WWW.DISSERTATION.XLIBX.INFO
FREE ELECTRONIC LIBRARY - Dissertations, online materials
 
<< HOME
CONTACTS



Pages:     | 1 | 2 ||

«People and Events Behind the Fractal Images When asked to write this article I, without space limitation, have unleashed a flood of recollections ...»

-- [ Page 3 ] --

Please come to my office, and I shall show you.” David came over and saw the algorithm that was eventually published as M 1983i, as told earlier. The exact words of our conversation are, of course, lost, but I should remember their substance as follows: “This is so simple, that Poincaré should have seen it, or someone else singe Poincaré. Why did this discovery have to wait for you?” – “Because no one before me has used a powerful new tool, the computer!” – “But one cannot prove anything with a computer!” – “Sure, but playing with the computer is a source of conjectures, often most unexpected ones. The conjecture it has suggested about Kleinian limit sets has been easy to prove; other are too hard for me.” – “In that case, would you help me learn to play with the computer?” – “With pleasure, but we would have to get help from my latest IBM assistant, Mark laff.” Soon afterwards, it became clear that Mumford had to seek associates closer by, in Cambridge. He was tutored by my course assistant Peter Moldave, and started working with David Wright, then a Harvard graduate student in mathematics, who ceased, at that point, to hide his exceptional programming skills. Eventually, Mumford became thoroughly immersed in computer, first as heuristic tools, then for their own sake.

He became instrumental in helping the awareness of the computer-as-tool idea that spread among mathematicians. The resulting needs grew so rapidly that, after barely eight years, the Figure National Science Foundation had established a Geometry Supercomputer Project! The charter members are F. Almgren (Princeton), J. Cannon (Brigham Young), D. Dobkin (Princeton), A. Douady (ENS, Paris), D. Epstein (Warwick), J. Hubbard (Cornell), B. Mandelbrot (IBM & Yale), A. Marden (Minnesota), J. Milnor (IAS, Princeton), D. Mumford (Harvard), R. Tarjan (Princeton & Bell Labs), and W. Thurston (Princeton). At the risk of sounding corny, let me confess that the opening of this project was a high point in my life. In 1991 it expanded, and its name changed to Geometry Center.

The next topic to be discussed concerning iteration is my fruitful interaction with V. Alan Norton, a Princeton mathematics Ph.D. in 1976, who was in my group as a postdoc in 1980-82, and stayed on the research staff at IBM Yorktown. He was one of the two principal “illustrators” of Figure The Fractal Geometry of Nature, as seen in that book’s very detailed picture credits. He has achieved great renown, starting with Siggraph 1982, for splendid quaternionic Julia set pictures.

Norton also worked on the end-papers, without legend, for The Fractal Geometry of Nature Figure. How these end-papers came about is a tale worth recounting. They involve an important problem from the theory of iteration of analytic functions, an artifact due to inherent limitations of the computer, and two decorative touches.

The original graph was unbounded, and Norton introduced a decorative touch:

inversion with respect to a circle. I loved the result; unfortunately, while bounded, it did not fit neatly on a double page spread. Hence I imposed a second and more arbitrary decorative touch: the horizontal stretching of the graph to fill the available space.

The serious mathematical problem that had led me to this graph was the use of Newton’s method to solve the equation exp(z0) = c. The solutions are known from calculus, but Gaston Julia had shown in 1917 that Newton’s method is a fruitful ground for the study of the iteration of functions of a complex variable z. Chapter 19 of The Fractal Geometry of Nature Figure examines the iteration of z2 + c and other polynomials. This end-paper relates to the iteration of the transcendental function z – 1 + Ce-z.

In Arthur Cayley’s pioneering global studies of iteration in 1879, the interest in iteration had arisen from the application of the Newton-Raphson method. (Peitgen et al.

tell the story, and illustrate it, in The Mathematical Intelligencer in 1984.) Cayley began by solving z2 = C, which proved easy, then went on to try z3 = C, which stumped him by exhibiting three “gray areas: that he found no way of resolving. Julia, in 1917, had found many acts about these areas, and John H. Hubbard had shown us his revealing earliest graph of the corresponding Julia set. It was natural for us, in late 1980, to play with zp = C, and then view exp(z) = C as a suitable limit of zp = C for p→inf. We made many interesting observations about this limit case, but the study was far form complete and publishable when we moved on to very different work.

Finally, and unfortunately, the non-fractal bold boundaries between the background and the solidly colored areas in the end-papers of The Fractal Geometry of Nature Figure are an artifact. The study of transcendental functions’ iterates leads very quickly to enormous integers, hence soon reaches intrinsic limits beyond which the computer takes its own arbitrary actions.

