Geometreks in Boston

 

Boston skyline as seen from Cambridge, Mass., across the Charles River. The tallest building in the photo is the Prudential Tower at 800 Boylston Street. The skyscraper to its left is 111 Huntington Avenue, sometimes called the "R2-D2" building.

Boston will host the Joint Mathematics Meetings (JMM) on January 4-7, 2012, bringing more than 5,000 mathematicians to the city.

One of the more notable mathematical sights that may greet visitors is the venerable "Mathematica: A World of Numbers . . . and Beyond" exhibit at the Boston Museum of Science (see "Endless Train Track"). Commissioned by IBM and designed by Charles and Ray Eames more than 50 years ago, the displays include a segmented arrow that travels (like a little train) along a track bisecting a Möbius strip to demonstrate the strip's one-sidedness.

A visit to a statue of Benjamin Franklin (1706-1790) also has a place on a mathematical itinerary. Franklin was born in Boston and spent his early years there. The mathematical connection involves Franklin's pastime of inventing magic squares, as described in the book Benjamin Franklin's Numbers: An Unsung Mathematical Odyssey by Paul C. Pasles.

"In my younger days, having once some leisure which I still think I might have employed more usefully, I had amused myself in making . . . magic squares," Franklin wrote in a letter more than 200 years ago.

"I could fill the cells of any magic square, of reasonable size, with a series of numbers as fast as I could write them disposed in such a manner, as that the sums of every row, horizontal, perpendicular, or diagonal, should be equal," he continued, "but not being satisfied with these, which I looked on as common and easy things, I had imposed on myself more difficult tasks, and succeeded in making other magic squares, with a variety of properties, and much more curious."

You can find an eight-foot bronze statue of Franklin in front of the Old City Hall, where it has stood since 1856. It was Boston's first public statue of a person. The statue overlooks the former site of the Boston Latin School, which Franklin attended, though he never graduated.

The Boston Art Commission offers a walking-tour brochure highlighting 100 examples of notable public art scattered throughout the city.

The Ayer Mansion at 395 Commonwealth Avenue, for example, features a wonderful array of colorful mosaics and stained-glass windows designed by Louis Comfort Tiffany, who often drew inspiration from Moorish and Islamic art.

Conceptual artist Sol LeWitt is represented by a colorful, geometric wall drawing (#1128) visible from the street at 100 Cambridge. On top of that, the Massachusetts Museum of Contemporary Art is hosting an extensive retrospective exhibition of LeWitt’s wall drawings.

Two minimalist artworks with a strong geometric presence that may be of interest are Trimbloid X by David Kibbey, on the Charles River Esplanade between Dartmouth Street and Clarendon Street, and Sudden Presence by Beverly Pepper, at New Chardon Street and Congress Street.

Across the Charles River, Cambridge has its own array of public art, and the city's Arts Council offers a series of activities guides for exploring these varied artworks.

At Harvard University, of particular interest is Topological III by Robert R. Wilson, which stands near an entrance to the Science Center. Wilson was not only a physicist and the founding director of Fermilab but also an accomplished sculptor.

Topological III by Robert R. Wilson.

Wilson's gleaming bronze sculpture is a three-dimensional analog of a Möbius strip. Its cross section is an equilateral triangle, and this triangle rotates through 120 degrees before the ends meet to form a complete loop. Instead of three surfaces, the final product has one continuous surface that runs three times around the loop.

The collection of intriguing public art and the array of striking instances of architectural design at the Massachusetts Institute of Technology deserves its own article. In the meantime, MIT provides a handy guide to its public art collection.

Designed by Eero Saarinen, MIT's Kresge Auditorium has a horizontal cross section that resembles an equilateral triangle with curved sides, close to the geometry of a Reuleaux triangle.

Photos by I. Peterson

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.

Split Strips

"Möbius bands (or strips) are beautiful as objects of art, and their mysterious qualities fascinate those who discover or encounter them," sculptor Larry Frazier wrote in an article titled "Möbius strips of wood and alabaster," published in the Journal of Mathematics and the Arts.

"I'm not a mathematician, but as a sculptor, I have been fascinated by the myriad forms that a Möbius band can take, especially when interpreted in a beautiful piece of wood or stone," he continued.

In recent years, Frazier has often brought his gracefully carved artworks to the annual Joint Mathematics Meetings, where he displays and sells them.

