.

The goal of this article is to show that the MRB constant is geometrically quantifiable. To “measure” the MRB constant, we will consider a set, sequence and alternating series of the nth roots of n. Then we will compare the length of the edges of a special set of hypercubes or n-cubes which have a content of n. (The two words hypercubes and n-cubes will be used synonymously.)

Finally we will look at the value of the MRB constant as a representation of that comparison, of the length of the edges of a special set of hypercubes, in units of dimension 1/ (units of dimension 2 times units of dimension 3 times units of dimension 4 times etc.). For an arbitrary example we will use units of length/ (time*mass* density*…).

A Countable Infinite Set

Consider, r, the set of roots of positive integers of the form r = n^(1/n). Of course the elements of this set are of the form x^(1/y). However what is not obvious is the geometric interpretation of x^(1/y). At least as far as natural number valued (x and y)>0 are concerned, x represents the content of an n-cube and y represents its dimension. That is a geometric interpretation of x^(1/y) as far as natural number valued (x and y)>0 are concerned. For instance we take a cube of any given volume and find the length of one of its sides. Let’s suppose the volume was 8 units^3. What would be the length of one of its sides? We might easily deduce that the length is 2. To confirm this answer we simply construct a cube of 2 linear units in length as in Diagram 1 and find its volume.

Diagram 1

The volume in units³ of the cube in Diagram 1 is indeed 2*2*2 = 8.
Now we look at the previous sentence with x^(1/y) in mind. 8^(1/3) = 2 implies the volume of the cube in Diagram 1 raised to the power of the reciprocal of its dimension equals the length of one of its sides. That is a geometric interpretation of x^(1/y) as far as natural number valued x and y > 0 are concerned.

Open Question

What is the geometric interpretation of x^(1/y) for all real values of x,y > 0?

Sequence and Alternating Series

Now we will consider the sequence of roots of positive integers of the form r = {n^(1/n)} = {1^(1/1), 2^(1/2), 3^(1/3), …}. Then we will add the elements of r in the alternating series

Concerning the partial sums of L, we remember

and we find, S(x) is divergent as x goes to infinity. However S(2x) and S(2x+1) are both convergent as x goes to infinity and the difference between S(2x) and S(2x+1) also converges.

Special Hypercubes

In geometric terms: As in Diagram 2, we give each n-cube a content equal to its dimension¹,

Diagram 2

so that we have a line segment of 1 linear unit, a square of 2 square units, and so forth.

The last cube in the diagram may be just an invention of the imagination because who has ever heard of a hypercube of unbounded dimension with unbounded content? Furthermore, there seems to be a paradox invoked when taking n-cubes as n->∞. While n is still a number the content of the n-cube or hypercube is defined in the specific unit of choice, whether the unit is inches, meters or what not and the resulting length of an individual edge is also defined in the same unit and is computed as shown above. “The [content] of the [n-]cube raised to the power of the reciprocal of its dimension equals the length of one of its [edges].”

However, when we arrived at the hypercube of unbounded dimension where n is no longer a number, the assigned “unbounded content” could be meant to be in units of feet, let’s say; while the resulting length of an individual edge is which could be in feet or any other unit of length because there are the same amount of inches or meters or any other unit of length in infinity feet as there are feet. To avoid the resulting ambiguity we will look at the sequence of roots of positive integers of the form r = {n^(1/n)} = {1^(1/1), 2^(1/2), 3^(1/3), …} as being analogous to the clopen interval [1, ∞).

A Sum of the Series

Above it is mentioned, "Add the elements of r in the alternating series

”

To show that geometrically we do the following: as in Diagram 3, on the y,z-plane line up an edge of each n-cube or hypercube. The numeric values displayed in the diagram are the partial sums of S(x) = S(2u) where u is a positive integer:

Diagram 3

Notice a directed line segment is moved from the origin down the (z or y=0)-axis. Then at the y=1δ axis another one is moved up 2^(1/2) units. Then at the y=2δ axis yet another one is moved down 3^(1/3) units, etc. It does not matter whether δ is one or any other real value; there still are an infinite number of y-valued axes with matching directed line segments.

