DOCSLIB.ORG

Non-Euclidean Geometry H.S.M

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

AMS / MAA SPECTRUM VOL AMS / MAA SPECTRUM VOL 23 23 Throughout most of this book, non-Euclidean geometries in spaces of two or three dimensions are treated as specializations of real projective geometry in terms of a simple set of axioms concerning points, lines, planes, incidence, order and conti- Non-Euclidean Geometry nuity, with no mention of the measurement of distances or angles. This synthetic development is followed by the introduction of homogeneous coordinates, begin- ning with Von Staudt's idea of regarding points as entities that can be added or multiplied. Transformations that preserve incidence are called collineations. They lead in a natural way to isometries or 'congruent transformations.' Following a NON-EUCLIDEAN recommendation by Bertrand Russell, continuity is described in terms of order. Elliptic and hyperbolic geometries are derived from real projective geometry by specializing in elliptic or hyperbolic polarity which transforms points into lines GEOMETRY (in two dimensions), planes (in three dimensions), and vice versa. Sixth Edition An unusual feature of the book is its use of the general linear transformation of coordinates to derive the formulas of elliptic and hyperbolic trigonometry. The area of a triangle is related to the sum of its angles by means of an ingenious idea of Gauss. This treatment can be enjoyed by anyone who is familiar with algebra up to the elements of group theory. The present (sixth) edition clari es some obscurities in the fth, and includes a new section 15.9 on the author's useful concept of inversive distance. H.S.M. Coxeter This book presents a very readable account of the fundamental principles of hyperbolic and elliptic geometries. The approach is by way of projective geometry, the three chapters immediately following a brief historical sketch being devoted to an excellent survey of the foundations of real projective geometry. —L. M. Blumenthal, Mathematical Reviews H.S.M. COXETER AMSMAA / PRESS 4-Color Process 355 pages on 50lb stock • Spine 11/16" • Trim size 6 x 9 10.1090/spec/023 NON-EUCLIDEAN GEOMETRY Copyright, Canada, 1942, 1947, 1957 First Edition 1942 Second Edition 1947 Third Edition 1957 Fourth Edition 1961 Fifth Edition 1965 Sixth Edition 1998 By the Same Author REGULAR POLYTOPES Dover, New York THE REAL PROJECTIVE PLANE Springer-Verlag, New York INTRODUCTION TO GEOMETRY Wiley, New York PROJECTIVE GEOMETRY Springer-Verlag, New York TWELVE GEOMETRIC ESSAYS Southern Illinois University Press ©1998 by The Mathematical Association of America (Incorporated) Library of Congress Number 98-85640 ISBN 978-0-88385-522-5 e-ISBN 978-1-61444-516-6 Printed in the United States of America Current Printing: 10 9 8 7 6 5 4 3 NON-EUCLIDEAN GEOMETRY H. S. M. COXETER, C.C, F.R.S., F.R.S.C. PROFESSOR EMERITUS OF MATHEMATICS UNIVERSITY OF TORONTO SIXTH EDITION THE MATHEMATICAL ASSOCIATION OF AMERICA Washington, D.C. 20036 Brianchon's Theorem and the concurrence of angle-bisectors, (See pages 59 and 200) SPECTRUM SERIES Published by THE MATHEMATICAL ASSOCIATION OF AMERICA Committee on Publications JAMES W. DANIEL, Chair Spectrum Editorial Board ARTHUR T. BENJAMIN, Editor DANIEL ASIMOV KATHLEEN BERVER DIPA CHOUDHURY JEFFREY NUNEMACHER ELLEN MAYCOCK PARKER JENNIFER J. QUINN EDWARD R. SCHEINERMAN HARVEY J. SCHMIDT, JR. SANFORD SEGAL The Spectrum Series of the Mathematical Association of America was so named to reflect its purpose: to publish a broad range of books including biographies, accessible expositions of old or new mathematical ideas, reprints and revisions of excellent out-of-print books, popular works, and other monographs of high interest that will appeal to a broad range of readers, including students and teach- ers of mathematics, mathematical amateurs, and researchers. 777 Mathematical Conversation Starters, by John dePillis All the Math That's Fit to Print, by Keith Devlin Circles: A Mathematical View, by Dan Pedoe Complex Numbers and Geometry, by Liang-shin Hahn Cryptology, by Albrecht Beutelspacher Five Hundred Mathematical Challenges, Edward J. Barbeau, Murray S. Klamkin, and William O. J. Moser From Zero to Infinity, by Constance Reid The Golden Section, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. / Want to Be a Mathematician, by Paul R. Halmos Journey into Geometries, by Marta Sved JULIA: a life in mathematics, by Constance Reid The Lighter Side of Mathematics: Proceedings of the Eugene Sirens Memorial Conference on Recreational Mathematics & Its History, edited by Richard K. Guy and Robert E. Woodrow Lure of the Integers, by Joe Roberts Magic Tricks, Card Shuffling, and Dynamic Computer Memories: The Mathematics of the Perfect Shuffle, by S. Brent Morris The Math Chat Book, by Frank Morgan Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical, by Steven G. Krantz Mathematical Carnival, by Martin Gardner Mathematical Circus, by Martin Gardner Mathematical Cranks, by Underwood Dudley Mathematical Fallacies, Flaws, and Flimflam, by Edward