Devaney, Barnsley, and the Bremen Book, The Beauty of Fractals

Out perception of the iteration of transcendental functions as a difficult and very rich topic was confirmed by several eminent mathematicians, such as Robert L. Devaney. No wonder, therefore, that one should see striking resemblances between our end-papers and his beautiful and widely seen illustrations and films. Bob’s papers on the iteration of transcendental functions had already brought admiring attention to him, but we did not become fast friends until we started bumping into each other constantly on the fractal Son et Lumière traveling shows.

The life orbit of Michael barnsley has also crossed mine, and then they stayed in the same fractal neighborhood. The amusing background, in this instance, is in the public record, and I will not repeat it. I first read about it in James Gleick’s book, Chaos: The Birth of a New Science. There I found out how it came to be that Michael burst into my house one day, full of enthusiasm, and lovely tales. Later, we held a few meetings at the Atlanta airport (of all places!), and since then it has been a pleasure to keep up with his work and that of his many associates.

Now back to the pictures of Julia and Mandelbrot sets in Figure The Fractal Geometry of Nature. During the summer of 1984, we were tooling up to redo them in color, with Eriko Hironaka as programmer, when mail brought in, hot off the press, the June issue of the German magazine Geo. We realized immediately that much of what we were proposing had already been achieved, in fact achieved beyond our aspirations, by Heainz-Otto Peitgen, Peter H. Richter, Dietmar Saupe and their associates. Their fractal pictures in The Mathematical Intelligencer earlier in 1984 had been most welcome, but those in color had not yet supplied ample reason for enthusiasm. The color pictures in the 1984 Geo showed a skilled and artistic eye, coupled with a sure hand, one that had gained experience but had not become lazy or hasty. They were unmistakably the outcome of the search for perfection I had admired earlier in the work of Voss, and always attempt in my own.

I wrote to the Geo authors at the University of Bremen to congratulate them, to tell them of the change of plans their success had provoked, and to express the hope of meeting them soon. They told me about a fractal exhibit they were planning, and for which they were preparing a catalogue that eventually led to their book Figure The Beauty of Fractals, and they asked me to write the personal contribution mentioned earlier in this paper. That this book was fated not to come from me or my associates, it is still a delight that it came from them. I gained these new friends when they invited me to Bremen in may 1985, to open the first showing of their exhibit, and I participated in this exhibit in several other cities as well. Our joint appearances have since then have become too numerous to count. There are no anecdotes to tell, only very pleasant events to remember.

Conclusion

I am reminded that, only a while ago (the sting has disappeared, but the memory remains), no one wanted to scan my early pictures for longer than a few minutes, and this would-be leader of a new trend had not a single follower. Then, very slowly in memory, yet almost overnight in present perception, fractals became of so wide interest that Siggraph started devoting lectures, then full day courses to them. The first makeshift appearance of fractals at Siggraph came in 1985 under my direction, the second in 1986 under Peter Oppenheimer, and the third in 1987 and led to The Beauty of Fractals and

Pages:     | 1 | 2 ||


Similar works:

«CENTRE FOR STUDIES IN SOCIAL SCIENCES, CALCUTTA M. Phil. in Social Sciences (Affiliated to Jadavpur University, Kolkata) HANDBOOK 2015-2017 R-1, BAISHNABGHATA PATULI TOWNSHIP KOLKATA : 700 094 Telephone: 2462-7252/5794/5795 FAX: 2462-6183 E-mail: info@cssscal.org cssscal@vsnl.net Website: http://www.cssscal.org ABOUT THE CENTRE The Centre for Studies in Social Sciences, Calcutta (CSSSC), was founded in 1973 by the Government of India and the Government of West Bengal to conduct and coordinate...»

«JURIES: AMERICA'S HISTORIC BARRIER AGAINST OPPRESSIVE LAWS AND GOVERNMENT TYRANNY. By Senator Wayne Stump Editor's Note: Former State Senator Wayne Stump of Arizona and a former member of the NRA board of directors. His article covers jury nullification of unjust laws. Alcohol prohibition failed and the politicians turned away from it because it was very difficult to get a conviction. NRA failure to publish this article was another in a long series of giving in to tyranny. It is another good...»

«Seraphim, p. 1 1/30/2013 Franziska Seraphim Associate Professor of Japanese History Boston College Department of History 158 Clark Street 140 Commonwealth Ave. Newton, MA 02459 Chestnut Hill, MA 02467-3859 Home phone: 617-527-8041 Office phone: 617-552-2142 E-mail: seraphim@bc.edu PROFESSIONAL APPOINTMENTS Boston College 2007Associate Professor of History 2006History Core Moderator 2001-2006 Assistant Professor of History Duke University 2000-2001 Visiting Assistant Professor of History...»

«Not balanced and hardly new: the European Commission’s quest for flexicurity Maarten Keune and Maria Jepsen WP 2007.01 European Trade Union Institute for Research, Education and Health and Safety (ETUI-REHS) Forthcoming in: Henning Jørgensen & Per Kongshøj Madsen (eds.): Flexicurity and Beyond, Copenhagen: DJØF Publishing. ISSN 1994-4446 (print version) ISSN 1994-4454 (online version) D/2007/10.574/20 The ETUI-REHS is financially supported by the European Community. Contents 1....»

