December 21, 2007

Flowery Sculptural Twists

For nearly three decades, sculptor Brent Collins has painstakingly fashioned wood and other materials into an astonishingly wide range of sinuous forms. A self-taught artist, he has twisted, stretched, and molded simple geometric motifs into one graceful sculpture after another.

Collins is not a mathematician. Yet his sculptural explorations, guided by intuition and aesthetic sense, have yielded patterns and forms that have an underlying mathematical logic. Many of his geometric sculptures echo with the saddle-shaped, soap-film economy of minimal surfaces.

"I've always tended in my work to distil form—pare it down to its essentials," Collins once remarked. "And those essentials are basically minimal mathematical relationships. It's a language of nature I've appropriated for aesthetic purposes."

Collins spoke recently at Knotting Mathematics and Art: Conference in Low Dimensional Topology and Mathematical Art, held at the University of South Florida.

In his latest pair of sculptures, Collins has twisted undulating ribbons into helical forms that dramatically flare into wavy petals, creating a pair of mathematical flowers. He has titled his sculptures Blosme 1 & 2, using the Middle English spelling for "blossom."


In these fanciful forms, the flower "stems" are helical bands. The edges of the "blossoms" that emerge from them follow helical pathways in space. Even the intersections of the blossom's undulating planes have a helical form.

"These compositions are sensuously organic landscapes of multiply embedded helicoid dimensions with an evocative resonance suggesting the blossoms of some flowers," Collins remarks. "The helicoid geometries as formulated in these sculptures have never been explored at a comparable level of sophistication in contemporary art and generate an unprecedented biomorphic aesthetic as sculpture."


"The impression is of an exquisite biomorphic surface," Collins adds, "which seemingly could have evolved somewhere, sometime in nature."

References:

Collins, B. 2005. Geometries of curvature and their aesthetics. In The Visual Mind II, M. Emmer, ed. MIT Press.

Francis, G.K. 1993. On knot-spanning surfaces: An illustrated essay on topological art (with an artist's statement by Brent Collins). In The Visual Mind: Art and Mathematics, M. Emmer, ed. MIT Press.

Peterson, I. 2003. A graceful sculpture's showy snow crash. MAA Online (Feb. 10).

______. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

______. 1999. Sculpture generator. MAA Online (Oct. 4).

November 18, 2007

Random Walks to Football Rankings

The national championship of U.S. college football is decided at the end of the season in a climactic game that is supposed to match the top two NCAA Division I (FBS) teams. Year after year, controversy has dogged the selection of those two teams.

The selection process is hampered by the fact that the 119 teams that belong to the division play only 10 to 13 games. Each team doesn't play every other team. Moreover, not all schedules are created equal. Some teams play against much stronger opponents than others do. And it's not even clear what "top two" means. Is it the two teams with the best overall record for the season or the two teams playing best at the end of the season?

The year 1998 saw the introduction of a complex mathematical formula to determine which two teams play for the national championship. The formula produced the Bowl Championship Series (BCS) standings, and the winner of the final bowl game matching the top two teams in the standings became the national champion.

Over the years, the formula has been tweaked and the system modified to remove flaws and better match human expectations. In its latest iteration, the BCS system simply averages a given set of polls and computer rankings. And the results continue to arouse skepticism.

Mathematically and computationally inclined football fans have proposed a variety of alternative ranking schemes—many quite complicated—that purportedly give "fairer" results. One interesting contender is a scheme developed by Thomas Callaghan, now a graduate student at Stanford University, Peter J. Mucha of the University of North Carolina at Chapel Hill, and Mason A. Porter of the University of Oxford.

"A simply-explained algorithm constructed by crudely mimicking the behavior of voters can provide reasonable rankings," Callaghan, Mucha, and Porter claim. They describe their scheme in the November American Mathematical Monthly.

In their ranking model, the mathematicians start with a collection of random walkers (voters), each of which declares its preference for a particular team. Each voter then repeatedly selects a game at random from its preferred team's schedule and decides whether to change its preference to the opposing team as biased by the game's outcome. So, if team i beats team j, the average rate at which a walker voting for team j changes its allegiance to team i is proportional to p, and the rate at which a walker already voting for team i switches to team j is proportional to 1 – p. The scheme hinges on the selection of an appropriate value for p. Note that the algorithm doesn't take into account the date on which games are played, a factor that some human analysts like to include.

