Ronald L. Graham (1935-2020)

Mathematician Ronald L. Graham died on July 6, 2020, at the age of 84. He was one of the principal architects of the field of discrete mathematics and made important contributions to computational geometry, Ramsey theory, notions of randomness, and other topics. He also served as president of the American Mathematical Society (1993-1994) and the Mathematical Association of America (2003-2004).

Ron Graham in August 1990 at the 75th anniversary of the Mathematical Association of America, held in Columbus, Ohio.

For more than 20 years while I was a reporter and writer at Science News, I relied on Ron Graham for advice, comments, and news tips. He was especially helpful in pinpointing significant developments in mathematics and patiently explaining their relevance and importance.

Graham's remarks on the difficulties of establishing mathematical certainty inspired the title of my second book: Islands of Truth: A Mathematical Mystery Cruise (W.H. Freeman, 1990).

I wrote about Graham's own research on a number of occasions.

Inside Averages

What can you deduce when you know certain averages but the original data from which the averages were computed are missing?

Considering that question in the 1980s, Ron Graham wondered, for example, to what extent secret data contained in confidential files can be uncovered if the right questions are asked. Such considerations led him to collaborate with Persi Diaconis to explore the discrete Radon transform and its inverse, an important mathematical tool in tomography.

Cracking a confidential database can be likened to the old parlor game of twenty questions. A player receiving only yes-or-no answers yet asking the right sequence of pertinent questions can often deduce the identity of some hidden object or person.

In the same way, the answers to a series of general questions addressed to a particular database could add up to a revealing portrait of something that is supposed to be secret.

A simple example shows how such a scheme might work. Suppose someone wants to find out Alice's salary. The inquisitor has access to information revealing that the average of Alice's and Bob's salaries is $30,000; the average of Alice's and Charlie's salaries is $32,000; and the average of Bob's and Charlie's salaries is $22,000. This provides enough information to deduce that Alice's salary is $40,000.

Researchers often face a situation in which certain averages are known but the original data are missing. If eight data points happen to be identified with the eight vertices of a cube and each of the eight numbers is the average of its three nearest neighbors, then it's possible to deduce the actual but currently hidden value associated with each vertex.

The actual value of vertex A is the sum of the averages shown at B, D, and E minus twice the average at G: (3 + 5 + 5) − 2(5) = 3. The values of the other vertices are 6, 3, 6, 9, 3, 12, and 9.

In this situation, the actual value at each vertex is equal to the sum of the nearest-neighbor averages minus double the average at the corner farthest from the point of interest. Curiously, the point that makes the biggest contribution to the answer is the one that's farthest away.

Diaconis and Graham developed a mathematical theory, based on the idea of discrete Radon transforms, that helps to decide how many and which averages are needed to crack a database or to analyze statistical data. Their results appear in the article "The Radon Transform on Z₂^k," published in the Pacific Journal of Mathematics.

At the root of their exercise is the mathematical concept of how completely a bunch of averages captures the mathematical relationship underlying a data set.

See also "Pennies in a Tray."

Pentagons for Floors

Tiling of identical pentagons, foyer floor. Mathematical Association of America (MAA) Headquarters, Washington, D.C., 2008.

See also "Tiling with Pentagons."

Photo by I. Peterson

Martin Gardner and Mathematics, Magic, and Mystery

One of my fonder memories of growing up in an isolated town in northwestern Ontario in the 1950s was my delight when the mail brought fresh issues of Humpty Dumpty's Magazine and Children's Digest. What I didn't appreciate until long afterward was that Martin Gardner was a key contributor to the early success of these magazines.

For Humpty Dumpty, Gardner was responsible for writing stories about the adventures of Humpty Dumpty Junior and poems of moral advice from Humpty senior to his son.

"For eight happy years, most of the time working at home, I wrote Junior and the poem, and also provided each of the year's ten issues (summer months were skipped) with the magazine's activity features of the sort that destroyed pages," Gardner writes in his posthumously published autobiography Undiluted Hocus-Pocus (Princeton University Press, 2013).

