, preserve and extend access to The American Mathematical Monthly. http://www.jstor.org This content downloaded from 140.180.251.176 on Tue, 3 Feb 2015 13:33:42 PM All use subject to JSTOR Terms and Conditions GEOMETRY AND EMPIRICAL SCIENCE C. G. HEMPEL, QueensCollege 1. Introduction. The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which ac- counts forits fame as the queen of the sciences,is no doubt the peculiar certainty and necessityof its results. No propositionin even the most advanced parts of empiricalscience can ever attain this status; a hypothesisconcerning "matters of empirical fact" can at best acquire what is loosely called a high probability or a high degree of confirmationon the basis of the relevant evidence available; but however well it may have been confirmedby careful tests, the possibility can never be precluded that it will have to be discarded later in the light of new and disconfirmingevidence.
    [Show full text]
  • From Geometry to Topology - J.J
    HISTORY OF MATHEMATICS - From Geometry To Topology - J.J. Gray and V.L. Hansen FROM GEOMETRY TO TOPOLOGY J.J. Gray Department of Mathematics, Open University, UK V.L. Hansen Department of Mathematics, Technical University of Denmark, Denmark Keywords: Geometry, topology, surface, Riemann surface, manifold, global differential geometry, curvature, dense subset, nowhere dense subset, Cantor set, dimension, homotopy, homology, metric space, compact, connected, topological space. Contents 1. Introduction 2. Three Visions of Geometry in the late Nineteenth Century 2.1. Projective Geometry as a Fundamental Geometry 2.2. Non-Euclidean Geometry and Physical Space 2.3. The Kleinian View of Geometry 2.4. Geometry in Spaces of any Number of Dimensions 2.5. The Search for an Axiomatic Foundation of Geometry 3. Main Roots to Topology 3.1. The Classification of Surfaces 3.2. Complex Function Theory and the Birth of Manifolds 3.3. Fourier Series and Topology of Point Sets 3.4. The Cantor Set 3.5. Construction of the Real Number System 3.6. The Dirichlet Problem and Topology of Planar Domains 4. Creating a Theory of Geometric Topology 4.1. The Alexander Horned Sphere 4.2. The Brouwer Fixed Point Theorem 4.3. Shaping the Notion of Dimension 4.4. Topology of Manifolds 5. Differential Geometry and Topology 5.1. Vector Fields on Manifolds 5.2. CurvatureUNESCO-EOLSS of Curves and Surfaces 5.3. Theorema Egregium 5.4. Surfaces of Constant Curvature and the Uniformization Theorem 5.5. The Gauss-SAMPLEBonnet Theorem and Generalizations CHAPTERS 6. Creating a Theory of Point-set Topology 6.1. The Beginnings to Point-set Topology 6.2.
    [Show full text]
  • II.2 Geometry
    i II.2. Geometry 83 Menninger, K. 1992. Number Words and Number Symbols: of geometry from most other kinds of science is its A Cultural History of Numbers. New York: Dover. (Trans- highly deductive nature. It really seems that by tak- lated by P. Broneer from the revised German edition of ing the simplest of concepts and thinking hard about 1957/58: Zahlwort und Ziffer. Eine Kulturgeschichte der them one can build up an impressive, deductive body Zahl. Göttingen: Vandenhoeck und Ruprecht.) of knowledge about space without having to gather Reid, C. 2006. From Zero to Infinity: What Makes Numbers Interesting. Natick, MA: A. K. Peters. experimental evidence. But can we? Is it really as simple as that? Can we have genuine knowledge of space without ever leaving our II.2 Geometry armchairs? It turns out that we cannot: there are other Jeremy Gray geometries, also based on the concepts of length and angle, that have every claim to be useful, but that dis- 1 Introduction agree with Euclidean geometry. This is an astonishing discovery of the early nineteenth century, but, before it The modern view of geometry was inspired by the novel could be made, a naive understanding of fundamental geometrical theories of hilbert [VI.63] and Einstein in concepts, such as straightness, length, and angle, had the early years of the twentieth century, which built in to be replaced by more precise definitions—a process their turn on other radical reformulations of geometry that took many hundreds of years. Once this had been in the nineteenth century. For thousands of years, the done, first one and then infinitely many new geometries geometrical knowledge of the Greeks, as set out most were discovered.
