August 29, 2010

Missing Numbers

When I travel, I have a habit of photographing numbers—telephone numbers, license plate numbers, street address numbers, identification numbers, room numbers, and so on. I then use the photos as illustrations for the daily entries in the MAA NumberADay blog.

I am always struck by how many numbers we live by. You can find them just about anywhere, signifying or representing one thing or another.

A license plate on a vehicle in Jaipur.

In the Indian cities of Hyderabad, Agra, and now Jaipur, I do see some numbers—the license plates and identifying numbers on the innumerable vehicles, a scattering of telephone numbers on billboards, the tolls on some highways. Numbers representing street addresses, however, are nearly completely absent.

A telephone number recorded on a billboard in Hyderabad.

Large, garish signs, crowded together along storefronts, blare their news of goods for sale, with nary a numerical address. Even telephone numbers rarely appear as a part of these identifiers.

Storefront signs in the Charminar neighborhood of Hyderabad.

I find that I am so used to relying on numerical addresses to locate and orient myself—to navigate from place to place—that their absence adds to the sense of traveling in another world here in India.

A street scene in Hyderabad.

Nonetheless, I found one notable exception in Jaipur—the first planned city in India. Founded in 1727 by Maharajah Sawai Jai Singh II, the original city was laid out at right angles in square blocks, with wide, straight streets. Shop fronts were numbered consecutively, and signs had a standardized appearance, largely preserved to this day.

Consecutively numbered shops and standardized signs in old Jaipur.

Photos by I. Peterson

August 26, 2010

Publications, (Mathematical) People, and Pearls

Hyderabad, India. The exhibits area at the International Congress of Mathematicians (ICM) had only about two dozen booths, but they included a variety of societies, publishers, institutions, and retailers.

The Cambridge University Press exhibit included an array of books published by the Mathematical Association of America.

Congress attendees could purchase autographed copies of the book Optimal Transport by 2010 Fields Medalist Cédric Villani at the Springer Verlag GMBH booth.

A mathematician at the Mathematical Society of Japan exhibit explains a fabric model of a five-dimensional surface—the moduli space of equilateral pentagons. It was created by sewing together 24 regular pentagons so that four faces meet at each vertex and the resulting surface has four tunnels.

Alice Peters holds up a copy of Mathematical People, edited by Donald J. Albers and Gerald L. Alexanderson and published by A K Peters. A publisher of mathematics books, A K Peters is now part of CRC Press, a member of the Taylor & Francis Group.

Colorful wooden toys were on display and available for purchase at the Crafts Council of Andhra Pradesh booth.

Strings of pearl were on sale not only in the exhibit hall but also in shops in the Hyderabad International Convention Centre and in the adjoining hotel. There was no shortage of customers.

The next ICM will be held in Seoul, Korea, in 2014.

Photos by I. Peterson

An Illusion of Understanding

Hyderabad, India, The panel discussion on "Communicating Mathematics to Society at Large" at the International Congress of Mathematicians attracted a large, attentive crowd. The presentations and audience comments all served to illustrate the many opportunities available for reaching out to the general public while acknowledging the diversity of that audience.

Of particular interest was a question from an Indian journalist in attendance, who asked whether achieving an "illusion of understanding" was enough, when reports for the general public have to leave out technical details and skim over complexities. Panel members, both journalists and mathematicians, replied that understanding develops step by step, whether in recreational reading or in mathematical research, along a road that approaches "truth" or "reality."

Other comments highlighted the difference between communicating mathematics and communicating about mathematics and whether a meaningful distinction exists between communicating mathematics and teaching mathematics.

Panel members (from left to right) Marianne Freiberger, Christiane Rousseau, Günter Ziegler, and R. Ramachandran.

Presentations by the panel members set the stage for the discussion. Christiane Rousseau (Université de Montréal) talked about why mathematicians should be interested in communicating mathematics to the general public and what message they ought to deliver and how best to do it.

I focused on the importance of understanding the intended audience and being aware of language barriers that can stand in the way of communication (see “Communicating Mathematics” and “The Mathematical Vocabulary Problem”).