For example, Gardner notes, you folded a page to change a picture, held it up to the light to see something on the back of a page, or moved a strip back and forth through slots.

Gardner also contributed to Children's Digest, writing both articles and filler material such as puzzles and brainteasers.

Among his many other writing activities, Gardner produced a series of articles on mathematical magic for the journal Scripta Mathematica, edited by Jekuthiel Ginsburg of Yeshiva University. These articles were later fashioned (with much additional material) into the book Mathematics, Magic, and Mystery, published in 1956 and still in print as a Dover paperback.

In the preface, Gardner writes: "So far as I am aware, the chapters to follow represent the first attempt to survey the entire field of modern mathematical magic. Most of the material has been drawn from the literature of conjuring, and from personal contacts with amateur and professional magicians rather than from the literature of mathematical recreations. It is the magician, not the mathematician, who has been the most prolific in creating mathematical tricks during the past half-century."

Around the same time, Gardner was introduced to a fascinating mathematical toy called a hexaflexagon. After learning as much as he could about flexagons, he submitted an article on the topic to Scientific American, and it was published in the December 1956 issue. The magazine had earlier published a Gardner article on logic machines.

Gardner's article on flexagons attracted widespread interest and led directly to his monthly "Mathematical Games" column in Scientific American, starting with an article on a curious type of magic square. By coincidence, Gardner notes in Undiluted Hocus-Pocus, the name that Scientific American chose for his column had the same initial letters as his name.

"One of the pleasures in writing the column was that it introduced me to many top mathematicians, which of course I was not," Gardner modestly insists. "Their contributions to my column were far superior to anything I could write, and were a major reason for the column's growing popularity."

Indeed, many mathematicians owe their start to Gardner's columns, and his writing was certainly an inspiration to me (see "Martin Gardner’s Generosity" and "Martin Gardner’s Möbius Surprise").

October 21, 2013, is the 99^th anniversary of Gardner’s birth, and many Gardner enthusiasts are commemorating the anniversary in a variety of ways (see "Celebration of Mind"). The MAA-sponsored event in Washington, D.C. features a presentation by mathemagician Art Benjamin.

The title of Gardner's book Mathematics, Magic, and Mystery also happens to be the theme chosen by the Joint Policy Board for Mathematics (JPBM) for Mathematics Awareness Month in 2014, just in time for the 100^th anniversary of Gardner’s birth. Watch for more news and announcements about the exciting activities planned for April 2014.

Two recent books, both written by mathematicians, expand considerably on Gardner's original mathematical magic writings: Magical Mathematics: The Mathematical Ideas That Animate Great Magic Tricks by Persi Diaconis and Ron Graham (Princeton University Press, 2012) and Mathematical Card Magic: Fifty-Two New Effects by Colm Mulcahy (A K Peters/CRC Press, 2013).

In his foreword to Magical Mathematics, Gardner writes: "If you are not familiar with the strange, semisecret world of modern conjuring you may be surprised to know that there are thousands of entertaining tricks with cards, dice, coins, and other objects that require no sleight of hand. They work because they are based on mathematical principles."

Twitter Math Journal

Courtesy of Frank Farris, Santa Clara University.
Published in Mathematics Magazine, Vol. 84, No. 4 (October 2009), p. 254.

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Beautiful Mathematics and Imaginative Words

Some years ago, David Wells surveyed readers of the Mathematical Intelligencer about what they consider to be beautiful in mathematics. The respondents ranked Euler's identity linking e, π, and i as the most beautiful expression in mathematics. See "Euler's Beauties."

In the intriguing new book Beautiful Mathematics (MAA, 2011), Martin Erickson expands on this theme, contemplating the esthetic appeal and elegance of selected mathematical words, geometric structures, formulas, theorems, proofs, solutions, and unsolved problems.