This is hard to understand; but we may say metaphorically that Diagram 3 is the path along the units of a particle moved 1 inch down in 2^(1/2) seconds, losing 3^(1/3) units of mass with density that increases 4^(1/4) units etc. The resulting position and condition of the particle is represented by

The MRB Constant

As the dimension and the content of a hypercube, both go to infinity we have the following: First, in Diagram 2 the difference between the length of an edge of the hypercube with content 2n+1, and an edge of the hypercube with content 2n, goes to the constant value,

So as n goes to infinity, the length of an edge of the hypercube with content 2n and an edge of the hypercube with content 2n+1 become closer to being the same. Second, in Diagram 3 an edge of each n-cube is arranged on y-valued axes in such a way that

M is the MRB constant².

A numerical approximation of M can be computed by the following summation

,

which sum converges³ (see Diagram 4), while

diverges⁴, as mentioned above.

Diagram 4

One should use acceleration methods when computing a numerical approximation of the MRB constant because it can be shown that one must sum a number in the order of 10^(n+1) iterations of (-1)^n*(n^(1/n)-1) to get n accurate digits of the MRB Constant. However, using a convergence acceleration of alternating series algorithm of Cohen-Villegas-Zagier one can compute the first 60 digits in only 100 iterations⁵.

In Diagram 4a both the lim sup⁶ and the lim inf converge upon the MRB constant, while in 4b only the lim sup converges upon it with the lim inf converging upon MRB constant-1. The MRB constant is Sloane’s On-Line Encyclopedia of Integer Sequences id:A037077⁷. More information, including a brief but documented history, can be found in Wikipedia⁸.

Summary

In retrospect, the geometry used here, particularly in diagram3, is transdimensional and thus we find it hard to understand through the previous experiences of our senses. (To examine its geometry we used edges from hypercubes of many dimensions.) However, considering the various temporal-spatial dimensions that affect our universe as proposed in some theories⁹ is there some significance to the MRB constant in our daily lives? Nevertheless, we have seen that the value of the MRB constant is geometrically quantifiable; it is the lim sup of the sequence that represents a particle traveling along a directed line segment that is moved 1 unit from the origin down the z-axis; at the y=1δ moved up 2^(1/2) units’ at the y=2δ axis moved down 3^(1/3) units etc. Whether the units are theoretical (as in units of length/ (time*mass* density*…)) or proposed temporal-spatial dimensions, the resulting z value of the particle’s position and condition

This article is released under the Creative Commons Attribution-Share Alike 3.0 Unported license.

About the author

Marvin Ray Burns has indulged in math research as a hobby since 1994. Having had only one college course, most of his discoveries were simply learning the basics of math. He has cataloged many of his early investigations at http://math2.org/mmb/search?query=Marvin. One of his ideas has served some purpose in the math world; that is the MRB constant. Since the discovery of the MRB constant, at least one major mathematics software company has found that it was useful to fix problems found while computing the MRB constant. Another major company changed the functionality of its sum function in such a way as to be able to compute the digits of the MRB constant shortly after its discovery. Mr. Burns has submitted a few integer sequences based on his explorations of the MRB constant; see http://www.research.att.com/~njas/sequences/?q=A037077. A siding applicator by profession, he presently takes various undergraduate courses at IUPUI in the hopes of obtaining a degree in pure math.

References

[1] http://www23.wolframalpha.com/input/?i=n-cube
[2] S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 450.
[3] http://mathworld.wolfram.com/MRBConstant.html
[4] http://mathworld.wolfram.com/notebooks/Constants/MRBConstant.nb
[5] http://arxiv.org/abs/0912.3844
[6] http://en.wikipedia.org/wiki/Upper_limit
[7] http://oeis.org/A037077
[8] http://en.wikipedia.org/wiki/MRB_constant
[9] http://arxiv.org/abs/hep-ph/9803466

Jonathan Borwein and Keith Devlin are well-known mathematicians who have a strong appreciation of, and expertise in, experimental mathematics. In this book they provide us with a concise, inviting introduction to the field.