Overall, the total number of votes cast for each team by all random-walking voters repeating this process indefinitely adds up to a ranking of the top teams. Remarkably, this simplistic ranking algorithm yields reasonable results.

Here's the current 2007 random walker top 10 (with p = 0.75), taking into account games played through Saturday, Nov. 10.


When compared with the BCS standings, the random walker algorithm shuffles the order a little (Oregon ranks higher than LSU, for example), and includes Florida instead of Virginia Tech. Nonetheless, there's a remarkably close match between the two rankings.

The following weekend, Oregon and Oklahoma lost. These results reshuffled the standings. The random walker model put LSU at the top, followed by Arizona State, Oregon, West Virginia, Georgia, Missouri, Kansas, Ohio State, Boston College, and Florida.

We'll see what happens as the regular season ends and the bowl games begin.

Last year, the random walker algorithm put Florida at the top at the end of the regular season, followed by USC. But the two teams didn't end up playing each other in the BCS national championship game. Instead, Florida defeated Ohio State in the championship game and USC beat Michigan.

"We remain committed to the proposition that the use of algorithmic rankings for determining the college football postseason will only become widely accepted when those rankings have been reasonably explained to the general public," Callaghan, Mucha, and Porter conclude. "In that context, the random walker rankings . . . provide reasonable ways to rank teams algorithmically with methods that can be easily explained and broadly understood."

References:

Callaghan, T., P.J. Mucha, and M.A. Porter. 2007. Random walker ranking for NCAA Division I-A football. American Mathematical Monthly 114(November):761-777.

______. 2004. The Bowl Championship Series: A mathematical review. Notices of the American Mathematical Society 51(September):887-893.

Peterson, I. 2004. College football, rankings, and wandering monkeys. MAA Online (Sept. 6).

______. 1998. Who's really no. 1? MAA Online (Dec. 14).

Information about rankings of U.S. college football teams can be found at http://homepages.cae.wisc.edu/~dwilson/rsfc/rate/.

November 7, 2007

Triangular Numbers and Magic Squares

It sometimes takes years—even decades—to solve a seemingly simple problem. The October American Mathematical Monthly features the solution to a problem originally posed 66 years ago—one that concerns triangular numbers and magic squares.

A triangular number can be represented as a triangular grid of points, in which the first row contains one element and each succeeding row has one more element than the previous one. The first few triangular numbers are 1, 3, 6, 10, 15, and 21. If you include 0 as the first number, the nth triangular number is given by the formula n(n – 1)/2.


Triangular numbers depicted as triangular arrays of dots. Courtesy of Christian Boyer.

A magic square consists of a set of distinct integers arranged in the form of a square so that the numbers in each row, column, and diagonal all add up to the same total.

In 1941, the American Mathematical Monthly published the following problem, posed by Royal Vale Heath, widely known for creating ingenious mathematical puzzles: "What is the smallest value of n for which the n2 triangular numbers 0, 1, 3, 6, 10, . . . n2(n2 – 1)/2 can be arranged to form a magic square?"

At that time, the journal's "Problems and Solutions" section was edited by Otto Dunkel, Orrin Frink Jr., and H.S.M. Coxeter.

A year later, Heath himself proposed a partial solution. He noted that a magic square that is still magic after the original entries are all squared (a bimagic square) can itself be used directly to construct a magic square of triangular numbers. Heath wrote, "Clearly, the magic property will still be retained if each of the original numbers is subtracted from its square. The resulting numbers are all even, and their halves are the triangular numbers."

Starting with a known bimagic square of order 8 (an eight-by-eight array), Heath then constructed a magic square of 64 triangular numbers, from 0 to 2016, with magic sum of 5460. He admitted, however, that a smaller set of distinct triangular numbers might also form a magic square.

So, Heath's puzzle remained unsolved until it finally came to the attention of Christian Boyer, who has recently put a lot of time and effort into exploring magic squares. Boyer proved that magic squares of triangular numbers are impossible for orders 3, 4, and 5. He did, however, discover magic squares using 36 triangular numbers from 0 to 630 (with a magic sum equal to 1295).

