Indra’s Pearls: The Vision of Felix Klein by David Mumford (PDF)

Description

Felix Klein, a great geometer of the nineteenth century, rediscovered an idea from Hindu mythology in mathematics: the heaven of Indra in which the whole Universe was mirrored in each pearl in a net of pearls. Practically impossible to represent by hand, this idea barely existed outside the imagination, until the 1980s when the authors embarked on the first computer investigation of Klein’s vision. In this extraordinary book they explore the path from some basic mathematical ideas to the simple algorithms that create delicate fractal filigrees, most appearing in print for the first time. Step-by-step instructions for writing computer programs allow beginners to generate the images.

User’s Reviews

Editorial Reviews: Review “[This book is] richly illustrated with these wonderful and mysterious pictures and gives detailed instructions for recreating them, right down to the level of computer programs (written in pseudo-code, and easy to translate into any computer language) … the reader who attempts any substantial subset of [the projects] will gain enormously … Even those who are convinced they have no ability to visualize may change their minds … It is almost required reading for the experts in the field … I truly love this book.” John H. Hubbard, The American Mathematical Monthly”It has been a great pleasure to read such a gracefully written, original book of mathematics … it is a flowing narrative, leavened with wit, whimsy, and lively cartoons by Larry Gonick. The three authors, with the support of Cambridge University Press, have produced a book that is as handsome in physical appearance as its content is stimulating and accessible. The book is an exemplar of its genre and a singular contribution to the contemporary mathematics literature.” Albert Marden, Notices (journal of the American Mathematical Society)”The production of the book leaves nothing to be desired. It is splendid. Printed entirely on glossy paper, with practically all of the many figures in glorious color, the book has a number of admirable design features: large type and wide margins wherein references are given and occasional comments (often quite talky) are made. Cambridge University Press has done a beautiful job, and David Tranah of the Press deserves special commendation for his role in pulling out all the stops.” Philip J. Davis, SIAM News”All of it is patiently explained … By the time you finish, you’ll know your way around the complex plane.” Brian Hayes, American Scientist”The book itself is a work of art … I am sure that [it] will have a major impact on the way we teach geometry and dynamics … a jewel that will more than repay the persistent reader’s efforts.” Michael Field, Science”I rarely feel a certain kind of euphoria by just looking at the cover of a mathematics book. But that happened with Indra’s Pearls: The Vision of Felix Klein … [contains] fantastic illustrations together with apparently well-founded mathematical explanations … [it is] presented in an accessible way which dares to prioritize general comprehension above a strict theoretical approach … As far as I know, this book is one of the most beautiful examples of the illustration of the inherent aesthetic beauty (which exists) within mathematics … the images are of the highest quality obtainable at present for mathematical structures. Everyone, who ever tried to create something comparable, knows how difficult it is.” Jürgen Richter-Gebert, Technische Universität München”This unique book can serve as a pedagogical and visual introduction to group theory for schoolchildren, and yet is just as suitable for professional mathematicians: I believe that both of them would read the book from the beginning to the end. Finally, it can be used as a book for popularising science, but is very different from most fashionable books on strings, black holes, etc: it gives you the joy of seeing, thinking and understanding.” European Mathematical Society”This is a beautifully presented book, rich in mathematical gems.” The Mathematical Gazette”One can browse through the numerous beautiful and fascinating pictures and marvel at them … Readers with widely different backgrounds will find something enjoyable in this unique book.” Acta Scientiarum Mathematicarum Book Description This book tells the story of the first computer exploration of Klein’s vision of infinitely repeated reflections, featuring extraordinary images. About the Author David Mumford has been University Professor in the Division of Applied Mathematics at Brown University since 1996. Before that he was at Harvard University for thirty-five years. He has been a contributor to the Calculus Textbook Project, led by Hughes-Hallett and Gleason. Over his long and distinguished career, Professor Mumford has received many awards and honours, including the Fields Medal in 1974, the Wolf Prize in 1998 and the National Medal of Science in 2010.Caroline Series is Professor of Mathematics at Warwick University where she currently holds an EPSRC Senior Research Fellowship. She was Kennedy Scholar at Harvard University from 1972 to 1974. In addition to technical publications, she has contributed widely to the popularization of mathematics. In 2014, she won the first Senior Anne Bennett Prize of the London Mathematical Society for contributions to mathematics and to the advancement of female mathematicians. She edits the new International Women in Maths website http://www.mathunion.org/wim/.David Wright is Professor at Oklahoma State University. He has held a guest professorship at the University of Göttingen, and was Sloan Fellow at the Institute for Advanced Study in Princeton from 1988 to 1990. From 1997 to 1999 he helped establish the famous William Lowell Putnam Mathematical Competition. In 2009 he received a Distinguished Teaching Award from the Mathematical Association of America. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is my first reading of this book. This is a book you can come back to. My first pass was to look through the beautiful images. I have somewhat superficially read the text. The book is well written. It has multiple levels and components. It is partitioned in a stepwise manner to proceed from Mobius transformations from circles to circles , then consideration iterated pair of Mobius transformation of disjoint circle pairs, then “kissing pairs” then more generalized approaches. The important properties (conformal orientation preserving mappings) are discussed. The classifications are explained and the relation of these properties to the dynamic (iterated) behaviour is discussed. This book fits well with Visual Complex Analysis.I found the later chapters difficult and hope to come back to them in the future when I have the happy confluence of time, mood and more education.Central to the book is the concept of the group and particular subgroups. Fractals, chaos, and other concepts from dynamical systems emerge from this book in a beautiful way. There are also historical vignettes of the key figures in the development of the subject.The pseudocode is well explained in words, then algorithm. I have only just started to play with this. On a second and subsequent readings I hope to work through the projects that the authors provide.This seems to have been a long labour of love for the three authors and their collaborators. My first read was a pleasure and motivator to learn more.