J. Barbeau Mathematical Magic Show, by Martin Gardner Mathematical Reminiscences, by Howard Eves Mathematics: Queen and Servant of Science, by E.T. Bell Memorabilia Mathematica, by Robert Edouard Moritz New Mathematical Diversions, by Martin Gardner Non-Euclidean Geometry, by H. S. M. Coxeter Numerical Methods That Work, by Forman Acton Numerology or What Pythagoras Wrought, by Underwood Dudley Out of the Mouths of Mathematicians, by Rosemary Schmalz Penrose Tiles to Trapdoor Ciphers ... and the Return of Dr. Matrix, by Martin Gardner Polyominoes, by George Martin Power Play, by Edward J. Barbeau The Random Walks of George Polya, by Gerald L. Alexanderson The Search for E.T. Bell, also known as John Taine, by Constance Reid Shaping Space, edited by Marjorie Senechal and George Fleck Student Research Projects in Calculus, by Marcus Cohen, Arthur Knoebel, Edward D. Gaughan, Douglas S. Kurtz, and David Pengelley Symmetry, by Hans Walser. Translated from the original German by Peter Hilton, with the assistance of Jean Pedersen. The Trisectors, by Underwood Dudley Twenty Years Before the Blackboard, by Michael Stueben with Diane Sandford The Words of Mathematics, by Steven Schwartzman MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 800-331-1622 FAX 301-206-9789 www.maa.org PREFACE TO THE SIXTH EDITION I am grateful to the Mathematical Association of America for rescuing my book from oblivion by agreeing to publish this new edition. I particularly appreciate Elaine Pedreira's careful editing and advice. I welcome the opportunity to correct FiG. 7.2A on page 133 and to make other small improvements and especially to add the new §15.9. That section contains, in particular, an 'abso- lute' sequence of circles (FiG. 15.9A) which involves a surprising application of Fibonacci numbers (15.94). For further discus- sion of inversive geometry, the reader may like to look at "The inversive plane and hyperbolic space," Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 28 (1965), pp. 217-242. The subject of §15.7 has been extended by E. B. Vin- berg in his paper, "The volume of polyhedra on a sphere and in Lobachevsky space," American Mathematical Society Translations (2), 148 (1991), pp. 15-27. Vinberg draws attention to W. P. Thurston, "Three-dimensional manifolds, Kleinian groups and hy- perbolic geometry," Bulletin of the American Mathematical Society, 6 (1986), pp. 357-382. When Hamlet exclaims (in Act II, Scene II) "I could be bounded in a nutshell, and count myself a king of infinite space," he is providing a poetic anticipation of Poincaré's inversive model of the infinite hyperbolic plane, using a circular "nutshell" for the Absolute. In fact, the hyperbolic plane can be filled with infinitely many congruent equilateral triangles, seven at each vertex, to form the regular tessellation {3, 7}. One ac- quires an intuitive feeling for hyperbolic geometry by packing a disc with a multitude of curvilinear triangles, becoming smaller and smaller as they approach the peripheral circle, which rep- ix X PREFACE resents the Absolute. This book's cover design shows the 63 largest of these triangular tiles, systematically coloured so that any two of them which are coloured alike are entirely disjoint, having no common vertex. In other words, every vertex belongs to seven or fewer triangles, all differently coloured. H.S.M.C. June, 1998. PREFACE TO THE THIRD EDITION Apart from a simplification of p. 206, most of the changes are additions. Accordingly, it has seemed best to put the ex- tra material together as a new chapter (XV). This includes a description of the two families of "mid-lines" between two given lines, an elementary derivation of the basic formulae of spherical trigonometry and hyperbolic trigonometry, a computation of the Gaussian curvature of the elliptic and hyperbolic planes, and a proof of Schäfli's remarkable formula for the differential of the volume of a tetrahedron. I gratefully acknowledge the help of L. J. Mordell and Frans Handest in the preparation of §15.6 (on quadratic forms) and §15.8 (on problems of construction), respectively. H.S.M.C. December, 1956. PREFACE xi PREFACE TO THE FIRST EDITION The name non-Euclidean was used by Gauss to describe a system of geometry which differs from Euclid's in its properties of parallelism. Such a system was developed independently by Bolyai in Hungary and Lobatschewsky in Russia, about 120 years ago. Another system, differing more radically from Euclid's, was suggested later by Riemann in Germany and Schläfli in Switzerland. The subject was unified in 1871 by Klein, who gave the names parabolic, hyperbolic, and elliptic to the respective systems of Euclid, Bolyai-Lobatschewsky, and Riemann-Shläfli. Since then, a vast literature has accumulated, and it is with some diffidence that I venture to add a fresh exposition. After a historical introductory chapter (which can be omitted without impairing the main development), I devote three chap- ters to a survey of real projective geometry.
Recommended publications
  • Chapter 13 the Geometry of Circles