"My approach to mathematics is as an art form, like painting, sculpture, or music," Erickson writes. "While the artist works in a tangible medium, the mathematician works in a medium with numbers, shapes, and abstract patterns."

Erickson's many examples include not only familiar beauties but also underappreciated wonders. His chapter on the fascinating names of mathematical objects, for example, ranges from the figure-eight curve known as the lemniscate to the waterfall of primes, which depicts the way that prime numbers fall into two classes (primes of the form 4n + 1 and primes of the form 4n + 3).

Among these "imaginative" terms, Erickson considers centillion, golden ratio, Borromean rings, sieve of Eratosthenes, transversal of primes, triangular numbers, determinant, and complex plane.

In the language of mathematics, what terms do you find particularly evocative, peculiar, or apt?

My nominee of the moment is the term "block monoid," to which I was recently introduced by Scott Chapman, Editor-Elect of the American Mathematical Monthly. Chapman and Paul Baginski have an article titled "Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory" in the December 2011 Monthly.

References:

Wells, D.1990. Are these the most beautiful? Mathematical Intelligencer 12(September): 37-41.

______.  1988. Which is the most beautiful? Mathematical Intelligencer 10(December): 30-31.

Oval Track Puzzles

The puzzle known as TopSpin consists of 20 circular pieces, numbered 1 to 20, filling and sliding along an oval track. TopSpin was introduced by ThinkFun (formerly Binary Arts) in 1988.

Pieces can be moved around the track in either direction, keeping their order. Or any four consecutive pieces can be maneuvered into reverse order. For example, consecutive pieces labeled 1, 2, 3, and 4, can be reversed into the order 4, 3, 2, 1.

TopSpin allows two types of moves. Numbered tokens can be moved around the track in either direction (left) or the order of four consecutive tokens can be reversed.

Interestingly, this puzzle has no impossible positions. Any possible arrangement (or permutation) of the pieces can be turned into any other arrangement. That would not be true if the puzzle had either 19 or 21 pieces.

This puzzle was the subject of a recent Numberplay blog, titled "From Sledgehammer to Scalpel," on the New York Times website. Pradeep Mutalik introduced the puzzle and noted: "The challenge is to create a way of moving a single unit without affecting the rest."

The puzzle has a long history and has been the subject of both research papers and books. In his article "TopSpin on the Symmetric Group," published in the September 2000 Math Horizons, Curtis D. Bennett illustrated how abstract algebra and group theory can be used to analyze the puzzle and develop a strategy for solving it.

In the book Oval Track and Other Permutation Puzzles—And Just Enough Group Theory to Solve Them (MAA, 2003), John O. Kiltinen describes TopSpin, in permutation group terms, as "a concrete realization of the subgroup of the symmetric group S₂₀ which is generated by the twenty-cycle (1, 2, 3, . . . , 20) and the product (1, 4)(2, 3) of two disjoint transpositions."

"Group-theoretically, the puzzle is simple to describe, but from a practical standpoint, it is nontrivial to solve," Kiltinen writes. "This makes it an excellent object of study for students of abstract algebra, giving them a concrete representation of a nontrivial and fruitful application of the theory of permutation groups."

Kiltinen's book comes with software (on a CD-ROM) to try and to study the puzzle, including variants that could not be realized in plastic. The book is currently available from the MAA bookstore at a special bargain price.

In his Numberplay blog, Mutalik poses the following problem: The puzzle's initial configuration has all the tokens in order, except that 19 and 20 are reversed. He asks: How can you move token 19 into its proper place without affecting the order of the others? Could you have done so if the tokens were arranged linearly and not in a loop?

Finally, Mutalik asks, "Can you analyze the analogous problem where you flip the order of five tokens at a time?"

"At ten I was fascinated by permutation puzzles like the fifteen puzzle," Bennett remarked. "At seventeen, I became enamored of the Rubik's cube, and today I still look for puzzles like these whenever I visit a toy store."

