Ecker, Michael W

What Is the Best Way to Lace Your Shoes?, Burkard Polster. Nature 420:6915 (5 December 2002) 476.

This article reports on one of the most common of human activities: tying one’s shoes. For a given configuration of 2n eyelets arranged in two symmetric vertical columns, the author’s goal was to minimize the length of lace and maximize the horizontal tension over all possible lacings such that at least one edge at each eyelet leads to an eyelet in the other column. If both edges at an eyelet end in the other column, the lacing is called “dense.” he gives (without proof) formulas for the possible number of n-lacings, for the possible number of dense n-lacings and he identifies the shortest n-lacing. For dense lacings, this is the traditional criss-cross pattern commonly found on shoes. The strongest lacings are the dense ones, and their strength depends on the ratio h of the vertical distance between adjacent rows of eyelets to the horizontal distance between the two columns. For ratios below a critical value, the traditional criss-cross lacing is strongest. If the ratio is above the critical value, a zig-zag pattern from the lowest eyelet on one side up to the highest on the other, followed by a long edge from that upper one back to the lowest one, is strongest. The illustrations accompanying the article show clearly what’s being discussed, and the formulas could provide interesting exercises. This is a rare application of elementary mathematics and mechanics to an everyday problem most of us overlook. [See also Kenneth Chang’s summary in the New York Times (December 10, 2002) D3: “Seeking Perfection in Shoe Lacing, With 43,200 Choices.”] NS

John W. Tukey: His Life and Professional Contributions, David R. Brillinger. Annals of Statistics 30:6 (December2002) 1535-1575.

John W. Tukey’s Contribution to Multiple Comparisons, Yoav Benjamini and Henry Braun. Annals of Statistics 30:6 (December 2002) 1576-1594.

John W. Tukey’s Work on Time Series and Spectrum Analysis, David R. Brillinger. Annals of Statistics 30:6 (December 2002) 1595-1618

John W. Tukey as “Philosopher”, A. P. Dempster. Annals of Statistics 30:6 (December 2002) 1619-1628.

John W. Tukey’s Work on Interactive Graphics, Jerome H. Friedmam and Werner Stuetzle. Annals of Statistics 30:6 (December 2002) 1629-1639.

John W. Tukey’s Contributions to Robust Statistics, Peter J. Huber. Annals of Statistics 30:6 (December 2002) 1640-1648.

John W. Tukey’s Contributions to Analysis of Variance, T. P. Speed. Annals of Staistics 30:6 (December 2002) 1649-1665.

John Tukey’s personal and professional contributions receive an extensive accounting in this series of memorial articles. Originally trained as a topologist, Tukey’s career shows an inquiring mind moving from subject to subject, making seminal contributions. His creations include the Fast Fourier Transform (also developed by others, but it’s the Cooley and Tukey article that catalyzed this industry) and exploratory data analysis. He was also amazingly effective in coining memorable terminology, “bit” and “software” being the two best known. Many of his numerous aphorisms are scattered among these articles. The best known is “Far better an approximate answer to the right question, which is often vague, than an exact answer to the wrong question, which can always be made precise.” The first article is of general interest. Among the rest, those about time series, Tukey as philosopher, and graphics have the broadest appeal to mathematicians. The remaining three will benefit readers more professionally involved in the areas. These articles offer beautiful testimony to Tukey’ s powerful and articulate insight. CMJ readers will be inspired by this special issue. NS

Waiting for the Pattern 123123 … 123 in Independent Trinomial Observations, Muniru Aderemi Asiru. International Journal of Mathematical Education in Science and Technology 33:5 (September-October 2002) 769-772.

In an earlier paper [“Waiting for HTHT… HT and Geometric Transforms,” International Journal of Mathematical Education in Science and Technology 25:5 (September-October 1994) 759-760], Lennart Rade used a novel method to calculate the expected number of coin tosses until the pattern HTHT occurs. The answer is 1/pq + 1/(pq)^sup 2^ and it easily generalizes for the pattern HTHT … HT of k heads and k tails. Earlier solutions involved manipulating infinite series and conditional expected values, or an ingenious use of Markov chains. Rade’s method involves a probability generating function and a weighted flow graph whose edge-weights are transition probabilities between states (vertices). Let X be the number of tosses to produce the pattern HTHT.