«During the past 5 years, the Department of Housing and Urban Development has expanded homeownership, increased access to affordable housing, strengthened communities through economic development, fought housing discrimination, and tackled chronic homelessness. HUD has implemented innovative solutions to address our nation’s housing needs and has achieved great results. Despite its many accomplishments, HUD recognizes that challenges remain to be addressed. Despite achieving the highest...»

«REHABILITATION DU CENTRE ARTISTIQUE DU COLLEGE STOLL D’AKONO REGION CENTRE CAMEROUN DOSSIER DE PRESENTATION Mélanine Mobile Vibe Pascale Quatela : 06 66 66 25 57 Licence d’entrepreneur de spectacles catégorie 2 n° 1046156 Et catégorie 3 n° 1046155 N° SIRET : 423 001 338 000 23 APE : 9001Z 1 SOMMAIRE  Historique de l’établissement  Kayou, Mélanine Mobile Vibe/Gondwana  Nos objectifs  Plan de réhabilitation 2015  Contact  Photos 2 Historique du collège Stoll Le...»

«Mark726’s EVE Lore Survival Guide Version 2.0 Table of Contents Foreword by Seismic Stan An Introduction In the Beginning A Sidenote (Ancient Civilizations) The Empires The Amarr Empire History Social Structure Roleplaying Tips The Minmatar Republic History Social Structure Roleplaying Tips The Gallente Federation History Social Structure Roleplaying Tips The Caldari State History Social Structure Roleplaying Tips CONCORD Frontier Organizations Ammatar Mandate Angel Cartel Blood Raider...»

«Einführungsseminar: The Power and the Glory – British Catholic Novelists Frau Dr. Julia Hoydis Universität zu Köln WS 2010/2011 Datum der Abgabe: 15.3.2011 Moral Decline and the Bankruptcy of Victorian Humanism in Evelyn Waugh’s A Handful of Dust Malte Biedermann XXX XXX LA Englisch u. Geschichte Fachsemester: 3 Table of Contents 1. Introduction 2 2. Two Types of Barbarism 4 2.1. The Decay of British Society 4 2.2. The Proto-Industrialist Savage 7 3. The Plight of Gothic Man 8 4....»

«IGLESIAS DEL SIGLO XII AL XIII CON COLUMNAS GEMELAS EN SUS PILARES* Hay períodos en la historia de la arquitectura en que las formas, en constante fluir, cambian y se modifican aparentemente de manera sencilla, con trabazón y continuidad perfectas. Tal es, por :ejemplo, o, por lo menos, así lo interpretan actualmente la mayoría de los :historiadores del arte y de la arquitectura, el que conduce desde las grandes iglesias.góticas francesas de fines. del siglo XII hasta las del XIV. Durante...»

«The Path to Statehood A lesson plan provided by the office of the Indiana Secretary of State IndianaCivics.com The Path to Statehood In this lesson, students will be exploring four primary documents as well as secondary sources that led to Indiana becoming a state on December 11, 1816. Students will also be exposed to several key historical figures that played a role in the path to statehood. Once the sources have been reviewed, students will be asked to synthesize the information into a...»

«WISDOM, HUMOR, AND FAITH: A HISTORICAL VIEW Walter G. Moss Table of Contents (with links)  Wisdom, Perspective, and Values Humor’s Contribution to Wisdom Humor and Wisdom in Europe: Some Highlights Renaissance Humor: Erasmus, Rabelais, Cervantes, Shakespeare Two European Russians: Anton Chekhov and Vladimir Soloviev Reflections on Humor from Nietzsche to the Theatre of the Absurd. 12  Humor and Wisdom in the United States: Lincoln, Beecher, Twain, Sandburg, and Buchwald Humor, Wisdom,...»

«DEPARTMENT OF HISTORY STATEMENT OF ASSESSMENT FOR UNDERGRADUATE STUDENTS Introduction 1. Formative Assessment 2. Summative Assessment 2.1 Forms of Assessment 2.2 Elective Modules 2.3 Study Abroad 2.4 Visiting Students 3. Mark Scales and Assessment Procedure 3.1 Mark Scale 3.2 Grade Descriptors 3.3 Marking Procedures 3.4 Penalties 3.5 Special Arrangements 3.6 Missed Exams and Other Assessments 3.7 Rubrics and Failure to Follow Rubrics 3.8 Academic Misconduct 3.9 Feedback (see also Department’s...»





 
<<  HOME   |    CONTACTS
2016 www.dissertation.xlibx.info - Dissertations, online materials

Materials of this site are available for review, all rights belong to their respective owners.
If you do not agree with the fact that your material is placed on this site, please, email us, we will within 1-2 business days delete him.