"For me today, however, the beauty of these puzzles is how easily they lead to deeper mathematics," he added.

Bennett's article was reprinted in the book The Edge of the Universe: Celebrating Ten Years of Math Horizons (MAA, 2006), edited by Deanna Haunsperger and Stephen Kennedy.

Further References:

Kaufmann, S. 2011. A mathematical analysis of the generalized oval track puzzle. Rose-Hulman Undergraduate Mathematics Journal 12(Spring):70-90.

Wilson, J.H. 1993. Permutation puzzles. College Mathematics Journal 24(March):163-165.

Mathematical Morsels I (Solutions)

THE FERRY BOATS

Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the river.

Solution: By the time of their first meeting, the total distance that the two boats have traveled is just the width of the river. It may take one mildly by surprise, however, to realize that, by the time they meet again, the total distance they have traveled is three times the width of the river. Since the speeds are constant, the second meeting occurs after a total time that is three times as long as the time for the first meeting. In getting to the first meeting, ferry A (say) traveled 700 feet. In three times as long, it would go 2100 feet. But, in making the second meeting, A goes all the way across the river and then back 400 feet. Thus the river must be 2100 – 400 = 1700 feet wide.

Source: American Mathematical Monthly, 1940, p. 111, Problem E366, proposed by C.O. Oakley, Haverford College, solved by W.C. Rufus, University of Michigan.

ROLLING A DIE

A normal die bearing the numbers 1, 2, 3, 4, 5, 6 on its faces is thrown repeatedly until the running total first exceeds 12. What is the most likely total that will be obtained?

Solution: Consider the throw before the last one. After the first throw the total must be either 12, 11, 10, 9, 8, or 7. If it is 12, then the final result will be either 13, 14, 15, 16, 17, or 18, with an equal chance for each. Similarly, if the next to last total is 11, the final result is either 13, 14, 15, 16, or 17, with an equal chance for each; and so on. The 13 appears as an equal candidate in every case, and is the only number to do so. Thus the most likely total is 13.

In general, the same argument shows the most likely total that first exceeds the number n (n > 5) is n + 1.

Source: American Mathematical Monthly, 1948, p. 98, Problem E771, proposed by C.C. Carter, Bluffs, Illinois, solved by N.J. Fine, University of Pennsylvania.

RED AND BLUE DOTS

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a segment of their common color; adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, none at the corners. There are 237 black segments. How many blue segments are there?

Solution: There are 19 segments in each of 20 rows, giving 19 x 20 = 380 horizontal segments. There is the same number of vertical segments, giving a total of 760. Since 237 are black, the other 523 are either red or blue.

Let r denote the number of red segments and let us count up the number of times a red dot is the endpoint of a segment. Each black segment has one red end, and each red segment has both ends red, giving a total of 237 + 2r red ends.

But, the 39 red dots on the border are each the end of 3 segments, and each of the remaining 180 red dots in the interior of the array is the end of 4 segments. Thus the total number of times a red dot is the end of a segment is 39(3) + 180(4) = 837. Therefore 237 + 2r = 837, and r = 300.

The number of blue segments, then, is 523 – 300 = 223.

Source: American Mathematical Monthly, 1972, p. 303, Problem E2344, proposed by Jordi Dou, Barcelona, Spain.

A PERFECT 4^TH POWER

Prove that the product of 8 consecutive natural numbers is never a perfect fourth power.

Solution: Let x denote the least of 8 consecutive natural numbers. Then their product P may be written

P = [x(x + 7)][(x + 1)(x + 6)][x + 2)(x + 5)][(x + 3)(x + 4)] = (x² + 7x)(x² + 7x + 6)(x² + 7x + 10)(x² + 7x + 12).

Letting x² + 7x + 6 = a, we have

P = (a – 6)(a)(a + 4)(a + 6) = (a² – 36)(a² + 4a) = a⁴ + 4a(a + 3)(a – 12).

