The Birth of Hyperbolic Geometry Carl Friederich Gauss
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Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E
BOOK REVIEW Einstein’s Italian Mathematicians: Ricci, Levi-Civita, and the Birth of General Relativity Reviewed by David E. Rowe Einstein’s Italian modern Italy. Nor does the author shy away from topics Mathematicians: like how Ricci developed his absolute differential calculus Ricci, Levi-Civita, and the as a generalization of E. B. Christoffel’s (1829–1900) work Birth of General Relativity on quadratic differential forms or why it served as a key By Judith R. Goodstein tool for Einstein in his efforts to generalize the special theory of relativity in order to incorporate gravitation. In This delightful little book re- like manner, she describes how Levi-Civita was able to sulted from the author’s long- give a clear geometric interpretation of curvature effects standing enchantment with Tul- in Einstein’s theory by appealing to his concept of parallel lio Levi-Civita (1873–1941), his displacement of vectors (see below). For these and other mentor Gregorio Ricci Curbastro topics, Goodstein draws on and cites a great deal of the (1853–1925), and the special AMS, 2018, 211 pp. 211 AMS, 2018, vast secondary literature produced in recent decades by the world that these and other Ital- “Einstein industry,” in particular the ongoing project that ian mathematicians occupied and helped to shape. The has produced the first 15 volumes of The Collected Papers importance of their work for Einstein’s general theory of of Albert Einstein [CPAE 1–15, 1987–2018]. relativity is one of the more celebrated topics in the history Her account proceeds in three parts spread out over of modern mathematical physics; this is told, for example, twelve chapters, the first seven of which cover episodes in [Pais 1982], the standard biography of Einstein. -
Riemann's Contribution to Differential Geometry
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematics 9 (1982) l-18 RIEMANN'S CONTRIBUTION TO DIFFERENTIAL GEOMETRY BY ESTHER PORTNOY UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801 SUMMARIES In order to make a reasonable assessment of the significance of Riemann's role in the history of dif- ferential geometry, not unduly influenced by his rep- utation as a great mathematician, we must examine the contents of his geometric writings and consider the response of other mathematicians in the years immedi- ately following their publication. Pour juger adkquatement le role de Riemann dans le developpement de la geometric differentielle sans etre influence outre mesure par sa reputation de trks grand mathematicien, nous devons &udier le contenu de ses travaux en geometric et prendre en consideration les reactions des autres mathematiciens au tours de trois an&es qui suivirent leur publication. Urn Riemann's Einfluss auf die Entwicklung der Differentialgeometrie richtig einzuschZtzen, ohne sich von seinem Ruf als bedeutender Mathematiker iiberm;issig beeindrucken zu lassen, ist es notwendig den Inhalt seiner geometrischen Schriften und die Haltung zeitgen&sischer Mathematiker unmittelbar nach ihrer Verijffentlichung zu untersuchen. On June 10, 1854, Georg Friedrich Bernhard Riemann read his probationary lecture, "iber die Hypothesen welche der Geometrie zu Grunde liegen," before the Philosophical Faculty at Gdttingen ill. His biographer, Dedekind [1892, 5491, reported that Riemann had worked hard to make the lecture understandable to nonmathematicians in the audience, and that the result was a masterpiece of presentation, in which the ideas were set forth clearly without the aid of analytic techniques. -
08. Non-Euclidean Geometry 1