⭐Extremely interesting and lucid discussions of new developments in an area of math, with lovely illustrations. The book is reader friendly with a sense of the people who are developing new and powerful ideas in recursive algorithms. The book can be read at many levels, from the casual reader to the research mathematician. These algorithms are underlying keys to graphics creations in computing. The book is highly enjoyable.

⭐This book is fantastic so far, I am really enjoying it, but the images are tiny and all of the formulas are images, not mathematics fonts. I am literally looking at a two line algebraic formula which is given the same amount of space as the printed word “symbols”. Full of low quality screen shots which have 1/2 the pixels they deserve at best. It’s really difficult to read the rather important parts of this book. Buy the physical version if you can.

⭐This a fantastic book, weel write. The exposition is simple and elegant.

⭐The book was jammed into a box smaller than the book itself, and with heavier items stacked on top of it.Reference the attached photo to see what I mean.I did not pay the price of a brand-new copy to received a damaged one.

⭐I have as contrast several books dealing with closely related material.My own approach using Iterated Function Systems isn’t actually the same asthe Limit set approach, yet sometime the results come out very similar.For comparison the graphics in :

⭐are very minimal and the presentation of the Mathematics makes no effort to be understandableto ordinary people.Another example is:

⭐which has some very awesome mathematics, but again has minimal graphicsand makes very little effort at really teaching the subject in an understandable manner.Don’t get me wrong, the Limit set Klein group approach in Indra’s Pearl’sisn’t easy, but for a very difficult subject the authors really try tomake the subject approachable to most anyone.For me the Klein group approach to geometry has been an eye opener,that put Teichmüller space into my mind alongside Reimannian conformal geometryand the quantum mechanical groups of physics. I think future generations of studentswill bless these authors for this representation.

⭐[this review shall replace the already existing one]Indras pearls provides a very well-made introduction to the basics of the theory of discrete groups acting on the complex plane. The whole discussion on the related limit sets had been accomplished in such a hand-by-hand method.The reader starts from complex numbers and after he is led into the deepest concepts: Möbius trasformations, limit sets of discrete groups (Schottky, Fuchsian, …).These limit sets are related to another interesting topic in today maths: complex dynamics on the Riemann sphere (Julia sets, …).As known, computer experiments had been fundamental for supporting complex dynamics and the successive success of this latter topic helped to promote and increase the interests for discrete groups too: in fact this book evinces already strong interest in the visualization and in the study of the properties of such limit sets since ’80s, due to the efforts of the same authors.One of the major points of attraction in Indra pearls is that all the theory had been helped by displaying a lot of detailed and colorful pictures which, aside the historical biography of the mathematicians that contributed to this theory, set this book as one of the masterpieces in this topic, for his lucidand fresh approach to basic concepts.In addition, the presence of amusing comic-strips, explaining some topological concepts on manifolds (for example), guarantees the easy-learning for the reader and also the approach, as imaginaed and completely accomplished by the authors. In this direction, it is clear how passion had been squandered by authors.The goal has been reached: finding an easy way to introduce the harsch theory of discrete groups.Interested readers will be rewarded and also excited.No doubts: this book strikes and it will be a corner-stone for present and future.

⭐Dieses Buch habe ich auf Wunsch für einen promovierten und habilitierten Mathematiker gekauft, der bei Mathe-Olympiaden schon mehrfach sehr erfolgreich war. Zum Glück für Studierende in NRW konnte er ein Jahrzehnt an einer NRW Universität auf der Basis von Halbjahresverträgen sein Wissen an Mathe-Studierende weitergeben. Stud. haben ihn (auch aus Dankbarkeit) für einen Lehrpreis an einer NRW Uni vorgeschlagen; er wurde auch von der Uni-Leitung für diesen Preis nominiert. Und das sogar in REINER (theoret.) Mathematik! Da er kein Deutscher (aber gut deutsch Sprechender) ist und auch kein Flüchtling, hat ihn das Mathematische Institut nicht eingestellt. Er darf wahrscheinlich ab Ende d.M. als Arbeitsloser sein Leben fristen.Un superbe livre pour découvrir le domaine des groupes de transformations et leur application aux groupes d’homographies. Les images générées par ces groupes sont absolument superbes (l’impression est excellente) et de plus l’auteur explique dans le détail comment générer ces images sur un ordinateur. Des “projets” sont également proposés pour développer certains aspects techniques. Bref un livre très pédagogique et en même temps qui nous fait comprendre un domaine de très haut niveau. J’ai particulièrement apprécié le côté très visuel des explications (nombreux schémas explicatifs et très nombreuses figures), qui fait souvent défaut dans les livres de mathématiques…

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