The first chapter tries to succinctly explain what experimental mathematics is and why it’s a fundamental tool for the modern mathematician. The following is their definition:

Experimental mathematics is the use of a computer to run computations—sometimes no more than trial-and-error tests—to look for patterns, to identify particular numbers and sequences, to gather evidence in support of specific mathematical assertions that may themselves arise by computational means, including search. Like contemporary chemists—and before them the alchemists of old—who mix various substances together in a crucible and heat them to a high temperature to see what happens, today’s experimental mathematician puts a hopefully potent mix of numbers, formulas, and algorithms into a computer in the hope that something of interest emerges.

They immediately address some of the possible objections and illustrate how an approach that doesn’t focus on formal proof, but rather on exploration and experimentation, ultimately leads to hypotheses which can then be, in many cases, proved analytically. The authors argue that in this sense, thanks to the aid of advanced computers, mathematics is becoming more and more similar to other natural sciences.

They also make a case for how great mathematicians like Euler, Gauss, and Reimann were doing experimental mathematics well before calculators where available. Their calculations on paper were far more limited than what computers afford us these days, yet they served them well when it came to sharpening and verifying their intuitions.

The rest of the book is a continuous series of examples that show the advantages of this approach in practice. The examples are highly interesting (some of them stunning) and tend to focus on calculus, analysis and analytical number theory.

Each chapter is accompanied by a section called “Explorations”. I found this section to be particularly valuable. Within it you’ll find exercises, and further examples and considerations. The answers/solutions to the actual problems are provided in the second to last chapter, just before the brief epilogue.

Chapter 2 discusses how to calculate an arbitrary digit for irrational numbers like , in certain bases. They illustrate how the so called BBP Formula (Bailey-Borwein-Plouffe formula, co-discovered by Jonathan Borwein’s brother) came to be.

The use of a program which implements the PSQL integer relation algorithm in high-precision, floating-point arithmetic was key to its discovery. The BBP Formula in turn allowed the calculation of the quadrillionth binary digit of back in 2000.

Chapter 3 focuses on identifying numbers, digits patterns, and sequences once you obtain a numeric result through your calculations and experimentation. They introduce the subject with relatively obvious values like the approximations of or , but the chapter quickly escalates to an example where a closed form for a seemingly random sequence needs to be found.

Chapter 4 analyzes the Reimann Zeta function from the eyes of an experimental mathematician, and shows us what kind of insight we can gain from this unique perspective.

In chapter 5 we learn how by numerically evaluating definite integrals, it is sometimes possible to identify the resulting value which will help us to analytically resolve those particular integrals. The examples presented in this chapter originate for the most part from physics and are very challenging if attempted without the aid of experimental methods. To better grasp the kind of integrals discussed in this chapter, here is an example:

The explorations section provides a few more interesting integrals, including some for which a closed form is not known. The authors even include an integral that intentionally stumps Mathematica 6 and Maple 11.

Chapter 6 is dedicated to serendipitous discoveries (“proof by serendipity”) with a few interesting examples of how “luck” met preparation, ultimately enriching the body of mathematical knowledge almost by chance.

In chapter 7 the authors go back to talk about , this time in base 10, to calculate its digits with efficient, fast converging formulas and methods. The chapter wraps up with a discussion about the normality of , which hasn’t been proved of course, but appears to be empirically supported by the statistical analysis of the first trillion digits. In the explorations section there is a nice discussion about the implementation of fast arithmetic through the Karatsuba multiplication, and the subject of Montecarlo simulations (a very inefficient method of calculating , but a great way to show the idea behind Montecarlo simulations).

Chapter 8 has a bold title, “The computer knows more math than you do”. This provocative title is quickly diminished to put it in context though. The authors start by approaching a tough problem posed by Donald Knuth (of TeX and The Art of Computer Programming fame) to the readers of the American Mathematical Monthly:

In an attempt to solve this the authors invite us to go on a journey involving the Lambert W function, the Pochhammer function, and Abel’s limit theorem. The rest of the chapter illustrates another difficult problem whose solution obtained through the aid of Maple has important implications not only for mathematics, but also for quantum field theory and statistical mechanics.

In chapter 9 a few infinite series are calculated in order to show how CAS systems and experimental methodology can still be useful when dealing with problems that involve infinite sequences, series, and products.