Günter M. Ziegler (Deutsche Mathematiker-Vereinigung), who chaired the session, described settings and occasions that could be used for communicating mathematics. As illustrations, he used examples from the many successful programs developed for “Mathematics Year 2008” in Germany, ranging from blogs and articles to quiz shows, lectures, and websites.

Marianne Freiberger (Plus magazine) gave examples of the kinds of mathematical topics that the public finds particularly interesting, as shown in visits to articles at the Plus magazine website. Most popular was an article titled “Mysterious number 6174” by Yutaka Mishiyama. Next was the article “The story of the Gömböc,” prepared by Freiberger. People respond to such topics, even when they may be fearful of mathematics and even when the subject may have no obvious practical value, she noted.

Journalist R. Ramachandran (The Hindu) lamented the unsatisfactory state of science and math communication in India and the reluctance of Indian mathematicians to get involved with communicating mathematics to the public.

The session concluded with remarks on several efforts to develop multinational programs for communicating mathematics. In North America, the major mathematics institutes have joined together to put a special emphasis on “Mathematics of Planet Earth” in the year 2013. Rousseau heads that effort and is looking for support from other math societies and institutes throughout the world.

Photo by I. Peterson

August 24, 2010

Signs of Hyderabad II

Hyderabad, India. The International Congress of Mathematicians has a tradition of breaking up its proceedings by offering a day with no programmed activities. So, after three intense, packed days of lectures, discussions, and other events, attendees got a chance to relax, explore Hyderabad, dine in exotic settings, go souvenir hunting, or pursue other interests.

For many, the break meant getting into a bus to tour some of Hyderabad's famous sites or hiring a taxi to go shopping in the city's many markets. My tour took me to Golconda Fort and the tombs of the Qutb Shahi kings, who ruled at Golconda Fort from 1512 to the end of the sixteenth century.

A view of the heights of Golconda Fort.

One of the most impressive forts in India, the vast complex of buildings and walls sprawls across a massive granite outcropping. I was fascinated by the fragments of patterns visible among the ruins, from intricate lattices to symmetrical lotus flower designs and other elaborate decorations.

A ceiling's lotus flower design.

But it was some of the quaintly expressed signs that we encountered as we ascended to the fort's high point at Durbar Hall that remain lodged in my memory.

The intent of the sign above is pretty clear. The one below is bit more ambiguous about the recommended action.

And some signs were quite comprehensive.

The tombs of the seven Qutb Shahi rulers cluster on a grassy area just a kilometer away from the fort's outer wall.

Now lacking the brilliant turquoise coloring and intricate tiling of the original structures, the tombs still loom impressively into the sky.

Here, too, some ingeniously emphatic signs added to the pleasure of the visit.

And the scummy pool really did look like one to avoid at all cost.

Other signs took a simpler, more direct approach to admonishing visitors..

Another sign simply instructed, "Do not spit on the walls."

Photos by I. Peterson

August 23, 2010

Signs of Hyderabad I

Hyderabad, India. The International Congress of Mathematicians is taking place at the Hyderabad International  Convention Centre (HICC, below).

Note the concentric circles that mark the plaza in front of the Hyderabad International Convention Centre.

A sign (below) outside the centre welcomes delegates from all over the world to the meeting.

The new Fields Medalists each get a chance to present their work. Winner Ngo Bao Chau also spent time autographing copies of a recently published article presenting some of his award-winning work (below). More than 200 mathematicians took advantage of the opportunity, purchasing offprints from the publisher for signing.

Security is tight. A security fence surrounds the entire area, including the conference hotel, and some roads are blocked off (below).

All vehicles travelling to the site must go through two checkpoints. Anyone entering the hotel or HICC must pass through a metal detector and be frisked by a guard. Armed soldiers stand on guard at various positions on the grounds.

Photos by I. Peterson.

English Play

Hyderabad, India. One of the highlights of the International Congress of Mathematicians was the chance to see the remarkable play A Disappearing Number, presented by the London-based theatre company Complicite. Like the plays Proof and Arcadia, it deals with mathematical genius; in this case, with Srinivasa Ramanujan, his relationship with Cambridge mathematician G.H. Hardy, and his unique contributions to mathematics.

Unlike other plays with a mathematical underpinning, A Disappearing Number forces the audience to face the reality of mathematics and to try to tangle with some its complexities, subtleties, and mysteries.