Topics: 08. Non-Euclidean Geometry 1. Euclidean Geometry 2. Non-Euclidean Geometry 1. Euclidean Geometry • The Elements. ~300 B.C. ~100 A.D. Earliest existing copy 1570 A.D. 1956 First English translation Dover Edition • 13 books of propositions, based on 5 postulates. 1 Euclid's 5 Postulates 1. A straight line can be drawn from any point to any point. • • A B 2. A finite straight line can be produced continuously in a straight line. • • A B 3. A circle may be described with any center and distance. • 4. All right angles are equal to one another. 2 5. If a straight line falling on two straight lines makes the interior angles on the same side together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles are together less than two right angles. � � • � + � < 180∘ • Euclid's Accomplishment: showed that all geometric claims then known follow from these 5 postulates. • Is 5th Postulate necessary? (1st cent.-19th cent.) • Basic strategy: Attempt to show that replacing 5th Postulate with alternative leads to contradiction. 3 • Equivalent to 5th Postulate (Playfair 1795): 5'. Through a given point, exactly one line can be drawn parallel to a given John Playfair (1748-1819) line (that does not contain the point). • Parallel straight lines are straight lines which, being in the same plane and being • Only two logically possible alternatives: produced indefinitely in either direction, do not meet one 5none. Through a given point, no lines can another in either direction. (The Elements: Book I, Def. -
Science and Fascism
Science and Fascism Scientific Research Under a Totalitarian Regime Michele Benzi Department of Mathematics and Computer Science Emory University Outline 1. Timeline 2. The ascent of Italian mathematics (1860-1920) 3. The Italian Jewish community 4. The other sciences (mostly Physics) 5. Enter Mussolini 6. The Oath 7. The Godfathers of Italian science in the Thirties 8. Day of infamy 9. Fascist rethoric in science: some samples 10. The effect of Nazism on German science 11. The aftermath: amnesty or amnesia? 12. Concluding remarks Timeline • 1861 Italy achieves independence and is unified under the Savoy monarchy. Venice joins the new Kingdom in 1866, Rome in 1870. • 1863 The Politecnico di Milano is founded by a mathe- matician, Francesco Brioschi. • 1871 The capital is moved from Florence to Rome. • 1880s Colonial period begins (Somalia, Eritrea, Lybia and Dodecanese). • 1908 IV International Congress of Mathematicians held in Rome, presided by Vito Volterra. Timeline (cont.) • 1913 Emigration reaches highest point (more than 872,000 leave Italy). About 75% of the Italian popu- lation is illiterate and employed in agriculture. • 1914 Benito Mussolini is expelled from Socialist Party. • 1915 May: Italy enters WWI on the side of the Entente against the Central Powers. More than 650,000 Italian soldiers are killed (1915-1918). Economy is devastated, peace treaty disappointing. • 1921 January: Italian Communist Party founded in Livorno by Antonio Gramsci and other former Socialists. November: National Fascist Party founded in Rome by Mussolini. Strikes and social unrest lead to political in- stability. Timeline (cont.) • 1922 October: March on Rome. Mussolini named Prime Minister by the King. -
Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing. -
Considerations Based on His Correspondence Rossana Tazzioli
New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence Rossana Tazzioli To cite this version: Rossana Tazzioli. New Perspectives on Beltrami’s Life and Work - Considerations Based on his Correspondence. Mathematicians in Bologna, 1861-1960, 2012, 10.1007/978-3-0348-0227-7_21. hal- 01436965 HAL Id: hal-01436965 https://hal.archives-ouvertes.fr/hal-01436965 Submitted on 22 Jan 2017 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. New Perspectives on Beltrami’s Life and Work – Considerations based on his Correspondence Rossana Tazzioli U.F.R. de Math´ematiques, Laboratoire Paul Painlev´eU.M.R. CNRS 8524 Universit´ede Sciences et Technologie Lille 1 e-mail:rossana.tazzioliatuniv–lille1.fr Abstract Eugenio Beltrami was a prominent figure of 19th century Italian mathematics. He was also involved in the social, cultural and political events of his country. This paper aims at throwing fresh light on some aspects of Beltrami’s life and work by using his personal correspondence. Unfortunately, Beltrami’s Archive has never been found, and only letters by Beltrami - or in a few cases some drafts addressed to him - are available. -
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. -
Beltrami's Models of Non-Euclidean Geometry