Chapter 10 is dedicated to the limits and the dangers of this approach. Several examples showcase how one can be misled into making assumptions, and how to avoid this from happening. The ad hoc example below is correct to over half a billion digits:

After having calculated a few hundred digits, it would be natural to assume that the series converges to a natural number, when in reality it’s an irrational and transcendental number.

In chapter 11, conscious of the selective focus on analysis and analytical number theory throughout the book, Borwein and Devlin introduce other examples such as a topology problem whose proof was reached thanks to a deeper insight gained through computer visualization of a surface, a knot theory problem, the Four Color Theorem, the Robbins Conjecture, the computation of , and so on.

In truth, I feel that such a thin book could have used more examples like the ones in chapter 11, in order to make a stronger case for the applicability of experimental mathematics to areas outside of analysis.

The book is well written and the tone is never heavy, despite the advanced mathematical examples within it. The authors include historical background and anecdotes which makes for a more interesting read and provides a human perspective behind the formulas presented. The (at times) funny illustrations and occasional jokes are definitely a pleasant addition.

This book is relatively tool agnostic; Maple and Mathematica are referenced throughout, and so are a few online tools to identify number sequences and known numeric values. Overall though, the emphasis in on the methodology rather than a particular CAS (Computer Algebra System) or programming language. In fact, with the exception of a snippet of Maple code in one of the explorations in the first chapter, the book describe the examples from a mathematical and algorithmic standpoint. You won’t find source code for the examples illustrated.

The ideal target audience for The Computer as Crucible is graduate students and researchers. A bright, motivated high-school student will get the gist of this book, but a more mature mathematical audience will actually be able to follow the steps within the examples and fully appreciate the insight on how an experimental approach can aid their research.

Despite the numerous examples employed to make their case, the authors start the book by explaining that it is not intended to be comprehensive. It’s meant to be thought provoking and to whet your appetite as to what is now possible in mathematical research thanks to computers.

As a computer programmer who’s passionate about mathematics, experimental mathematics fascinates me greatly. As such, I hope to work my way through the actual textbooks that are generally suggested as a follow up to this book. Namely, I’ve already started reading Mathematics by Experiment: Plausible Reasoning in the 21st Century (Second Edition), which is co-authored by Jonathan Borwein himself. Other textbooks referenced in this introduction are Experimental Mathematics in Action and Experimentation in Mathematics: Computational Paths to Discovery.

In conclusion, The Computer as Crucible is a lovely little book which builds a strong case for experimental mathematics. Any practicing mathematician or serious amateur should consider checking out this introduction to a topic that will no doubt transform mathematics.

Full disclosure: We received this book for free from the publisher, but we’re under no obligation to review or endorse it. We routinely receive a fair number of books from several publishers that never make the cut for an actual review. The links have our Amazon referral id which gives us a tiny percentage if you buy a book. In turn this helps support this site.

The reasonable man adapts himself to the world; the unreasonable one persists in trying to adapt the world to himself. Therefore all progress depends on the unreasonable man. — George Bernard Shaw (attributed)

On November 11, 2002, Grigory Perelman, a Russian mathematician known to his friends as “Grisha”, posted a research paper to the www.arXiv.org preprint server containing, amongst other things, the outline of a proof of the Poincaré Conjecture, a famous conjecture in topology first articulated in 1904 by the great mathematician Henri Poincaré. Dr. Perelman also e-mailed a few selected mathematicians directly, drawing attention to his somewhat curious paper. This rapidly created a stir as the mathematicians realized that he might well have proven the Poincaré Conjecture, an extremely difficult problem that had eluded the talents of many top mathematicians including Poincaré. Perelman went on to post two more papers to arXiv.org elaborating his proof. The Clay Institute, which had offered a prize of $1 million for the proof (or disproof) of the Poincaré Conjecture, funded two teams of mathematicians to verify Perelman’s proof. The National Science Foundation also funded efforts to verify and expand upon the proof. By 2006, the “consensus” in the mathematical community was that Dr. Perelman had proved the Poincaré Conjecture. Dr. Perelman was offered the prestigious Fields Medal, close to the Nobel Prize of mathematics. He became the first mathematician to decline the Fields for reasons that remain somewhat unclear.