Statisticans Advise Math Association On Undergraduate Curriculum, Tom Moore, Roxy Peck, and Allan Rossman. Amstat News 306 (December 2002) 10-11.

This report gives statisticians’ views of an important MAA initiative that relates to the two disciplines. The initiative concerned the curricular relations between mathematics and statistics. Two main questions are featured in this article: “What Do Statisticians Need From Mathematics?” and “What Can Statistics Contribute?” The answer to the first question is twofold: “(1) Develop skills and habits of mind for problem solving and for generalization. Such development toward independent learning is deemed more important than coverage of any specific content area. (2) Focus on conceptual understanding of key ideas of calculus and linear algebra, including function, derivative, integral, approximation, and transformation.” Six basic recommendations vital to attaining these two goals are listed. These should be valuable for readers who teach calculus or linear algebra. There are several answers to the second question, reflecting the different needs of mathematics majors, majors in other quantitative fields, and majors in less quantitative disciplines. Solid reasons for the necessity of statistics are given for all three audiences. In particular, acquaintance with modern data analysis is far more likely to be relevant to our students in both their personal and professional lives than is calculus or precalculus. Valuable references are given to articles and reports on the interplay between mathematics and statistics and on the subject of statistical literacy. NS

The Generalized Jug Problem, Thomas J. Pfaff and Max M. Tran. Journal of Recreational Mathematics 312 (2002-2003) 100-103.

The classic jug problem involves pouring water from one jug into another without any measuring devices. The problem is to get exactly four liters of water, given an unlimited supply of water, a three-liter jug, and a five-liter jug. (This problem reached the popular culture when Bruce Willis’s character had to solve it in the movie Die Hard 3.) The authors consider the natural generalization: Given a p-liter jug and a q-liter jug, which amounts of water can be obtained (from either or both jugs) by transferring among jugs and water source? If p or q is 1, the puzzle is trivial. If 1

Sold to the Latest Bidder, Erica Klarreich. SIAM News 36:2 (March 2003) 12, 10.

A common annoying phenomenon of public auctions at eBay is sniping, in which bidders place their bids in the last few seconds of an auction, leaving rivals no time to respond. eBay’s web site recommends that people not snipe. Some researchers have considered it an irrational approach. But economists Alvin Roth (Harvard University) and Axel Ockenfels (University of Magdeburg in Germany) have used game theory to show that sniping makes sense, There are two Nash equilibria in an eBay-type auction. The first is called proxy bidding, in which a bidder simply bids the highest price he or she is willing to pay. In the second, a player places a bid, using the proxy system, only in response to other bids. If there are no other bids, the player snipes. Roth and Ockenfels showed that this second strategy results in a lower selling price on average. Sniping is ineffective in an auction, such as Amazon uses, in which the auction does not end (even if the stated closing time has passed) unless ten minutes have passed with no bids. RNG

Master-Keyed Mechanical Locks Fall to Cryptographic Attack, Sara Robinson. SIAM News 36:2 (March 2003) 1, 7.

Matt Blaze, a cryptologist at AT&T Labs-Research, recently showed that anyone with access to a single lock and key in a typical master-keyed system of locks can easily create a master key by using a few blank keys and a file. This discovery followed from his attempt to apply cryptography theory to areas outside of computers and electronics. His method works with the “pin tumbler” locks in common use, where the number of blank keys needed equals the number of pins, usually between four and seven. This article briefly discusses Blaze’s method. Blaze wondered whether it was ethical to publish his method; he finally did so after a New York Times reporter wrote about it in the January 23,2003 issue. Blaze received angry letters frommany locksmiths, who said they already knew the method, but it’s too dangerous to talk about. RNG

Know When to Fold ‘Em: Mathematicians Get Wrapped Up In the World of Origami, Lila Guterman. The Chronicle of Higher Education 49:44 (July 11, 2003) A13-A14.