BELTRAMI'S MODELS OF NON-EUCLIDEAN GEOMETRY NICOLA ARCOZZI Abstract. In two articles published in 1868 and 1869, Eugenio Beltrami pro- vided three models in Euclidean plane (or space) for non-Euclidean geometry. Our main aim here is giving an extensive account of the two articles' content. We will also try to understand how the way Beltrami, especially in the first article, develops his theory depends on a changing attitude with regards to the definition of surface. In the end, an example from contemporary mathe- matics shows how the boundary at infinity of the non-Euclidean plane, which Beltrami made intuitively and mathematically accessible in his models, made non-Euclidean geometry a natural tool in the study of functions defined on the real line (or on the circle). Contents 1. Introduction1 2. Non-Euclidean geometry before Beltrami4 3. The models of Beltrami6 3.1. The \projective" model7 3.2. The \conformal" models 12 3.3. What was Beltrami's interpretation of his own work? 18 4. From the boundary to the interior: an example from signal processing 21 References 24 1. Introduction In two articles published in 1868 and 1869, Eugenio Beltrami, at the time pro- fessor at the University of Bologna, produced various models of the hyperbolic ver- sion non-Euclidean geometry, the one thought in solitude by Gauss, but developed and written by Lobachevsky and Bolyai. One model is presented in the Saggio di interpretazione della geometria non-euclidea[5][Essay on the interpretation of non- Euclidean geometry], and other two models are developed in Teoria fondamentale degli spazii di curvatura costante [6][Fundamental theory of spaces with constant curvature]. -
Appendix a Differential Forms and Operators on Manifolds
Appendix A Differential Forms and Operators on Manifolds In this appendix, we introduce differential forms on manifolds so that the integration on manifolds can be treated in a more general framework. The operators on manifolds can be also discussed using differential forms. The material in this appendix can be used as a supplement of Chapter 2. A.1 Differential Forms on Manifolds Let M be a k-manifold and u =[u1, ··· ,uk] be a coordinate system on M.A differential form of degree 1 in terms of the coordinates u has the expression k i α = aidu =(du)a, i=1 1 k 1 where du =[du , ··· , du ]anda =[a1, ··· ,ak] ∈ C . The differential form is independent of the chosen coordinates. For instance, assume that v = [v1, ··· ,vk] is another coordinate system and α in terms of the coordinates v has the expression n i α = bidv =(dv)b, i=1 i n 1 ∂v ∂v 1 ··· ∈ where b =[b , ,bk] C .Write = j .By ∂u ∂u i,j=1 T ∂v dv =du , ∂u we have T ∂v dvb =du b, ∂u 340 Appendix A Differential Forms and Operators on Manifolds and by T ∂v a = b, ∂u we have (du)a =(dv)b. From differential forms of degree 1, we inductively define exterior differential forms of higher degree using wedge product, which obeys usual associate law, distributive law, as well as the following laws: dui ∧ dui =0, dui ∧ duj = −duj ∧ dui, a ∧ dui =dui ∧ a = adui, where a ∈ R is a scalar. It is clear that on a k-dimensional manifold, all forms of degree greater than k vanish, and a form of degree k in terms of the coordinates u has the expression 1 k 1 ω = a12···kdu ∧···∧du ,a12···k ∈ C . -
Nonsense Version of This Geometry, Independently Derived
Nonsense version of this geometry, independently derived. Carl SEE ALSO CONTINUUM;EXTENSION;INFINITY;MATHEMATICS, Friedrich Gauss (1777–1855), often called the “princeps PHILOSOPHY OF;MEASUREMENT. mathematicorum” (prince of mathematicians), simulta- neously developed this geometry, calling it “non- BIBLIOGRAPHY Coffa, Alberto. “From Geometry to Tolerance: Sources of Euclidean,” but he kept his results private, fearing that Conventionalism in Nineteenth-century Geometry.” In From the learned public was unprepared for this development. Quarks to Quasars: Philosophical Problems of Modern Physics, This type of geometry is now called hyperbolic geometry. edited by Robert Colodny, 3–70. Pittsburgh, PA: University In 1766, Johann Heinrich Lambert (1728–1777) of Pittsburgh Press, 1986. showed that hyperbolic geometry could be envisioned as Greenberg, Marvin Jay. Euclidean and Non-Euclidean the geometry that is valid on a sphere of imaginary Geometries: Development and History. 