Two recent books attempt to tell the story of Grigory Perelman and the Poincaré Conjecture. Masha Gessen’s Perfect Rigor is the first biography of the elusive and enigmatic Perelman. It gives a great deal of information about the world of Soviet mathematics in which Perelman grew up and Perelman’s life to date. The author was unable to interview Perelman who has declined nearly all interviews; he has given an interview to Sylvia Nasar and David Gruber for their New Yorker article “Manifold Destiny“, about which more later. The book suffers from an unremittingly hostile, perhaps jealous, view of the unusual Dr. Perelman, who is variously portrayed as extremely naive, weird, and possibly mentally ill.

Dr. Perelman’s father was an electrical engineer and his mother a mathematics teacher at a Soviet trade school. His mother apparently had a strong interest in mathematics and almost pursued a doctorate before marrying his father. Perelman appears to have been involved in mathematics at an early age and joined a competitive math club. He competed and won a gold medal at the International Math Olympiad in Budapest, Hungary in 1982 at the age of 16. He attended a special math and physics school, Leningrad Secondary School #239, usually identified as “School 239″ in Perfect Rigor. He then became a student at Leningrad State University. In 1987, he became a graduate student at the Leningrad (subsequently the St. Petersburg) branch of the Steklov Mathematical Institute, the mathematics division of the Soviet (now Russian) Academy of Sciences. The mathematician Yuri Burago was his adviser. Perelman defended his dissertation in 1990. He continued to work at the Steklov Institute until 1992, publishing a number of papers in Russian and American mathematical journals.

In the fall of 1992, Perelman came to the United States for a semester at the Courant Institute at New York University and then another semester at the State University of New York Stony Brook in early 1993. At New York University, he met and may have become friends with the mathematician Gang Tian. Perelman and Gang Tian traveled together from NYU to the Institute for Advanced Study at Princeton to listen to mathematics lectures. Then, Perelman became a prestigious Miller Fellow at Berkeley. During this time he proved the Soul Conjecture, a difficult problem in topology. His Miller Fellowship ended in 1995. He received several job offers from a number of top universities. However, he wanted a tenured position. His job offers appear to have been untenured, tenure-track positions. He returned to Russia and the Steklov Institute in 1995 where he was part of the Mathematical Physics group, dropping almost entirely out of sight, publishing nothing. He appears to have spent the next seven years working on the Poincaré conjecture. In 2002, he stunned the mathematical world by posting his proof to the Internet, flouting tradition by declining to submit the proof to a peer reviewed mathematics journal. The Clay Institute would fund mathematicians John Morgan and Gang Tian (Perelman’s friend or acquaintance at NYU) as well as a separate team at the University of Michigan to verify Perelman’s work in the form of a peer reviewed academic book.

In 2006, the prominent mathematician Shing-Tung Yau and two of his former students argued that Perelman had published an incomplete proof which they “fixed” in a lengthy paper published in the Asian Journal of Mathematics. At this point the elusive Dr. Perelman appears to have struck back with a vengeance, possibly exhibiting something other than the naivete imputed in Pefect Rigor. Perelman granted a rare interview to Sylvia Nasar, best known as author of A Beautiful Mind about the mathematician John Forbes Nash, and David Gruber for an article in the New Yorker magazine, “Manifold Destiny,” which all but openly accused Yau and his former students of blatant plagiarism.

The article quotes Perelman attributing his decision to decline the Fields medal and withdraw from the mathematics profession to the low ethical standards of the profession (in his opinion). The article also discusses the alleged rivalry between Yau and his former student Gang Tian, Perelman’s acquaintance from NYU and co-author with John Morgan of the book on Perelman’s proof. Yau threatened legal action against the New Yorker which stood by its story. Yau soon appears to have retreated under a storm of negative publicity and criticism within the mathematics “community”.