The play opens with a professor, Ruth Minnen, writing strings of numbers on a stage equivalent of a blackboard: the sequence 2, 4, 6, 8, . . . , the beginning of the sequence of prime numbers, and more. A narrator shatters the staged illusion, pointing out that everyone on stage is an actor, demonstrating that a door doesn't lead anywhere, showing that the glasses worn by the professor have no lenses. The mathematics on the blackboard is the only thing that is real, he insists.

Yet this reality poses its own puzzles. The sequence of even numbers is completely predictable, but it is also infinite. The sequence of primes is also infinite, but it doesn't have the same predictability. These simple examples lead to more profound mysteries, as Ruth begins describing zeta functions and the Riemann hypothesis, which concerns the distribution of primes.

So begins this imaginatively produced account of Ramanujan's life, which intermingles pieces of his story with the modern-day tale of the romance between Ruth and hedge-fund trader Al Cooper, who manipulates numbers with no sense of the impact of those manipulations on people and society or of the underlying mathematics.

The striking production vividly integrates text, image, and action. It scrambles past, present, and future, fragmenting and reassembling time at a pace so rapid that it is sometimes bewildering and disconcerting.

As it proceeds, the play touches on a variety of mathematical themes, from the startling infinitude of infinities to the more commonplace notion that math is everywhere. It incorporates several famous stories about Ramanujan and Hardy, including Ramanujan's prompt rejoinder to Hardy's comment that 1729 is an uninteresting number: 1729 is the smallest number that is the sum of two positive cubes in two distinct ways.

The collaboration between Ramanujan and Hardy spanned World War I, and one of the more startling figures that the play dramatically cites is the more than 34,000,000 casualties (killed, missing, or wounded) of that war. Towards the end, the play brings Ramanujan's research on mock theta functions into the realm of modern physics via quantum mechanics and string theory.

A Disappearing Number is also about clashes between different cultures—in Ramanujan's perspective on life in an alien England, in the differing viewpoints of a math professor and a bond trader trying to find common ground, in the peaceful and destructive uses of mathematics.

Indeed, so much is packed into the play that it sometimes seems to skip too quickly from one thought or one scene to the next. Yet it also successfully conveys some of the wonder and appeal of mathematical activity for its own sake.

Seeing the play was a fitting conclusion to my first day in India—a memorable day that offered a deliciously chaotic mixture of the old and new and of the commonplace and exotic. The conference setting in a new, high-tech convention center seemed familiar from the innumerable meetings I have attended over the years, with some distinctive variants—a pearl shop open for business, rice-based meals for registered participants, a mathematician lecturing on Indian classical music, and going through security checks for every entry to both the center and the nearby hotel. No aimless wandering allowed!

But the seemingly anarchic traffic was something else entirely. The endlessly weaving and honking gaggles of vehicles—trucks, buses, cars, scooters, auto rickshaws, and more—presented an intriguing lesson in cooperative management. I saw practically no traffic signs or lights, yet drivers somehow manage to get where they want to be, with judicious honking and deft maneuvering—and nary a scratch or dent.

Any trip around the city was a breathtaking, nerve-wracking affair, and I was glad I wasn't the driver, but somehow we arrived. I began to relax a bit—as long as I was in a bus. It opened my eyes to the possibility that tight regulation and firm adherence to rules (hence, imposed predictability) may not always be the only way to go.

As the play demonstrated, seeing things in new ways is an important component of creativity—a jolt out of the familiar and routine.  Now that I am out of my usual comfort zone (but not too far out), I can hardly wait for day two.

My title, "English Play," refers to the curious heading under which the play was advertised and described on the ICM India website and in printed material about the meeting. To get to and from the theater, we boarded buses bearing signs that also read "English Play." And, yes, the play was in English, and it was performed by an English theatre company.

August 18, 2010

A Diamond-Lattice Exoskeleton

Among the tall buildings of Pittsburgh, the United Steelworkers Building stands out not for its height (only 13 stories) but for its dramatic diamond-lattice façade.

The diamond grid is more than just decorative. The steel exterior walls represent the building's supporting structure; they serve as a load-bearing exoskeleton.