Since a = x² + 7x + 6 and x ≥ 1, we have a ≥ 14 and a – 12 is positive.

Hence P > a⁴.

However, P = a⁴ + 4a³ – 36a² – 144a reveals that P is less than (a + 1)4 = a⁴ + 4a³ + 6a² + 4a + 1.

Hence a⁴ < P < (a + 1)⁴, showing that P always falls between consecutive fourth powers and never coincides with one.

Source: American Mathematical Monthly, 1936, Problem 3703, proposed by Victor Thébault, Le Mans, France, solved by the Mathematics Club of the New Jersey College for Women, New Brunswick, New Jersey.

Mathematical Morsels I

The American Mathematical Monthly has a long tradition of publishing problems, going all the way back to its first issue in 1894.

In a letter that appeared in the debut issue, Monthly coeditors B.F. Finkel and J.M. Colaw argued the value of posing and solving mathematical problems.

"While realizing that the solution of problems is one of the lowest forms of Mathematical research . . . its educational value cannot be over estimated," they wrote. "It is the ladder by which the mind ascends into the higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem."

Readers of the Monthly continue to look forward to fresh doses of perplexity and ingenuity with the arrival of each new issue, and the problems sections of past issues remain a treasure house of mathematical gems to revisit and ponder anew.

Several decades ago, Ross Honsberger (University of Waterloo) chose scores of "elementary" problems, originally posed in the Monthly, to appear in a volume titled Mathematical Morsels (Mathematical Association of America, 1978). He wanted to illustrate that "all kinds of simple notions are full of ingenuity."

"Mathematics abounds in bright ideas," Honsberger wrote. "No matter how long and hard one pursues her, mathematics never seems to run out of exciting surprises. And by no means are these gems to be found only in difficult work at an advanced level."

Here are four classic problems from this selection for you to try.

THE FERRY BOATS

Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the river.

Source: American Mathematical Monthly, 1940, p. 111, Problem E366, proposed by C.O. Oakley, Haverford College, solved by W.C. Rufus, University of Michigan.

ROLLING A DIE

A normal die bearing the numbers 1, 2, 3, 4, 5, 6 on its faces is thrown repeatedly until the running total first exceeds 12. What is the most likely total that will be obtained?

Source: American Mathematical Monthly, 1948, p. 98, Problem E771, proposed by C.C. Carter, Bluffs, Illinois, solved by N.J. Fine, University of Pennsylvania.

RED AND BLUE DOTS

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a segment of their common color; adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, none at the corners. There are 237 black segments. How many blue segments are there?

Source: American Mathematical Monthly, 1972, p. 303, Problem E2344, proposed by Jordi Dou, Barcelona, Spain.

A PERFECT 4^TH POWER

Prove that the product of 8 consecutive natural numbers is never a perfect fourth power.

Source: American Mathematical Monthly, 1936, Problem 3703, proposed by Victor Thébault, Le Mans, France, solved by the Mathematics Club of the New Jersey College for Women, New Brunswick, New Jersey.

Reference:

Honsberger, R. 1978. Mathematical Morsels. Mathematical Association of America.

SOLUTIONS

Riding on Square Wheels

No visit to Macalester College can be complete without a quick spin on a square-wheeled trike. It's a weird contraption, but you can ride it quite smoothly, without the sequence of jarring bumps that you might expect. The secret is in the shape of the road over which the square wheels roll.

The author aboard the Macalester square-wheeled tricycle. Photo by P. Zorn.

A square wheel can roll smoothly, keeping its axle moving in a straight line and at a constant velocity, if it travels over evenly spaced bumps of just the right shape. This special shape is called an inverted catenary.

A square rolling across a sequence of linked inverted catenaries. Wikipedia.

