Geometrekking in Lexington

MAA MathFest will be held Aug. 4-6 in Lexington, Kentucky, bringing more than 1,400 mathematicians and students to the city. This will be my first visit to Lexington since the early 1980s, and I am looking forward to exploring and photographing its mathematical and architectural sights. My main memory of my previous visit is of an excursion to the Kentucky Horse Park (I still have the souvenir mug).

I have noticed that Lexington has a lively public art scene, some of it well documented on the web.

One especially intriguing artwork is "Surface Reflections" by Bill Fontana, which combines a sound sculpture with a video installation. It's located in the landscaped passageway between the Lexington Financial Center and its adjacent parking garage.

Another noteworthy sculpture is "Exponential Symmetry" by Michael Martinez. It stands in front of the College of Education at the University of Kentucky (Dickey Hall at 251 Scott Street).

Museum Without Walls of Central Kentucky offers a handy Android and iPhone application as a guide to Lexington's public art. The "TakeItArtside!" app includes maps, tours, games, lesson plans, and lots of images. The app’s Facebook page has additional information and updates.

Endless Train Track

One of my favorite demonstrations of the one-sidedness of a Möbius strip is a model train that loops endlessly along a track bisecting a Möbius strip.

Such a demonstration is one of the prime displays in the "Mathematica: A World of Numbers. . . and Beyond" exhibit currently at the Boston Museum of Science. In this case, the model train is a segmented arrow.

Commissioned by IBM and created by the famous design team of Charles and Ray Eames 50 years ago, the "Mathematica" exhibit still draws museum visitors to some of the wonders of mathematics. The interactive exhibit wanders through a wide range of mathematics, offering snapshots of probability, topology, celestial mechanics, minimal surfaces, projective geometry, calculus, Boolean algebra, and logic. A History Wall provides a timeline of mathematical achievements. 

The History Wall is visible beyond the entrance to the "Mathematica" exhibit at the Museum of Science in Boston.

Möbius strips make a second appearance among the displays in the section on minimal surfaces, where a closed wire loop dipped in a soap solution emerges with a glistening Möbius-strip surface.

In an age of multimedia extravaganzas and elaborate simulations, the "Mathematica" exhibit now seems a bit subdued and static, though it continues to intrigue. It will be interesting to see how the new Museum of Mathematics, slated to open in 2012 in New York City, manages to blend the old with the new to create an interactive, appealing vision of mathematics for the 21^st century.

A train on a Möbius strip also plays a key role in the entertaining short story "A Subway Named Mobius" (1950) by A.J. Deutsch. The author imagines that the addition of new train line to the Boston subway system creates a system so tangled that a train vanishes into a higher dimension.

This short story became the basis of a 1996 movie, titled Moebius, by Argentinian director Gustavo Mosquero and his film school students. In the film, a train vanishes in the closed circuit of the Buenos Aires underground, and a young topologist leads the investigation of the mysterious disappearance.

Photos by I. Peterson

A Passion for Tossing Dice

Ordinary dice—those sold in novelty stores and with many board games—have rounded edges and little hollows for each of the pips denoting the numbers from one to six.

But the hollows mean that such ordinary dice are somewhat biased. A little more material has been removed from sides with a larger number of pips, so a die with six hollows on one side and only one hollow on the opposite would have a slightly greater tendency to end up with the six side up.

Casino dice differ in crucial ways from everyday dice. Their edges are generally perfectly square and sharp. Moreover, the sides are flat, with no recesses.

Magician and gambling expert John Scarne once described the process of making casino dice in the following terms: Casino dice are often made by hand, each cube typically 0.75 inch wide and precisely sawed from a rod of cellulose or some other plastic. Pits are drilled about 0.017 inch deep into the faces of a cube, and the recesses are filled in with paint of the same weight as the plastic that has been drilled out. The resulting transparent cubes are then buffed and polished.

How fair are casino dice? A cubic die produces six possible outcomes. It makes sense to use a mathematical model in which each face has an equal probability of showing up. You can then calculate other probabilities, including the number of times a certain number is likely to come up in a row.

Several decades ago, Harvard statistician Fred Mosteller had a chance to test the model against the behavior of real dice tossed by a person. A man named Willard H. Longcor, who had an obsession with throwing dice, came to Mosteller with an offer to record the results of millions of tosses.

Mosteller accepted the offer, and, some time later, received a large crate of manila envelopes. Each envelope contained the results of 20,000 tosses with a single die and a written summary showing how many runs of different kinds had occurred. Altogether, Longcor had tested 219 dice of four different brands for a total of 4,380,000 throws.