    [Show full text]
  • A History of Greek Mathematics
    CORNELL UNIVERSITY LBRAaY Cornell University Library QA 22.H43 V.1 A history of Greek mathematics, 3 1924 008 704 219 A HISTORY OF GREEK MATHEMATICS VOLUME I A HISTORY OF GKEEK MATHEMATICS BY SIR THOMAS HEATH K.C.B., K.C.V.O.. F.R.S. Se.D. CAMI). ; HON. D.SC. OXFORD HONORARV FEt.r.OW (FORMFRLV FELLOw) OF TRI>fITY COLI.FHF, CAAIBRIDGE ' . An independent world, Created out of pnre intelligence.' Wordsworth. VOLUME I FROM THALES 'JO EUCIJD OXFORD AT THE CLARENDON PRESS 1921 OXFORD UNIVERSITY PRESS London Edinburgh Glasgow Copenhagen New York Toronto Melbourne Cape Town Bombay Calcutta Madras Shanghai HUMPHREY MILFORD Publisher to the University PREFACE The idea may seem quixotic, but it is nevertheless the author's confident hope that this book will give a fresh interest to the story of Greek mathematics in the eyes both of mathematicians and of classical scholars. For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ah initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring. The interest of the subject for the classical scholar is no doubt of a different kind. Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight. Most people, when they think of the Greek genius, naturally call to mind its master- pieces in literature and art with their notes of beauty, truth, freedom and humanism.
    [Show full text]
  • A Quarta Dimens˜Ao: Da Sua Natureza Espacial Na Geometria Euclidiana
    Revista Brasileira de Ensino de F´ısica, vol. 43, e20210034 (2021) Artigos Gerais www.scielo.br/rbef c b DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0034 Licenc¸aCreative Commons A quarta dimens˜ao:da sua natureza espacial na geometria euclidiana `acomponente tipo tempo de variedades n˜aoeuclidianas The fourth dimension: from its spatial nature in Euclidean geometry to a time-like component of non-Euclidean manifolds Jos´eMaria Filardo Bassalo1,2, Francisco Caruso*3,4 , Vitor Oguri4 1Academia Paraense de Ciˆencias, Bel´em,PA, Brasil. 2Funda¸c˜aoMinerva, Bel´em,PA, Brasil. 3Centro Brasileiro de Pesquisas F´ısicas,Coordena¸c˜aode F´ısicade Altas Energias, 22290-180, Rio de Janeiro, RJ, Brasil. 4Universidade do Estado do Rio de Janeiro, Instituto de F´ısica Armando Dias Tavares, 20550-900, Rio de Janeiro, RJ, Brasil. Recebido em 27 de janeiro de 2021. Revisado em 12 de fevereiro de 2021. Aceito em 13 de fevereiro de 2021. Neste artigo, apresenta-se a evolu¸c˜aodas ideias sobre a quarta dimens˜aoespacial, partindo daquelas que surgem da geometria euclidiana cl´assicae abordando, em seguida, as que resultam do ˆambito das geometrias n˜ao euclidianas, como as de Riemann e Minkowski. Particular aten¸c˜ao´edada ao momento no qual o tempo real passa efetivamente a ser considerado como uma quarta dimens˜ao,conforme introduzido por Einstein. Palavras-chave: Quarta Dimens˜ao,Espa¸co,Geometrias n˜aoEuclidianas, Hist´oriada Geometria, Hist´oriada F´ısica. In this article, the evolution of the ideas about the fourth spatial dimension is presented, starting from those which come out within classical Euclidean geometry and going through those arose in the framework of non- Euclidean geometries, like those of Riemann and Minkowski.
    [Show full text]