Particularly intriguing examples arise from slicing a Möbius strip along its length. Slicing it lengthwise down the middle produces a single, longer band. Slicing a Möbius strip about a third of the way in from its edge produces two linked bands. In effect you've sliced off its edge to produce a narrower version of the original band. The outer piece has two half-twists, and the inner piece has one half-twist, just as the original, uncut band did.

Doubleslice, by Larry Frazier.

Frazier performs this trick in wood. Unlike paper strips, the two wood components can be easily reassembled into the original Möbius band.

Larry Frazier displays his split Möbius strip.

"It's especially nice in wood, because you can put the pieces back together, and you can see the Möbius it came from," Frazier said. "Making the same two cuts in a paper Möbius just produces two loops of paper, and you can't see the original Möbius they came from."

A reassembled sliced Möbius.

Sculptor Keizo Ushio performs similar magic in split stone (see "Sculpting with a Twist").

References:

Frazier, L., and D. Schattschneider. 2008. Möbius strips of wood and alabaster. Journal of Mathematics and the Arts, 2(No. 3):3, 107-122.

Peterson, I. 1991. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

LeWitt's Complex Form

To bring a trace of irregularity to his pristine geometric structures, conceptual and minimalist artist Sol LeWitt (1928-2007) started with a polygon sketched on a flat surface. He placed dots at various positions within the polygon. These points were then elevated to different heights, dictating the edges of the resulting three-dimensional, faceted object that eschewed the right angles typical of his earlier constructions (see, for example, "Incomplete Open Cubes").

Constructed from painted aluminum and titled Complex Form 6, this LeWitt sculpture was on display in New York City's City Hall Park as part of the "Sol LeWitt: Structures, 1965-2006" outdoor exhibition.

Photos by I. Peterson

Incomplete Open Cubes

Examples of Sol LeWitt's Incomplete Open Cubes on display at City Hall Park in New York City.

A cube has six faces, eight vertices, and twelve edges. In his series titled Incomplete Open Cubes, conceptual and minimalist artist Sol LeWitt (1928-2007) chose to work with cubes represented as frameworks.

LeWitt started by removing one edge from an open cube, then two edges, and so on, as an exploration of how many variations of an incomplete open cube exist and what they look like. One key constraint was that the remaining edges had to be joined.

Removing one edge results in just one possible configuration for an incomplete open cube.

By experiment, LeWitt identified 122 unique variations of open cubes with three edges (the minimum number needed to suggest three dimensions) to eleven edges. Nine of these frameworks, rendered in painted aluminum, were recently on display in the exhibition "Sol LeWitt: Structures, 1965-2006" at City Hall Park in New York City.

Three connected edges is the minimum number needed to suggest three dimensions.

In 2000, the San Francisco Museum of Modern Art exhibited all 122 incomplete open cubes. A representation of these variations is shown here.

One interesting mathematical question is whether LeWitt found all the possible variations of incomplete open cubes that met his criteria? Did he miss any? How would you find all the possibilities and prove that no others exist?

Scott Kim wondered the same thing in a "Bogglers" column in the April 2003 issue of Discover. As an "easy" question, he asked, "Can you make six different shapes that each contain four edges? The edges in each shape must all connect to form a single figure. Mirror images of the same shape are considered different, but rotations are not."

One example of a configuration made from four connected edges.

More difficult: How many distinct shapes can you make using five connected edges of a cube? The answer is 14. Can you find them? Is there a formula or some systematic way to determine the number of possibilities, from three to eleven edges.

It turns out that LeWitt did not include examples in which three or four edges are all lie in the same plane. Are other configurations missing?

For another mathematical puzzle concerning a LeWitt artwork, see "Puzzling Lines." For articles about other LeWitt artworks, see "Thirteen Geometric Shapes" and "LeWitt’s Pyramid."

Photos by I. Peterson

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

A Tetrahedral Forest

A neon-framed model of a regular tetrahedron hangs in an entrance to the Milan Central Train Station (Stazione di Milano Centrale).

Defined by four triangular faces, the tetrahedron is the simplest of all polyhedra. Any four points in space that are not in the same plane mark its corners.

Despite its apparent simplicity, a variety of artists have used the tetrahedron as the inspiration for artworks (see, for example, "Three Sentinels"). Part of the visual appeal of these constructions is that a tetrahedron is so angular that its aspect can change abruptly as a viewer moves around to see it from different angles.

One enthusiast of the tetrahedron was Philadelphia-based artist Robinson Fredenthal (1940-2009). His large, angular sculptures take advantage of the tetrahedron's amazing rigidity and its ability to resist an incredible amount of force from the outside.