A catenary is the curve describing a rope or chain hanging loosely between two supports. At first glance, it looks like a parabola. In fact, it corresponds to the graph of a function called the hyperbolic cosine. Turning the curve upside down gives you an inverted catenary.

The Exploratorium in San Francisco exhibits a model of such a roadbed and a pair of square wheels joined by an axle to travel over it.

Square wheel exhibit at the Exploratorium in San Francisco.

When Macalester mathematician Stan Wagon saw the Exploratorium model, he was intrigued. The exhibit inspired him to investigate the relationship between the shapes of wheels and the roads over which they roll smoothly. These studies also led Wagon to build a full-size bicycle with square wheels. "As soon as I learned it could be done, I had to do it," Wagon says.

The resulting bicycle (actually a trike) went on display at the Macalester science center, where it could be seen and ridden by the public. In 2004, the science center obtained a new, improved square-wheeled trike. "The old one was falling apart," Wagon says. "The new one's ride is much, much smoother."

The improved Macalester square-wheeled trike. Photo by I. Peterson.

Ken Moffett briefly describes in a YouTube video how he reengineered Wagon's trike to improve its performance.

Steering remains difficult, however. If you turn the square wheels too much, they get out of sync with the inverted catenaries. Nonetheless, people ride the trike. It has an odometer, which logs about 15 miles per year, an average of eight rides per day.

A view of the rear of Wagon's square-wheeled trike. Photo by I. Peterson.

In 2007, students in a mathematical modeling course at St. Norbert College in Wisconsin successfully built a square-wheeled bicycle. This bicycle was a popular attraction in 2008 at MAA MathFest in Madison, Wisconsin.

The St. Norbert square-wheeled bicycle in the exhibit hall at MAA MathFest in Madison, Wisconsin. Photo by R. Miller.

It turns out that for just about every shape of wheel there's an appropriate road to produce a smooth ride, and vice versa. Wagon and Leon Hall described many of the possibilities in the article "Roads and Wheels," published in the December 1992 Mathematics Magazine.

Just as a square rides smoothly across a roadbed of linked inverted catenaries, other regular polygons, including pentagons and hexagons, also ride smoothly over curves made up of appropriately selected pieces of inverted catenaries. As the number of a polygon's sides increases, these catenary segments get shorter and flatter. Ultimately, for an infinite number of sides (in effect, a circle), the curve becomes a straight, horizontal line.

Interestingly, triangular wheels don't work.

As an equilateral triangle rolls over one catenary, it ends up bumping into the next catenary (above). However, you can find roads for wheels shaped like ellipses, cardioids, rosettes, teardrops, and many other geometric forms.

A cardioid rolls on an inverted cycloid.

You can also start with a road profile and find the shape that rolls smoothly across it. A sawtooth road, for instance, requires a wheel pasted together from pieces of an equiangular spiral.

An equiangular spiral on a sawtooth road.

There's certainly more than one way to ride a bike!

Originally posted July 13, 1998.
Updated May 25, 2011.

References:

Hall, L., and S. Wagon. 1992. Roads and wheels. Mathematics Magazine 65(December):283-301.

Rathgen, D., P. Doherty, and the Exploratorium Teacher Institute. 2002. Square wheels. In Square Wheels and Other Easy-to-Build, Hands-On Science Activities. San Francisco: Exploratorium.

Robison, G.B. 1960. Rockers and rollers. Mathematics Magazine 33(January-February): 139-144.

Wagon, S. 1999. The ultimate flat tire. Math Horizons 5(February):14-17.

______. 2010. Mathematica in Action, 3rd ed. New York: Springer.

Geometreks in New Orleans

The Joint Mathematics Meetings (JMM) will be held Jan. 6-9, 2011, in New Orleans, bringing together nearly 6,000 mathematicians. Famous for its French Quarter, jazz, food, and more, the city also has a claim to fame in the realm of public art, some of it mathematical in nature.

River Stones by Terry Weldon.