4th ed. New York: radius. But being imaginary, this reasoning did not Freeman, 2008. convince Lambert of the consistency of hyperbolic Stillwell, John. Sources of Hyperbolic Geometry. Providence, RI: geometry. Lambert also recognized that the great circles American Mathematical Society, 1996. Includes a critical on a sphere behave like lines without parallels, since introduction and English translations of key articles by Bel- trami, Felix Klein, and Henri Poincaré. every two “great circles” (paths on the sphere dividing it into two equal hemispheres) meet in two points. This idea gave rise to another type of non-Euclidean geometry Andrew Arana Associate Professor of Philosophy and Mathematics called elliptic geometry, which was investigated more University of Illinois at Urbana-Champaign (2013) systematically by Bernhard Riemann (1826–1866). -
Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels
University of Montana ScholarWorks at University of Montana Graduate Student Theses, Dissertations, & Professional Papers Graduate School 2007 Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels Seth Braver The University of Montana Follow this and additional works at: https://scholarworks.umt.edu/etd Let us know how access to this document benefits ou.y Recommended Citation Braver, Seth, "Lobachevski Illuminated: Content, Methods, and Context of the Theory of Parallels" (2007). Graduate Student Theses, Dissertations, & Professional Papers. 631. https://scholarworks.umt.edu/etd/631 This Dissertation is brought to you for free and open access by the Graduate School at ScholarWorks at University of Montana. It has been accepted for inclusion in Graduate Student Theses, Dissertations, & Professional Papers by an authorized administrator of ScholarWorks at University of Montana. For more information, please contact [email protected]. LOBACHEVSKI ILLUMINATED: CONTENT, METHODS, AND CONTEXT OF THE THEORY OF PARALLELS By Seth Philip Braver B.A., San Francisco State University, San Francisco, California, 1999 M.A., University of California at Santa Cruz, Santa Cruz, California, 2001 Dissertation presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematical Sciences The University of Montana Missoula, MT Spring 2007 Approved by: Dr. David A. Strobel, Dean Graduate School Dr. Gregory St. George, Committee Co-Chair Department of Mathematical Sciences Dr. Karel Stroethoff, Committee Co-Chair Department of Mathematical Sciences Dr. James Hirstein Department of Mathematical Sciences Dr. Jed Mihalisin Department of Mathematical Sciences Dr. James Sears Department of Geology Braver, Seth, Ph.D., May 2007 Mathematical Sciences Lobachevski Illuminated: The Content, Methods, and Context of The Theory of Parallels. -
Hyperbolic Geometry
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln MAT Exam Expository Papers Math in the Middle Institute Partnership 7-2007 Hyperbolic Geometry Christina L. Sheets University of Nebrask-Lincoln Follow this and additional works at: https://digitalcommons.unl.edu/mathmidexppap Part of the Science and Mathematics Education Commons Sheets, Christina L., "Hyperbolic Geometry" (2007). MAT Exam Expository Papers. 39. https://digitalcommons.unl.edu/mathmidexppap/39 This Article is brought to you for free and open access by the Math in the Middle Institute Partnership at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in MAT Exam Expository Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. HYPERBOLIC GEOMETRY Christina L. Sheets July 2007 Hyperbolic Geometry p.2 HISTORY In general, when one refers to geometry, he or she is referring to Euclidean geometry. Euclidean geometry is the geometry with which most people are familiar. It is the geometry taught in elementary and secondary school. Euclidean geometry can be attributed to the Greek mathematician Euclid of Alexandria. His work entitled The Elements was the first to systematically discuss geometry. Since approximately 600 B.C., mathematicians have used logical reasoning to deduce mathematical ideas, and Euclid was no exception. In his book, he started by assuming a small set of axioms and definitions, and was able to prove many other theorems. Although many of his results had been stated by earlier Greek mathematicians, Euclid was the first to show how everything fit together to form a deductive and logical system. In mathematics, geometry is generally classified into two types, Euclidean and non-Euclidean.