Origamists have been folding paper for centuries, but mathematicians only recently have applied their craft to this art. Over the past ten years, new theorems on the mathematical underpinnings of origami have inspired applications beyond the art of paper folding. Much of this material is accessible to undergraduates. For example, if you flatten an origami figure and then unfold it, you will see two types of creases in the paper: mountain folds (where the crease is folded away from you) and valley folds (where the crease is folded towards you). At any intersection of creases, the number of mountain folds and valley folds will differ by two. A proof of this theorem, along with other results on “flat” origamis, can be found in Thomas Hall’s paper “On the Mathematics of Flat Origamis,” Congressus Numerantium 100 (1994) 215-224. Expanding on Hull’s work, Marshall Bern and Barry Hayes have shown that, given a crease pattern, the problem of assigning the mountain and valley folds necessary to fold the pattern into a flat origami is NP-hard (“The Complexity of Flat Origami,” Proceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms (1996) 175-183). These mathematical insights inspired Robert Lang, an origami designer and laser-physics consultant, to create a computer program that generates a crease pattern for an origami, given a stick figure version of the desired shape. His program, which he cells TreeMeker, is available at Lang has also worked with physicists at Lawrence Livermore National Laboratory, where ideas from mathematical origami are being used to build an enormous folding lens for the next generation of space telescopes (“A Giant Leap for Space Telescopes,” Arnie Heller, Science and Technology Review (March 2003)12-18). DH

Out of the Ivory Tower: The Significance of Dirk Struik as Historian of Mathematics, Henk J. M. Bos. Historia Mathematical 29:4 (November 2002) 363-368.

The author, himself a distinguished historian of mathematics, was a friend of Dirk Struik before Struik’s untimely death-he was only 106-in October 2000. Bos begins this commemorative article by observing that Struik’s classic A Concise History of Mathematics (Dover, New York, 1948) determined the image of the history of mathematics for mathematicians of the past half century. Why were Struik’s historical writings so captivating and influential, and what were their great themes? Bos finds three central themes and elaborates upon each: Mathematics does not, and did not, live in an ivory tower, History of mathematics is written for the mathematician of today; History of mathematics reflects the great global developments of mankind. Struik’ s more recent writings illustrate these themes. The following sociological observation fits the second theme: “This may sound a little facetious, but one of the advantages in the study of the history of mathematics is to bring colleagues together and improve the harmony of the department.” [“Why Study the History of Mathematics?”, Undergraduate Mathematics and Applications Journal 1 (1980) 3-28.] Regarding the third theme, Struik wasn’t afraid to speculate on what he called the “great questions” of the history of mathematics. One such question is, what accounted for the “Greek wonder” that formal mathematics seemed to develop out of nowhere in Thales’ and Pythagoras’s times? Another great question was, why was it “precisely in Western Europe in the 17th century that the ‘scientific revolution’ occurred, which took experiment and mathematics as the basis for understanding the phenomena of nature?” Struik’s ruminations on the story of mathematics extended to prehistoric times. For example, he speculated that as artifacts such as axes became more geometrically regular, mathematical concepts like symmetry originated in the experiences of generations of craftspeople. PR

The Largest Small n-Dimensional Polytope with n + 3 Vertices, Andreas Klein and Markus Wessier. Journal of Combinatorial Theory Series A 102:2 (May 2003) 401-409.

This paper investigates the following problem about polytopes (that is, n-dimensional generalizations of polygons in the plane and polyhedra in 3-space): Among all polytopes in n-space that have diameter 1 and having n + 3 vertices, find the one of maximal volume. The case n = 3 is of particular interest since the argument here generalizes to higher dimensions. So first consider the problem of finding the polyhedron of unit diameter with 6 vertices that has maximal volume. There are two types of such polyhedra. (1) Each vertex has degree at most 4, and the polyhedron is topologically equivalent to the regular octahedron. (2) There is a vertex of degree at least 5, and the polyhedron is topologically equivalent to a pyramid with a pentagonal base. In the first case (the easier one), the maximal volume is 1/6, achieved by the regular octahedron. In the second (and harder) case, the maximal volume is 0.1954 … , attained by a pentagonal pyramid. CHJ

Prime Suspects, James Alexander. The New York Times (July 6, 2003) 18.