One noteworthy site is the Sydney and Walda Besthoff Sculpture Garden at the New Orleans Museum of Art. The garden is free to the public and well worth visiting.

The largest sculpture in the garden is Kenneth Snelson's tensegrity structure, titled Virlaine Tower. Set in a lagoon and rising 45 feet into the air, this gravity-defying construct consists of stainless steel tubes held together and supported by cables (see "Tensegrity Tower in New Orleans").

Visitors intrigued by curious or striking geometries may also be interested in seeing the following artworks.

Joel Shapiro's untitled work is a striking assemblage of angular blocks that resembles a contorted torso with straining limbs.

Untitled by Joel Shapiro.

In Castle of the Eye, II, Minoru Niizuma created a set of four stacked cubes, with a repeated square-within-square pattern on four sides of each block, to produce a structure reminiscent of a medieval Japanese castle.

Castle of the Eye, II by Minoru Niizuma.

Menashe Kadishman's massive but seemingly unstable sculpture, Open Suspense, features a precariously balanced collection of geometric forms fashioned from distinctively colored Cor-Ten steel.

Open Suspense by Menashe Kadishman.

Sunyatta, by Linda Howard, is made from arrays of aluminum strips fanning out from a central vertical axis to create a grid of light and shadow embodying the transformation of matter into energy (light).

Sunyatta by Linda Howard.

Outside the garden, New Orleans artist Arthur Silverman has based more than 400 sculptures on the tetrahedron, stretching, slicing, skewing, and assembling copies of this form in myriad ways (see "Three Sentinels," "Art of the Tetrahedron," "Art of the Tetrahedron, Revisited," and "Four Corners, Four Faces").

About 20 of Silverman's sculptures are on public display throughout New Orleans, many within walking distance in the downtown portion of the city.

Located at the corner of Poydras and Loyola, Echo consists of a pair of elongated tetrahedra that rise dramatically 60 feet into the air.

Interlocking Boxes, Closed stands outside the front entrance of City Hall. Near the rear entrance, on Poydras Street, another of Silverman's sculptures consists of welded tubing that delineates the edges of stacks of interlocking tetrahedra.

A sculpture near the corner of Poydras and Magazine, titled Painted Trio, consists of three colorful tetrahedra standing on edge.

Painted Trio by Arthur Silverman.

The lobby of a building called Place St. Charles, near Canal Street, features a Silverman sculpture based on the notion of removing tetrahedra from a rectangular block.

The office building at 1555 Poydras Street (directly across from the Superdome) has a large relief on the far wall as you enter. Its basic elements are sections made through a group of tetrahedra attached to each other. A slowly changing light plays over the piece, continually highlighting different areas of the relief.

Further uptown, a streetcar ride away, Silverman has two sculptures on the campus of Tulane University, one in front of the Tulane Law School and another at the A.B. Freeman School of Business. He also created a large outdoor menorah for Temple Sinai, at 6227 St. Charles Avenue.

Sculptor Clement Meadmore is known for his twisted rectangular prisms (see "Bending a Square Prism"). A prime example of his work, titled Out of There, can be found in front of the Hale Boggs Federal Building in New Orleans.

A median park along Diamond Street, near the New Orleans Convention Center, has a number of interesting sculptures. Giro Naito's Diamond is a massive polyhedron with identical facets. Terry Weldon's River Stones features five distorted, intriguingly fingerprinted spheres.

Diamond by Giro Naito.

Looking down as you walk along the streets of New Orleans, you might notice the distinctive tiling of hexagons and rhombuses that decorates local manhole covers.

It's also worth noting the mathematical significance of some curious signs around the city. The Tulane campus has a sign that specifies a speed limit of 23 (a prime number) miles per hour.  If you take the St. Charles streetcar to the end of the line, you'll find an intersection where south meets south. And the warehouse district has a pizza and pasta restaurant that, perhaps inevitably, is named πie.

For those attending JMM, the meeting will itself host a juried exhibition of mathematical art.