By most accounts, Perelman is an unusual person. He left his job at the Steklov Institute and apparently resides with his aging mother in her apartment in St. Petersburg. He has reportedly indicated that he is no longer interested in mathematics and generally refuses interviews, prizes, and so forth. It is not unlikely that many prominent research universities and institutions would fall over themselves to offer him a tenured professorship or something similar if he expressed any interest. It remains to be seen whether he will decline the Clay Institute’s $1 million prize if offered. Without knowing more about Perelman and his adventures in mathematics than can be found in Perfect Rigor or other accounts to date, it is difficult to draw firm conclusions about the man or even his discovery.

Notwithstanding, a few thoughts come to mind. Perfect Rigor and some other accounts implicitly criticize Perelman for his decision to turn down the job offers in 1995 and return to the Steklov Institute, imputing arrogance or just plain nuttiness. Some mathematicians and scientists would kill for some of the offers that Perelman turned down. Most major breakthroughs take a long time, usually five years or more. Perelman spent at least seven years on the Poincaré conjecture and he probably was working on it while in the United States. Most tenure track positions involve a seven year period. The assistant professor is up for review typically in six years; he or she usually must produce allegedly ground breaking work within six years. If he or she is denied tenure, he or she has one year, the seventh year, to find another job. Most assistant professors have acquired a spouse and small children by this time. There is considerable pressure to produce research papers, write grant proposals and raise money. Perelman apparently published nothing from 1995 until 2002. He most likely would not have gotten tenure had he tried to do this at any of the jobs that he turned down in 1995.

There appears to be a long history of mathematicians developing serious psychological problems. The aforementioned John Forbes Nash succumbed to mental illness, diagnosed as paranoid schizophrenia, and was well known to Princeton students for wandering around campus scribbling incomprehensible formulas on blackboards. Kurt Gödel developed psychological problems and allegedly starved himself to death. Georg Cantor became increasingly erratic as he got older. There are many anecdotal accounts of high levels of concentration and mental efforts sustained over months or years resulting in a kind of mental exhaustion and other problems. Both the western and eastern literature of meditation, which often involves prolonged concentration, contain warnings about various adverse psychological effects including anxiety attacks and hallucinations. Disillusioned former adherents of various meditation movements or “cults” have alleged serious adverse effects of heavy meditation, meaning many hours per day every day, similar to those recounted in ancient traditional sources on meditation. Although computer programming can be exhilarating, many programmers appear to experience mental exhaustion and “burnout” after lengthy programming projects involving high levels of sustained concentration.

In engineering there is an adage: “if you are one step ahead, you are a genius; if you are two steps ahead, you are an idiot!” Perfect Rigor portrays Perelman as astonishingly naive, protected from the “real world” by the bizarre Soviet mathematical system. While this may have some truth, a number of Perelman’s actions may exhibit much foresight, like a champion chess player sacrificing a piece for subsequent gain. Is pretending not to notice the alleged anti-Semitism (Perelman is a Russian Jew) in the Soviet mathematical system naive or politically astute? Declining the Fields medal, as some have noted, attracted enormous attention to Perelman. He is now one of the best known recipients (or non-recipients in this case) of the Fields Medal. It also gave him a great deal of moral authority which he seems to have used effectively to fend off Shing-Tung Yau’s alleged attempt to steal credit for proving the Poincaré Conjecture. Refusing to grant interviews also means that Perelman probably has a great deal of leverage with journalists in the rare cases when he grants an interview, as he did with such great effect in The New Yorker in 2006.

Perelman was a math prodigy, returning home with a gold medal and a perfect score from the 1982 International Math Olympiad. Prodigies are often not as successful as one might expect. Math and physics prodigies often flame out, sometimes catastrophically. While prodigies are more common among people who make major inventions and scientific discoveries than in the general population, they are not nearly as common as most people probably think. Perfect Rigor portrays Perelman’s success in proving the Poincaré Conjecture as a logical consequence of his youthful training and competition in the sometimes bizarre Soviet mathematical system. Since Perelman has revealed little about the process of his discovery, this is difficult to evaluate.

Prodigies often run into problems and don’t realize their seeming potential later in life. This has been observed in math, physics, and other fields for many generations. There are probably several causes. Some prodigies are probably frauds, manufactured by ambitious parents; that such people fail to make major breakthroughs is not surprising. Some prodigies are probably the product of a hothouse environment, driven or manipulated by parents or others to practice heavily and perform at an unusually high level that is difficult to sustain. As they get older and establish their own lives, other interests or needs intervene. Some prodigies undoubtedly fall afoul of politics that they are ill-prepared to deal with.