Designed by the architectural firm Curtis and Davis and constructed in the early 1960s, this structure (then known as the IBM Building) was one of the first to have its load-bearing frame visible on the outside—a style regarded then as daring and innovative.

The design is a modern adaptation of a wooden framing scheme for covered bridges, patented in 1820 by U.S. architect Ithiel Town. His lattice trusses, made from diagonally positioned planks, replaced the heavy timbers required for more traditional, rectilinear bridge designs.

Photos by I. Peterson

August 15, 2010

Recycling Arrows

The triangle of three bent arrows that signifies recycling is a fixture of the world in which we live. This recycling symbol appears in newspapers and magazines and on signs, bottles, envelopes, cardboard cartons, trash receptacles, and many other containers.

If you look closely, however, you'll see all sorts of variants of this ubiquitous symbol.

In the version of the recycling symbol on this trash container, the two arrows with the same twist are in the lower two corners.

The original design was the result of a contest sponsored by the Container Corporation of America (CCA) as a special event in response to the first Earth Day in 1970. Art and design students were invited to create a symbol to represent paper recycling. The winning logo, selected from more than 500 entries, was submitted by Gary Dean Anderson, then an art student at the University of Southern California.

"The figure was designed as a Möbius strip to symbolize continuity within a finite entity," Anderson recounted in an article published in the May 1999 issue of the trade magazine Resource Recycling.

"I used the [logo's] arrows to give directionality to the symbol. I envisioned it with the small edge or the point of the triangle at the bottom," Anderson said. "I wanted to suggest both the dynamic (things are changing) and the static (it's a static equilibrium, a permanent kind of thing). The arrows, as broad as they are, draw back to the static side."

Anderson's original design was then refined by William J. Lloyd, design manager in CCA's public relations department. He sharpened the lines and rotated the symbol so that the stylized outline of a tree can be seen in its center.

Initially, CCA licensed the design to trade associations for a nominal fee. The company later dropped its application to register the logo as a service mark, leaving it in the public domain.

In the 1970s, the American Paper Institute and the American Forest and Paper Association started promoting use of this symbol to describe recyclable and recycled paper products. Its use spread rapidly and expanded to many other items.

The original design had three bent arrows—two bent in the same direction and one bent in the opposite direction. That still leaves room for variants—the lone, differently twisted arrow can be in any one of three positions, and the arrow portion can be on top or on the bottom.

Some users of the symbol even invert the design, with a point of the triangle toward the bottom.

A more striking variant has three identical bent arrows. Perhaps it was introduced accidentally, when someone failed to notice that the direction of the twists in the arrows makes a difference.

Set within a hexagon, this version of the recycling symbol features three arrows with the same twist, with the arrow heads in the foreground rather than the background.

One possibility is that an illustrator drew just one bent, twisted arrow, made two copies of it, and put the arrows in a triangle pattern, never realizing that the original symbol was meant to conform to the shape of a standard half-twist Möbius band.

A Möbius surface has only one side and one edge. You can make a Möbius strip by gluing together the two ends of a long strip of paper after giving one end a half twist.

Interestingly, the mutant recycling symbol is based on a somewhat different surface—a one-sided band formed by gluing together the two ends of a long strip of paper after giving one end three half-twists instead of just one (as you would have in a standard Möbius strip).

The standard recycling symbol (top left) and an alternative version (top right) can be represented by continuous folded ribbons, showing that the standard form is a Möbius band made with one half-twist (bottom left) and the alternative is a one-sided band with three half-twists (bottom right).

If you were to lay a string along the strip's edge until the string's ends meet and pulled the string tight, you would end up with a trefoil knot in the string. If you did this with a standard Möbius band, you wouldn't get a knot.

The version of the recycling symbol used in Santa Clara, Calif., joins the arrows of the mutant three-arrow form to create a trefoil knot, with a Möbius-like twist.

A few years ago, the original design was the dominant form. Nowadays, you're as likely to see the mutant version as you are to see the original.

In this mutant form, the arrow heads are in the background rather than the foreground.

What's fascinating about the entire recycling-symbol episode is how a geometric shape that came out of pure mathematical research, done in the 19th century by August Ferdinand Möbius (1790–1868), has become a modern cultural icon.