Photos by I. Peterson

Paul Halmos on Writing Mathematics

As a mathematician, Paul R. Halmos (1916-2006) made fundamental contributions to probability theory, statistics, functional analysis, mathematical logic, and other areas of mathematics. He was also known and widely recognized as a masterly mathematical expositor. And he served as editor (1981-1985) of the American Mathematical Monthly.

Halmos described his approach to writing in an essay published in the book How to Write Mathematics (American Mathematical Society, 1973). One paragraph presents the essence of the process:

"The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order that you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.”

Halmos adds, “That’s all there is to it.”

Halmos then expands on what he sees as the key elements of good mathematical writing.

1. Say something. To have something to say is by far the most important ingredient of good exposition.

2. Speak to someone. Ask yourself who it is that you want to reach.

3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.

4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.

5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.

6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.

7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.

8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.

9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.

10.Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

Halmos concludes: “The basic problems of all expository communication are the same. . . . Content, aim, and organization, plus the vitally important details of grammar, diction, and notation—they, not showmanship, are the essential ingredients of good lectures, as well as good books.”

The 44-minute  film I Want to Be a Mathematician: A Conversation with Paul Halmos is based on an interview with Paul Halmos, in which he discusses various aspects of writing, teaching, and research (Trailer).

Pythagoras at the Plate

Baseball is played on a field of geometric regularity. The baseball "diamond," for instance, is properly a square, 30 yards on each side.

Official league rules also specify the size and shape of home plate: Home base shall be marked by a five-sided slab of whitened rubber. It shall be a 12-inch square with two of the corners filled in so that one edge is 17 inches long, two are 8 1/2 inches and two are 12 inches.

But something isn't quite right. The diagram implies the existence of a right triangle with sides 12, 12, and 17. If it were truly a right triangle, the Pythagorean theorem would hold, and 12² + 12² would be the same as 17². It's not: 12² + 12² = 288 and 17² = 289.

So, the dimensions of home plate (an irregular pentagon) are not mathematically correct.

But there's a difference between measured numbers (accurate to a certain number of significant digits) and purely mathematical numbers. To the degree of accuracy required to construct a workable home plate, 17 is as good as (and certainly more measurable than) the more exact value of 12 times the square root of 2.

The history of baseball sheds some light on how the dimensions of home plate came about.

The playing field has been the same shape and size since the rules of baseball were first published more than 140 years ago. The size, placement, and shape of the bases, however, have changed over the years.

Initially, the rules insisted that bases be 1 square foot in area (most simply, a 1 foot by 1 foot square). Out on the field, the center of each base sat directly over a corner of the infield square. Home plate started as a circular iron plate, painted white, with a diameter not less than 9 inches. By the 1870s, however, home plate had become a square just like the other bases.

In 1877, the width of the bases was increased to 15 inches but home plate stayed at 12 inches. First and third base were moved to their present positions, where they fit snugly inside the corners of the square that defines the infield. This change was made so that umpires could call foul balls more easily. Second base, however, still stuck out of the square, where it remains to this day.

The year 1900 saw the introduction of the five-sided home plate, with a flat side rather than a point facing the pitcher. The extra rubber made it easier for both umpires and pitchers to judge when a ball "cut the corner," especially when dirt happened to cover the corners of home plate.

Original version posted March 25, 1996.

Updated July 12, 2004; June 30, 2010.

References:

Bradley, M.J. 1996. Building home plate: Field of dreams of reality. Mathematics Magazine 69 (February): 44-45.

Kreutzer, P., and T. Kerley. 1990. Little League's Official How-to-Play Baseball Book. New York: Doubleday.

Peterson, I. 2002. Pythagoras plays ball. In Mathematical Treks: From Surreal Numbers to Magic Circles. Mathematical Association of America.

Thorp, J., and P. Palmer, eds. 1995. Total Baseball, 4th ed. New York: Viking.