    Chapter 13 the Geometry of Circles

    Chapter 13 TheR. Connelly geometry of circles Math 452, Spring 2002 Math 4520, Fall 2017 CLASSICAL GEOMETRIES So far we have been studying lines and conics in the Euclidean plane. What about circles, 14. The geometry of circles one of the basic objects of study in Euclidean geometry? One approach is to use the complex numbersSo far .we Recall have thatbeen the studying projectivities lines and of conics the projective in the Euclidean plane overplane., whichWhat weabout call 2, circles, Cone of the basic objects of study in Euclidean geometry? One approachC is to use CP are given by 3 by 3 matrices, and these projectivities restricted to a complex projective the complex numbers 1C. Recall that the projectivities of the projective plane over C, line,v.rhich which we wecall call Cp2, a CPare, given are the by Moebius3 by 3 matrices, functions, and whichthese projectivities themselves correspondrestricted to to a 2-by-2 matrices.complex Theprojective Moebius line, functions which we preservecall a Cpl the, are cross the Moebius ratio. This functions, is where which circles themselves come in. correspond to a 2 by 2 matrix. The Moebius functions preserve the cross ratio. This is 13.1where circles The come cross in. ratio for the complex field We14.1 look The for anothercross ratio geometric for the interpretation complex field of the cross ratio for the complex field, or better 1 yet forWeCP look= Cfor[f1g another. Recall geometric the polarinterpretation decomposition of the cross of a complexratio for numberthe complexz = refield,iθ, where r =orjz betterj is the yet magnitude for Cpl = ofCz U, and{ 00 } .Recallθ is the the angle polar that decomposition the line through of a complex 0 and znumbermakes with thez real -rei9, axis.
  • Diameter, Width and Thickness in the Hyperbolic Plane

    Diameter, Width and Thickness in the Hyperbolic Plane

    DIAMETER, WIDTH AND THICKNESS IN THE HYPERBOLIC PLANE AKOS´ G.HORVATH´ Abstract. This paper contains a new concept to measure the width and thickness of a convex body in the hyperbolic plane. We compare the known concepts with the new one and prove some results on bodies of constant width, constant diameter and given thickness. 1. Introduction In hyperbolic geometry there are several concepts to measure the breadth of a convex set. In the first section of this paper we introduce a new idea and compare it with four known ones. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. In Euclidean space more or less these classes are agree but in the hyperbolic plane we need to differentiate them. Among others we find convex compact bodies of constant width which size essential in the fulfilment of this property (see Statement 7) and others where the size dependence is non-essential (see Statementst:circle). We prove that the property of constant diameter follows to the fulfilment of constant shadow property (see Theorem 2), and both of them are stronger as the property of constant width (see Theorem 1). In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness. 1.1. The hyperbolic concepts of width (breadth) introduced earlier. 1.1.1. width ( ): Santal´oin his paper [14] developed the following approach.
  • Squaring the Circle in Elliptic Geometry

    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].
  • Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence

    Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence

    Elliptic Geometry Hawraa Abbas Almurieb Spherical Geometry Axioms of Incidence • Ax1. For every pair of antipodal point P and P’ and for every pair of antipodal point Q and Q’ such that P≠Q and P’≠Q’, there exists a unique circle incident with both pairs of points. • Ax2. For every great circle c, there exist at least two distinct pairs of antipodal points incident with c. • Ax3. There exist three distinct pairs of antipodal points with the property that no great circle is incident with all three of them. Betweenness Axioms Betweenness fails on circles What is the relation among points? • (A,B/C,D)= points A and B separates C and D 2. Axioms of Separation • Ax4. If (A,B/C,D), then points A, B, C, and D are collinear and distinct. In other words, non-collinear points cannot separate one another. • Ax5. If (A,B/C,D), then (B,A/C,D) and (C,D/A,B) • Ax6. If (A,B/C,D), then not (A,C/B,D) • Ax7. If points A, B, C, and D are collinear and distinct then (A,B/C,D) , (A,C/B,D) or (A,D/B,C). • Ax8. If points A, B, and C are collinear and distinct then there exists a point D such that(A,B/C,D) • Ax9. For any five distinct collinear points, A, B, C, D, and E, if (A,B/E,D), then either (A,B/C,D) or (A,B/C,E) Definition • Let l and m be any two lines and let O be a point not on either of them.
  • David Hilbert's Contributions to Logical Theory

    David Hilbert's Contributions to Logical Theory

    David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques.
  • + 2R - P = I + 2R - P + 2, Where P, I, P Are the Invariants P and Those of Zeuthen-Segre for F

    + 2R - P = I + 2R - P + 2, Where P, I, P Are the Invariants P and Those of Zeuthen-Segre for F

    232 THE FOURTH DIMENSION. [Feb., N - 4p - (m - 2) + 2r - p = I + 2r - p + 2, where p, I, p are the invariants p and those of Zeuthen-Segre for F. If I is the same invariant for F, since I — p = I — p, we have I = 2V — 4p — m = 2s — 4p — m. Hence 2s is equal to the "equivalence" in nodes of the point (a, b, c) in the evaluation of the invariant I for F by means of the pencil Cy. This property can be shown directly for the following cases: 1°. F has only ordinary nodes. 2°. In the vicinity of any of these nodes there lie other nodes, or ordinary infinitesimal multiple curves. In these cases it is easy to show that in the vicinity of the nodes all the numbers such as h are equal to 2. It would be interesting to know if such is always the case, but the preceding investigation shows that for the applications this does not matter. UNIVERSITY OF KANSAS, October 17, 1914. THE FOURTH DIMENSION. Geometry of Four Dimensions. By HENRY PARKER MANNING, Ph.D. New York, The Macmillan Company, 1914. 8vo. 348 pp. EVERY professional mathematician must hold himself at all times in readiness to answer certain standard questions of perennial interest to the layman. "What is, or what are quaternions ?" "What are least squares?" but especially, "Well, have you discovered the fourth dimension yet?" This last query is the most difficult of the three, suggesting forcibly the sophists' riddle "Have you ceased beating your mother?" The fact is that there is no common locus standi for questioner and questioned.
  • The Dual Theorem Relative to the Simson's Line

    The Dual Theorem Relative to the Simson's Line

    THE DUAL THEOREM RELATIVE TO THE SIMSON’S LINE Prof. Ion Pătrașcu Frații Buzești College, Craiova, Romania Translated by Prof. Florentin Smarandache University of New Mexico, Gallup, NM 87301, USA Abstract In this article we elementarily prove some theorems on the poles and polars theory, we present the transformation using duality and we apply this transformation to obtain the dual theorem relative to the Samson’s line. I. POLE AND POLAR IN RAPPORT TO A CIRCLE Definition 1. Considering the circle COR(,), the point P in its plane PO≠ and uuur uuuur the point P ' such that OP⋅= OP' R2 . It is said about the perpendicular p constructed in the point P ' on the line OP that it is the point’s P polar, and about the point P that it is the line’s p pole. Observations M 1. If the point P belongs to the circle, its polar is the tangent in P to the circle U COR(,). uuur uuuur P' Indeed, the relation OP⋅= OP' R2 gives that PP' = . OP 2. If P is interior to the circle, its polar p is a V line exterior to the circle. 3. If P and Q are two points such that p m(POQ )= 90o , and p,q are their polars, from the definition results that p ⊥ q . Fig. 1 Proposition 1. If the point P is external to the circle COR( , ) , its polar is determined by the contact points with the circle of the tangents constructed from P to the circle Proof Let U and V be the contact points of the tangents constructed from P to the circle COR( , ) (see fig.1).
  • Understand the Principles and Properties of Axiomatic (Synthetic