Academic homework, exams, competitions like the International Math Olympiad, admissions exams such as the SAT or GRE exams in the United States, specialized exams and competitions such as the famous Putnam math examinations, and so forth do not necessarily either teach or measure some of the skills required in actual invention or discovery. Exams and homework in math and physics tend to test the ability to accurately and quickly perform certain calculations or apply certain known mathematical methods to a problem. Some people either through heavy practice or rare natural ability can learn to perform these calculations rapidly with negligible error. This does not translate directly into the ability to handle unsolved research problems which often seem to require large amounts of frustrating trial and error and often deeper understanding of concepts, mental visualization, and so forth.

Many topics taught at a high school, college, and even beginning graduate school level are quite mature. Logical and technical flaws that abound in original research papers have been cleaned up and eliminated. Teachers and textbook writers have learned how to present the material clearly so that a bright or highly motivated student may be able to easily master the material quickly. Prodigies can sometimes read a textbook and immediately start performing the methods described in the textbook very accurately. This becomes more difficult as one reaches the “bleeding edge” where the available learning materials are original research papers or badly written textbooks that may contain errors, impenetrable jargon, opaque language, and even deliberate obfuscation of logical or technical flaws. Prodigies may encounter a sudden drop off of their remarkable abilities which they may inaccurately attribute to a lack of the magic “ability” required for the field rather than the immature state of the bleeding edge knowledge. Perelman presumably navigated these difficulties as he progressed in mathematical research.

One is reminded of the old sayings “actions speak louder than words” and “talk is cheap”. If Perelman’s proof stands the test of time, he has done much. If he is sincere in declining prizes, honors, and adulation, he sets an example by his actions. In reading Perelman’s story, one also cannot shake the impression that he may have had some unhappy experiences during his stay in the United States and went home silently vowing “I’ll show them,” which he apparently has.

The Poincare Conjecture

Donal O’Shea’s The Poincare Conjecture is a more pleasant book to read than Perfect Rigor, lacking the hostile tone of Perfect Rigor and sugar coating a number of topics. Perelman is “eccentric”. Little is said about “Manifold Destiny” or the ugly priority dispute. O’Shea focuses on the history of geometry, the Poincaré Conjecture, mostly inspiring stories about great mathematicians, and tries to explain the mathematics of the Poincaré Conjecture to a general audience.

On the whole, The Poincare Conjecture is an enjoyable and informative book to read. O’Shea carefully debunks the myth that scholars in the Middle Ages and the ancient world believed the Earth was flat. He gives an interesting account of Columbus, the slow discovery of the exact shape and geography of the Earth, confirming the ancient theory of the spherical Earth. He slowly and deftly leads the reader through the history of mathematics and geometry to the Poincaré Conjecture, the many failed attempts to prove it, and the seeming final solution by Perelman.

Some of the illustrations leave a bit to be desired. In discussing mathematics in the ancient world, O’Shea uses modern CIA maps of the modern world to show the ancient Greek kingdom of Ionia where Pythagoras was born and to show the Middle East. One map, for example, shows modern Bulgaria which did not exist in the time of Pythagoras. Similarly, O’Shea is discussing ancient Babylonia and Persia but the associated map shows modern Iraq and Iran. Hopefully, this will be fixed in a future edition.

Some of the discussion of hyperbolic geometry and most of the chapter on Poincaré’s topology papers, which presents the actual Poincaré conjecture, could be improved. The diagrams and explanation on page 27 in the chapter “Possible Worlds” showing how the surface of a two-holed torus can be mapped to an octagon is hard to follow. O’Shea returns to the two-holed torus and the octagon in Chapter 10, “Poincaré’s Topological Papers”. Probably many readers will have forgotten the discussion on page 27 by then. The term “natural geometry” is used in this chapter but not defined clearly. A number of diagrams in this chapter are small and difficult to follow. Interested readers can find a better explanation of some of the relevant aspects of hyperbolic geometry in the second chapter of Roger Penrose’sThe Road to Reality which features some entertaining Escher prints showing the so-called “Poincaré disc model” of hyperbolic geometry (first discovered not by Poincaré, but by Eugenio Beltrami, Penrose carefully points out).