"I can't think of anything more perfect than a tetrahedron," Fredenthal once remarked. If visitors came from outer space, "I'd hand them a tetrahedron, and they would understand."

Fredenthal manipulated tetrahedra in a variety of ways, creating peculiarly balanced, leaning towers of tetrahedra, and great bridges of these remarkable forms.

Fredenthal's Black Forest (above) is located on the campus of the University of Pennsylvania.

A bridge of tetrahedra forms the basis for Fredenthal's sculpture White Water, found at 5^th and Market Streets in Philadelphia.

Huge tetrahedra representing Fire Water Ice loom tall in a three-part sculpture at 1234 Market Street in Philadelphia.

Afflicted with Parkinson's disease for much of his life, Fredenthal spent his time in geometric exploration, crafting thousands of small-scale cardboard models of variations of simple forms such as cubes and tetrahedra. Most of his models are now in the Penn architectural archives.

Reference:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

Block Patterns in Blue and White

Block quilt pattern: Double Irish Chain.

The Charles Hotel in Cambridge, Mass., prides itself on its extensive collection of early American quilts, many of which are on display throughout the hotel. One particularly striking array is a set of nine blue-and-white quilts by the hotel's grand staircase.

Created in the 1880s and 90s, these hand-crafted quilts feature striking traditional designs based on geometric block patterns. Each block is usually a square or rectangle with a distinctive geometric pattern. Identical blocks are then sewn together to create a quilt.

A quilt's pattern name often gives clues about where and when a quilt was sewn, and it may say something about the interests or preoccupations of a particular quilt's creator.

Corn and Beans.

Evoking tidy garden rows, Churn Dash and Corn and Beans, for example, are interpretations of everyday items and chores. Names such as Geese in Flight, Ocean Waves, and Swallows in Flight reflect the underlying geometry of natural forms.

Geese in Flight.

Drunkard's Path is not only a representation of repeated fragments of a random walk but also social commentary dating to a period of U.S. history when the Temperance Movement was strengthening.

Drunkard's Path.

The characteristic blue (indigo dyes) and white of these quilts and their geometric designs meant they could fit into just about any setting.

Chinese Blocks.

Additional information about these quilts and other artworks in the hotel's collection are available in the walking tour brochure "Art at The Charles" (pdf).

Churn Dash.

Can you identify the block unit in each quilt? How would you characterize the symmetry of the block pattern that serves as the basis of each quilt?

Double Ninepatch.

You can find a variety of teaching aids to lead students through characterizing block patterns: Shape and Space in Geometry: Quilts (Annenberg/CPB Math and Science Project); Shapes, Lines, Angles & Quilts (Franklin Institute); Quilt Geometry (Steven H. Cullinane); Quilt Block Patterns (Math Forum).

Irish Chain.

For a colorful, animated tour of geometric quilt designs, see the National Film Board of Canada's wonderful production "Quilt."

Swallows in Flight.

Photos by I. Peterson

Pythagorean Fractal Tree

Pythagorean Fractal Tree was designed by Koos Verhoeff, cast in bronze by Anton Bakker and  Kevin Gallup, and displayed at the first art and mathematics conference in Albany, New York, in 1992.

Born in Holland in 1927, Verhoeff studied mathematics and computer science. He worked for a time at the Mathematical Center in Amsterdam, where he encountered the Dutch artist M.C. Escher, who often came to the center to research mathematical ideas he applied to his artworks.

Inspired by Escher, Verhoeff ended up pursuing the application of mathematics and computers to art. One of his main interests after his retirement in 1988 was the discovery and development of artistic structures based on geometric principles.

Fractal formations (trees), in which small pieces of a structure echo the appearance of the entire structure, inspired the branched structure shown above.

Reference:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photo by I. Peterson

Spiral Arms in Lexington

Ragged spiral arms, enclosed within a square, define a large metal sculpture standing in front of the College of Education at the University of Kentucky (Dickey Hall at 251 Scott Street). Titled "Exponential Symmetry," the artwork was created by Michael Martinez, a University of Kentucky alumnus.

The two-sided, boxed structure references the spiral geometry of a chambered nautilus shell, which in an idealized form incorporates the golden ratio in an equiangular or logarithmic spiral. (Real nautilus shells, when measured, typically fall short of this ideal. See "Sea Shell Spirals.")

According to the artist, the sculpture celebrates rationality and the human search for understanding, inspired and guided by the beautiful forms of nature. It embodies the notion that with language, writing, and mathematics, people learn from one another to organize, label, and classify the world around them to build an understanding of the planet.

Photos by I. Peterson

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