Riemann’s Riddle. The Economist (July 12-18, 2003) 74-75.

The Indivisible Man, Enrico Bombieri. American Scientist 91:4 (July-August, 2003) 360-364.

The Riemann Hypothesis has recently burst on the popular scene through the coincidental appearance of three popular books: John Derbyshire’s Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (Joseph Henry Press, 2003); Marcus du Sautoy’s The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics (Harper/Collins Publishers, 2003); Karl Sabbagh’s The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics (Farrar, Strauss Giroux, 2002). This outpouring has inspired numerous informative and exciting reviews. Individually, and together, they give a rare glimpse of subtle landscapes within mathematics. James Alexander’s review is the most coherent and comprehensive, and the least technical. The anonymous review in The Economist is mathematically wide ranging, though the technical allusions may confuse nonmathematicians. Bombieri’s review is the most mathematically informative and detailed and will much enrich CMJ readers’ appreciation for the issues involved. However, he discusses only two of the books, omitting du Sautoy’s. This is unfortunate since du Sautoy’s is the most mathematically sophisticated of the three books (du Sautoy is a mathematician). NS

Notable Properties of Specific Numbers webpage, pub/matli/numbers.html

Robert Munafo’s selection of notable numbers for his website is eclectic and idiosyncratic, but up front he admits, “Other people have compiled similar lists, but this is my list-it includes the numbers that I think are important.”) The extensive website consists of eleven webpages, the first of which alone printed out to eight ordinary pages, packed with notable numbers and their properties. The numbers are listed in increasing order, with the first entry being the Planck time in seconds, 5.390×10^sup -44^, the shortest measurable period of time. The site ends with a link to Munafo’s “Large Numbers” page, where the properties for the numbers range from mundane to sophisticated but are often quite interesting. For 2.54,: “Since July 1, 1959 by international agreement, the inch has been defined to be exactly 2.54 centimeters or 0.0254 meters.” (In 1957, as an MIT student, I had to memorize that one inch was approximately 2.54 centimeters.) For 17: “There are 17 planar crystallographic groups.” And, 17 is a “psychologically interesting random number” since it is the most common response to the request, “Pick a random number from 1 to 20.” For 41: Euler’s polynomial n^sup 2^ + n + 41 that yields prime numbers for n = 0, 1, 2, … , 39 is given, along with mention of Goldbach’s proof in 1752 that no polynomial in n yields primes for all n. However, a polynomial is given in 26 variables-it takes five lines!-that does yield all prime numbers. (When I showed this polynomial to a colleague, he noted that the Greeks couldn’t have found it since their alphabet had only 24 letters.)

Numerous links on the website connect notable numbers to integer sequences at Neil Sloane’s On-Line Encyclopedia of Integer Sequences at http://www.research One of the approximately 80,000 sequences tabulated there begins 2, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, … . The notable number 7 links to this sequence of “long period primes” since the repeating part of the decimal expansion of 1/7 has maximal length, 7 – 1 = 6. You might escape from the Notable Numbers site after a brief visit, but the On-Line Encyclopedia seems to be a black hole. PR

Patterns of Collaboration in Mathematical Research, Jerrold W. Grossman. SIAM News 35:9 (November 2002) 1, 8-9.