One can only go so far with analogies to rubber sheets or cloth fabric in describing topology and especially differential geometry. This is a problem many popular mathematics and science books encounter. If we had a better way of explaining and introducing differential calculus to a general audience, this would improve the general public’s ability to follow issues in mathematics and science and also improve our educational system.

Pure mathematics today suffers from a particularly opaque and confusing language. It now typically takes several months for a skilled person to master the arcane language of modern pure mathematics. Abstraction has been taken to an extreme. Words and phrases such as “algebra”, “ring”, “module”, “field”, and so forth have meanings in pure mathematics that differ both from common usage and the language of applied mathematics used in most engineering and also much physics. The Poincare Conjecture suffers in places from terms like “natural geometry” that have a special meaning in pure mathematics.

Conclusion

Both books focus on the genius of Perelman and famous mathematicians such as Gauss, Riemann, Poincaré, and others. Indeed, the subtitle of Perfect Rigor is “A Genius and the Mathematical Breakthrough of the Century”. This superman theory of scientific progress and a strong focus on extreme intelligence is common in popular science and math books and articles.

The story of the Poincaré Conjecture, at least until Perelman, is a story of large amounts of trial and error (lots of error) as both books allude to. Henri Poincaré formulated the conjecture in 1904 and published an incorrect proof. Almost every year has seen publication or presentation of attempts to prove the Poincaré Conjecture. Numerous mathematicians, including very top mathematicians, have published incorrect proofs. Many different approaches to the problem have been developed. Most failed. Richard Hamilton developed the basic approach that Perelman built upon but apparently stopped making progress in the 1980′s or early 1990′s. It is common to find large amounts of trial and error in the detailed history of inventions and discoveries, including discoveries in pure and applied mathematics.

It is clear that Perelman spent at least seven years on the Poincaré conjecture. We have no idea how much trial and error and how much failure took place during those seven years. Perelman reportedly fixed two minor errors in his first paper in the subsequent two papers posted to www.arXiv.org in 2002 and 2003. Other inventors and discovers have frequently gone through long periods of trial and error and repeated failure before their “breakthrough”. While respecting Perelman’s accomplishments, we should also be interested in the precise process used to reach the answer and avoid attributing it to magical genius alone.

Both Perfect Rigor and The Poincare Conjecture are interesting and informative books for general audiences. Even practicing mathematicians may gain some insights and new information from Perfect Rigor. Yet, Grigory Perelman remains an enigma. A definitive biography remains to be written. The world might learn a lot from more details on how he discovered his proof of the Poincaré Conjecture.

(C) Copyright 2010, John F. McGowan, Ph.D.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

The first narrates the story of Andrew Wiles, who proved Fermat’s last theorem in 1994. It’s a relatively short documentary, coming in at about 45 minutes, but I find it to be both inspiring and a nice aid to better understanding Andrew “as a person”, before thinking of him as a superb mathematician. This documentary is based on Simon Singh’s excellent book
Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem.

The second documentary focuses on the obsessive quest for knowledge shared by Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing. The basic idea behind “Dangerous Knowledge” is that the genius of these outstanding mathematicians and their obsession ultimately lead to their madness and tragic deaths. In truth, I feel that the underlying thread that tries to tie the four stories together is forced.

For example, Alan Turing was persecuted for his homosexuality, and it is believed that this had a significant impact on his eventual suicide. The filmmakers are trying to lead the viewers to come to the certain conclusion that the quest for understanding infinity is what led these mathematicians to insanity, which is entirely unsupported. Nevertheless, if you’re aware of the agenda behind this film, you’ll get a beautiful 1h 29m documentary that is absolutely worth watching. It poses interesting questions about the nature of knowledge, our understanding of nature, and other puzzling dilemmas that encompass mathematics, physics and philosophy.

What other mathematical documentaries are you fond of? If various titles are suggested, we could definitely start a nice must-watch math documentary list here on Math-Blog.com.