"The only way to check the work was by checking the runs and then comparing the results with theory," Mosteller once explained. "It turned out [Longcor] was very accurate." Indeed, the results even highlighted some errors in the then-standard theory of the distribution of runs.

"The main formulas were correct, but the endpoints of the formulas were not quite right," Mosteller observed.

"We found some aberrant results that suggest that things a little unusual happen more often than the classical theory would suggest," he added. "Consequently maybe we should be a little more careful than we are when we interpret tests."

Because the data had been collected using both casino dice from Las Vegas and ordinary, store-bought dice, it was possible to compare their performance not only with theory but also with each other and with a computer that simulated dice tossing.

As it turned out, the computer proved to have a flawed random-number generator, whereas the Las Vegas dice were very close to perfect in comparison with theory.

Longcor's data were important enough that his name appears on the paper that Mosteller and his colleagues eventually published recounting these investigations: "Bias and runs in dice throwing and recording: A few million throws" by Gudmund R. Iversen, Willard H. Longcor, Frederick Mosteller, John P. Gilbert, and Cleo Youtz, published in Psychometrika, Vol. 36, No., 1, pp. 1-19.

References:

Albers, D.J., G.L. Alexanderson, and C. Reid, eds. 1990. More Mathematical People: Contemporary Conversations. Academic Press.

Mosteller, F. 2010. The Pleasures of Statistics: The Autobiography of Frederick Mosteller. Springer.

Scarne, J. 1986. Scarne's New Complete Guide to Gambling. Simon & Schuster.

Photos by I. Peterson

Mobile of the Fourth Dimension

An intricate geometric framework of linked pentagons hangs in the atrium of the Fields Institute for Research in Mathematical Sciences in Toronto. The five-foot-diameter construction resembles a giant soccer ball stripped of its skin to reveal an elaborate supporting structure.

Created by Marc Pelletier, the stainless-steel sculpture represents a mathematical object known as the 120-cell. It is a three-dimensional shadow, or projection, of a four-dimensional dodecahedron.

A regular dodecahedron has 30 edges and 12 faces, each of which is a regular pentagon. Its four-dimensional analog—a polydodecahedron or hyperdodecahedron—contains 120 dodecahedra, three to an edge. The resulting 120-cell consists of 720 pentagons and has 600 vertices and 1200 edges.

Pelletier's sculpture embodies one possible, particularly symmetric projection of this four-dimensional object in three dimensions. In this projection, not all of the 120 dodecahedra of the 120-cell are visible explicitly. As it slowly rotates, it shows off its various symmetries.

The sculpture features an undistorted dodecahedron at its center. This dodecahedron is surrounded by 12 others, which are only slightly distorted by foreshortening. Proceeding outward, the next layer has 20 dodecahedra, then 12 more that are considerably flattened by foreshortening. The final layer consists of 30 dodecahedra that are seen edge-on and so appear flat, delineating the sculpture's outer surface. Steel rods define the edges.

Installed and dedicated in 2002, the Fields Institute sculpture honors geometer H.S.M. Coxeter, who described the 120-cell in his classic book Regular Polytopes. Coxeter died in 2003 at the age of 96.

Pelletier later produced a copy of this sculpture for Princeton mathematician John H. Conway. It was on display in 2006 and 2007 at a temporary outdoor art space known as Quark Park, in Princeton, N.J. (see "Quark Park").

Photos by I. Peterson

Box on Stilts

One of the stranger sights in downtown Toronto is a massive rectangular box, seemingly hovering in the air amid older, somewhat more conventional brethren. It looks like an alien import seeking a place to land.

This curious structure houses the Sharp Centre for Design at the Ontario College of Art and Design (OCAD University).

Designed by Will Alsop (Alsop Architects), the structure is a parallelepiped 84 meters long, 31 meters wide, and 9 meters high.  It has two floors of teaching, studio, and office space. An elevator and stair core near one end connects the centre to the remainder of the college.

The box stands on 12 slender legs, which barely seem capable of supporting the structure. Indeed, it is built like half of a suspension bridge, so it is held up mainly by its central core, which acts like the tower at one end of a suspension bridge. Its legs serve as the equivalent of a suspension bridge's cables. The walls are nearly two feet thick to contain the massive, supporting steel framework.

Its entire surface is pixellated with a seemingly random smattering of black squares and rectangles, which play with the regular array of square windows. The red exit "tube" houses an extra stairwell for emergency use.

Photos by I. Peterson