The creator of the Erdos Number Project Web site ( erdosnp.html) has used the Mathematical Reviews database to study trends of authorship of mathematical publications. The number of authors in the database has been growing at an annual rate of 6% since the 1950s while the mean number of authors per paper has increased from 1.10 to 1.45 and the percent of one-author papers has decreased from 91 % to 54%. According to the database, 42.7% of authors published just one paper and fewer than 0.1% have published more than 200 (Paul Erdos has 1500 papers). In the “collaboration graph,” each author is represented by a vertex, and two vertices are joined by an edge if the two authors have published a joint paper. Previous research shows that the number of vertices of degree x is proportional to x^sup -[beta]^, where [beta] is around 3. The graph has one large component with 208200 vertices and 461643 edges; the remaining 45139 vertices (excluding isolated vertices) have 16883 components, with between 2 and 39 vertices each. In the large component, the average distance between two vertices is 27. For a fixed vertex in the large component, the distribution of the distances to the other vertices in the component is usually bell-shaped with a long right tail. The mean is usually between 6 and 11, but can be as high as 17.5 and as low as 4.7 (for Paul Erdos). The standard deviations are remarkably consistent, indicating that people far from the heart of a component might have a long path to get to the heart, but once there, their pattern of connectedness is the same as those close to the heart. RNG

Columella’s Formula, Jean-Marc Levy-Leblond. The Mathematical Intelligencer 25:2 (Spring 2003) 51-54.

Ancient mathematicians had difficulty with the area of a segment of a circle. Both the Egyptians and the Chinese used the formula A = ½h(c + h), where c is the length of the chord of the segment and h is the height of the segment. This equals the area of a trapezoid with lower base c, height h, and upper base h. If you draw a circle’s segment with this trapezoid superimposed, you’ll see that the areas look similar. However, the formula gives an area that is always too small; the error is more than 10% when the central angle [theta] of the segment is less than 110°.

In the first century C.E., the Roman agronomist Columella of Cadiz gave the revised formula A = ½h(C + h) + (1/14)(c/2)^sup 2^ in his De re rustica. (The article does not mention that the same formula appeared at about the same time in Heron of Alexandria’s Metrica.) The improvement the extra term yields is stunning; for values of [theta] between 75° and 225°, the error is less than 1%. Levy-Leblond offers an explanation for that extra term, with its mysterious 1/14. Suppose we want a formula that involves only the lengths c and h, which would be readily measurable on an agricultural field (but how many fields were circle segments?), as opposed to the radius r of the circle and the central angle [theta] of the segment, which would not be so easily measured. Since areas should be products of lengths, our formula should have the form [alpha]h^sup 2^ + [beta]hc + [gamma]c^sup 2^. The “formula of the ancients” gives [alpha] = [beta] = 1/2; if we ask that the formula be exact for [theta] = 90° and [theta] = 180°, we find that [gamma] must be ([pi] – 3)/8. Archimedes’ approximation of 22/7 for [pi] gives Columella’s formula. PDS

Boyd: The Fighter Pilot Who Changed The Art of War, Robert Coram. Little, Brown and Company (2002).

This is a gripping account of how a legendary fighter pilot’s insight and integrity wrought major changes in aircraft design, aerial combat tactics, and military management. No matter what your views are on war and peace, the book gives a fine account of how an individual can affect large organizations. It’s permeated with references to mathematics. The hero taught himself calculus, learned the calculus of variations, asymptotic expansions, etc., and used these to develop and codify innovative air fighter tactics. The author also notes the importance of mathematics and statistics to other key players in this dramatic account, one of whom “was the youngest person ever to receive a Ph.D. in math from the University of Mississippi.” Mathematical adjectives and metaphors crop up from time to time (“in the Mobius strip that is the congressional-media relationship”). In one conversation Boyd tells his colleague, “I’m plotting some E-M data and I need to know how to take a derivative,” which draws the astounded reply, “You’re plotting … in your head?” (“E-M” stands for Energy- Maneuverability.) The author then notes “in the mathematics doctoral program at MIT, Ricciont had studied … theorems that Boyd had never heard of. But Riccioni could not plot E-M data in his head.” There is even a discussion of Godel’s theorem and its vaguely described consequences for the theories Boyd developed. On at least 31 pages scattered throughout this 457 page book, there are references to mathematics, training in mathematics, or closely related fields. For most readers of this book, mathematics is foreign territory. Yet they will recognize that Boyd learned this subject on the run, as needed, and used it to great effect. This is a fascinating story about a driven individual. There’s no explanation of just how the mathematics is used, yet its pervasiveness and effectiveness are unmistakable.

Copyright Mathematical Association Of America Nov 2003

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