Dyer, J.C. The history of the recycling symbol. Dyer Consequences.

Jones, P., and J. Powell. 1999. Gary Anderson has been found! Resource Recycling (May).

Long, C. 1996. Möbius or almost Möbius. College Mathematics Journal 27(September):277.

Photos by I. Peterson

August 14, 2010

Manhole Cover Geometry

Why is the cover of a manhole nearly always round? Why isn't it oval or square?

Manhole covers are nearly always circular, but they may feature distinctive geometric designs, like the ring-and-spoke pattern shown here..

One answer is that a circular lid, unlike a square or an oval cover, won't fall through the opening. There's no way to position a round lid so that it can slip through a slightly smaller hole of the same shape. That's because a circle has a constant diameter. It has the same width all the way around.

In contrast, an oval or an ellipse is longer than it is wide. You can always find a way to slip an oval lid through a hole of the same shape. That's also true of a square or a hexagonal cover.

Plus signs add an arithmetical touch to this manhole cover in Pittsburgh.

A 12-sided (dodecagonal) manhole cover shouldn't work, but if the hole is sufficiently smaller than the lid, the cover's shape may be close enough to a circle to work effectively.

This 12-sided manhole cover near the U.S. Capitol features a triangular grid and a curious pattern of holes.

Amazingly, the circle isn't the only shape that would work safely as a manhole cover. In fact, any shape of constant width would do, and there are infinitely many such shapes. The simplest example is the Reuleaux triangle, named after distinguished mechanical engineer Franz Reuleaux (1829–1905), who was a teacher in Berlin more than 100 years ago.

A Reuleaux triangle (solid lines), based on an equilateral triangle (dashed lines) and set inside a square, is a geometric shape that has a constant width.

One way to draw a Reuleaux triangle is to start with an equilateral triangle, which has three sides of equal length. Draw three arcs of circles, with each arc having as its center one of the triangle's corners and as its endpoints the other two corners. The resulting “curved triangle," as Reuleaux termed it, has a constant width equal to the length of the interior triangle's side.

I haven't yet seen a manhole cover in the shape of a Reuleaux triangle, but I have heard that they exist.

One can construct a curve of constant width not only from an equilateral triangle but also from any polygon with an odd number of sides. Thus, you can readily obtain a curved pentagon, heptagon, and so on.

A Reuleaux pentagon and a Reuleaux heptagon (outer boundaries) are both curves of constant width.

Some coins have a rounded heptagonal shape that allows their use in slot machines designed for ordinary coins.

The Reuleaux curves described so far have corners—points where two sides meet at an angle. However, curves of constant width having rounded corners can be readily constructed from the angular forms.

Moreover, a curve of constant width need not be symmetrical or even consist of circular arcs. Therefore, an unlimited number of curves of constant width are possible, and the Reuleaux triangle happens to be the family member of least area.

Hexagons adorn a circular manhole cover in New Orleans.

In principle, manhole cover designers have an infinite array of shapes at their disposal, but circular covers remain the norm.

Photos by I. Peterson

August 12, 2010

Suspension Roof

The swooping roof of the David L. Lawrence Convention Center in Pittsburgh hangs from a series of 15 enormous cables tied to tall masts.

Designed by architect Rafael Viñoly, the structure mimics the curves of Pittsburgh's many suspension bridges.

The cable-suspended roof owes its graceful shape to the natural curve formed by a flexible cable or chain hanging between its fixed ends—a catenary.

The cables end in exposed anchors (above) inside and on the roof, visible to passersby.

Photos by I. Peterson

August 10, 2010

Unit Signs

As you walk along the streets in Washington, D.C., you'll notice that signs at corners give not only the street name but also a numerical indicator representing the numbers assigned to a given block—typically in multiples of 100.

So, as you approach the U.S. Capitol, you're likely to see signs saying 500, 400, 300, 200, 100. Then what happens? Do you get 000 or 00?

Curiously, the chosen designation for the zero-order block is UNIT.

I wonder how this designation came about (and whether it even makes sense in this context). So far, I haven't been able to find anything on its history. And what happens in other cities?

My thanks to Joshua Zucker for pointing out this unusual feature of the Washington streetscape.

Photos by I. Peterson