    Understand the Principles and Properties of Axiomatic (Synthetic

    Michael Bonomi Understand the principles and properties of axiomatic (synthetic) geometries (0016) Euclidean Geometry: To understand this part of the CST I decided to start off with the geometry we know the most and that is Euclidean: − Euclidean geometry is a geometry that is based on axioms and postulates − Axioms are accepted assumptions without proofs − In Euclidean geometry there are 5 axioms which the rest of geometry is based on − Everybody had no problems with them except for the 5 axiom the parallel postulate − This axiom was that there is only one unique line through a point that is parallel to another line − Most of the geometry can be proven without the parallel postulate − If you do not assume this postulate, then you can only prove that the angle measurements of right triangle are ≤ 180° Hyperbolic Geometry: − We will look at the Poincare model − This model consists of points on the interior of a circle with a radius of one − The lines consist of arcs and intersect our circle at 90° − Angles are defined by angles between the tangent lines drawn between the curves at the point of intersection − If two lines do not intersect within the circle, then they are parallel − Two points on a line in hyperbolic geometry is a line segment − The angle measure of a triangle in hyperbolic geometry < 180° Projective Geometry: − This is the geometry that deals with projecting images from one plane to another this can be like projecting a shadow − This picture shows the basics of Projective geometry − The geometry does not preserve length
  • ROBERT L. FOOTE CURRICULUM VITA March 2017 Address

    ROBERT L. FOOTE CURRICULUM VITA March 2017 Address

    ROBERT L. FOOTE CURRICULUM VITA March 2017 Address/Telephone/E-Mail Department of Mathematics & Computer Science Wabash College Crawfordsville, Indiana 47933 (765) 361-6429 [email protected] Personal Date of Birth: December 2, 1953 Citizenship: USA Education Ph.D., Mathematics, University of Michigan, April 1983 Dissertation: Curvature Estimates for Monge-Amp`ere Foliations Thesis Advisor: Daniel M. Burns Jr. M.A., Mathematics, University of Michigan, April 1978 B.A., Mathematics, Kalamazoo College, June 1976 Magna cum Laude with Honors in Mathematics, Phi Beta Kappa, Heyl Science Scholarship Employment 1989–present, Wabash College Department Chair, 1997–2001, 2009–2012 Full Professor since 2004 Associate Professor, 1993–2004 Assistant Professor, 1991–1993 Byron K. Trippet Assistant Professor, 1989–1991 1983–1989, Texas Tech University, Assistant Professor (granted tenure) 1983, Kalamazoo College, Visiting Instructor 1976–1982, University of Michigan, Graduate Student Teaching Assistant 1976, 1977, The Upjohn Company, Mathematical Analyst Research Visits 2009, Korea Institute for Advanced Study (KIAS), Visiting Scholar Three weeks at the invitation of C. K. Han. 2009, Pennsylvania State Univ., Shapiro Visitor Four weeks at the invitation of Sergei Tabachnikov. 2008–2009, Univ. of Georgia, Visiting Scholar (sabbatical leave) 1996–1997, 2003–2004, Univ. of Illinois at Urbana Champaign, Visiting Scholar (sabbatical leave) 1991, Pohang Institute of Science and Technology, Pohang, Korea Three months at the invitation of C. K. Han. 1990, Texas Tech University Ten weeks at the invitation of Lance D. Drager. Current Fields of Interest Primary: Differential Geometry, Integral Geometry Professional Affiliations American Mathematical Society, Mathematical Association of America. Teaching Experience Graduate courses Differentiable manifolds, real analysis, complex analysis.
  • Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg

    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg

    Old and New Results in the Foundations of Elementary Plane Euclidean and Non-Euclidean Geometries Marvin Jay Greenberg By “elementary” plane geometry I mean the geometry of lines and circles—straight- edge and compass constructions—in both Euclidean and non-Euclidean planes. An axiomatic description of it is in Sections 1.1, 1.2, and 1.6. This survey highlights some foundational history and some interesting recent discoveries that deserve to be better known, such as the hierarchies of axiom systems, Aristotle’s axiom as a “missing link,” Bolyai’s discovery—proved and generalized by William Jagy—of the relationship of “circle-squaring” in a hyperbolic plane to Fermat primes, the undecidability, incom- pleteness, and consistency of elementary Euclidean geometry, and much more. A main theme is what Hilbert called “the purity of methods of proof,” exemplified in his and his early twentieth century successors’ works on foundations of geometry. 1. AXIOMATIC DEVELOPMENT 1.0. Viewpoint. Euclid’s Elements was the first axiomatic presentation of mathemat- ics, based on his five postulates plus his “common notions.” It wasn’t until the end of the nineteenth century that rigorous revisions of Euclid’s axiomatics were presented, filling in the many gaps in his definitions and proofs. The revision with the great- est influence was that by David Hilbert starting in 1899, which will be discussed below. Hilbert not only made Euclid’s geometry rigorous, he investigated the min- imal assumptions needed to prove Euclid’s results, he showed the independence of some of his own axioms from the others, he presented unusual models to show certain statements unprovable from others, and in subsequent editions he explored in his ap- pendices many other interesting topics, including his foundation for plane hyperbolic geometry without bringing in real numbers.
  • Hyperbolic Geometry

    Hyperbolic Geometry

    Flavors of Geometry MSRI Publications Volume 31,1997 Hyperbolic Geometry JAMES W. CANNON, WILLIAM J. FLOYD, RICHARD KENYON, AND WALTER R. PARRY Contents 1. Introduction 59 2. The Origins of Hyperbolic Geometry 60 3. Why Call it Hyperbolic Geometry? 63 4. Understanding the One-Dimensional Case 65 5. Generalizing to Higher Dimensions 67 6. Rudiments of Riemannian Geometry 68 7. Five Models of Hyperbolic Space 69 8. Stereographic Projection 72 9. Geodesics 77 10. Isometries and Distances in the Hyperboloid Model 80 11. The Space at Infinity 84 12. The Geometric Classification of Isometries 84 13. Curious Facts about Hyperbolic Space 86 14. The Sixth Model 95 15. Why Study Hyperbolic Geometry? 98 16. When Does a Manifold Have a Hyperbolic Structure? 103 17. How to Get Analytic Coordinates at Infinity? 106 References 108 Index 110 1. Introduction Hyperbolic geometry was created in the first half of the nineteenth century in the midst of attempts to understand Euclid’s axiomatic basis for geometry. It is one type of non-Euclidean geometry, that is, a geometry that discards one of Euclid’s axioms. Einstein and Minkowski found in non-Euclidean geometry a This work was supported in part by The Geometry Center, University of Minnesota, an STC funded by NSF, DOE, and Minnesota Technology, Inc., by the Mathematical Sciences Research Institute, and by NSF research grants. 59 60 J. W. CANNON, W. J. FLOYD, R. KENYON, AND W. R. PARRY geometric basis for the understanding of physical time and space. In the early part of the twentieth century every serious student of mathematics and physics studied non-Euclidean geometry.
  • Foundations of Geometry

    Foundations of Geometry

    California State University, San Bernardino CSUSB ScholarWorks Theses Digitization Project John M. Pfau Library 2008 Foundations of geometry Lawrence Michael Clarke Follow this and additional works at: https://scholarworks.lib.csusb.edu/etd-project Part of the Geometry and Topology Commons Recommended Citation Clarke, Lawrence Michael, "Foundations of geometry" (2008). Theses Digitization Project. 3419. https://scholarworks.lib.csusb.edu/etd-project/3419 This Thesis is brought to you for free and open access by the John M. Pfau Library at CSUSB ScholarWorks. It has been accepted for inclusion in Theses Digitization Project by an authorized administrator of CSUSB ScholarWorks. For more information, please contact [email protected]. Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino In Partial Fulfillment of the Requirements for the Degree Master of Arts in Mathematics by Lawrence Michael Clarke March 2008 Foundations of Geometry A Thesis Presented to the Faculty of California State University, San Bernardino by Lawrence Michael Clarke March 2008 Approved by: 3)?/08 Murran, Committee Chair Date _ ommi^yee Member Susan Addington, Committee Member 1 Peter Williams, Chair, Department of Mathematics Department of Mathematics iii Abstract In this paper, a brief introduction to the history, and development, of Euclidean Geometry will be followed by a biographical background of David Hilbert, highlighting significant events in his educational and professional life. In an attempt to add rigor to the presentation of Geometry, Hilbert defined concepts and presented five groups of axioms that were mutually independent yet compatible, including introducing axioms of congruence in order to present displacement.