Here's an example.


Boyer also found that there are magic squares of order 7, which use the first 49 triangular numbers, starting with 0. Furthermore, you can obtain magic squares of orders 6 and 7 when you start with 1 instead of 0.

What is the smallest order n possible if you allow the use of any triangular numbers rather than consecutive ones? Boyer has found the first four-by-four and five-by-five magic squares of distinct triangular numbers.


But it is still unknown whether a three-by-three magic square of distinct triangular numbers exists.

References:

Boyer, Christian. 2007. Better late than never (Solution to problem E 496). American Mathematical Monthly 114(October):745-746. Expanded version of this solution.

Heath, R.V. 1942. A magic square of triangular numbers (Solutions: E 496). American Mathematical Monthly 49(August-September):476.

______. 1941. Problems for solution: E 496. American Mathematical Monthly 48(December):699.

Peterson, I. 2005. Magic squares of squares. MAA Online (June 27).

October 31, 2007

Optimal Fetching

For a Welsh corgi named Elvis, merely fetching a ball isn't enough. The dog also works out the optimal path he should take to retrieve the ball in the shortest possible time—all the better to squeeze in some extra rounds.


Welsh corgi Elvis lives to fetch. Photo courtesy of Tim Pennings.

Having established that Elvis apparently solves a calculus problem each time he fetches a ball thrown from a sandy beach into the water, mathematician Tim Pennings of Hope College in Holland, Mich., has since studied his dog's problem-solving ability when Elvis has to choose between two qualitatively different options on which path to take.

While playing fetch at the beach, Elvis sometimes starts in the water. When a ball is thrown parallel to the shore, Elvis has the option of swimming directly to the ball or heading for the shore, running on the sand, then swimming back out to the ball. If the distance to the ball is short, the best strategy is to swim directly to the ball. For longer distances, the swim-run-swim option is Elvis's best bet to minimize retrieval time, assuming that Elvis runs faster than he swims.

"Such a situation induces a bifurcation in his optimal strategy," Pennings and colleague Roland Minton of Roanoke College note in the November College Mathematics Journal. In other words, at some distance, the optimal strategy must change from swim to swim-run-swim.

The mathematicians ask, "What is the bifurcation point at which the optimal strategy changes?" And does Elvis change his strategy at this optimal point? In other words, does Elvis know bifurcations?

To find the optimal path, suppose that Elvis starts x1 meters out in the water and races to a ball that is z meters downshore and x2 meters out into the water. His swimming speed is s meters per second to a point y1 meters downshore; he then runs along the beach at speed r meters per second to a point y2 meters upshore from the ball, before finally swimming out to the ball.

Minimizing the total time for the swim-run-swim pathway reveals that, for the optimal path, the incoming swim leg makes the same angle with the shore as the outgoing swim leg does (like light reflecting from a mirror or a billiard ball bouncing off a rail). Pennings and Minton speculate that Elvis may somehow take advantage of this symmetry in determining the quickest path to the ball.

The mathematicians also worked out a formula for the bifurcation point at which the nature of the optimal solution switches from an all-swimming mode to the swim-run-swim strategy. Clearly, there's no bifurcation point if swimming is faster than running. For shorter distances, when running is faster than swimming, the small advantage that running provides can't compensate for the extra swimming distance to get to and from shore. As the ratio of running speed to swimming speed approaches infinity, the optimal path gets closer and closer to a trapezoid.

In the case of a ball thrown parallel to the shore, x2 = x1, and the bifurcation point is:


Because Elvis lives to fetch, it was possible to set up an experiment to test his bifurcation prowess. Pennings and two undergraduate students took Elvis to the beach and first conducted some time trials. They found that Elvis's running speed is about 6.39 meters per second and his swimming speed about 0.73 meter per second. However, when later chasing a ball, Elvis's running speed dropped to a leisurely 3.02 meters per second (it was, after all, a lazy July afternoon—and swimming is tiring). In this case, according to the mathematical model, the optimal bifurcation point is 2.56x.

The researchers conducted nine trials. Each time, Pennings stood 4 meters out in the water with Elvis and threw a ball various distances, but still 4 meters from the shore. One student measured the distance of the throw and the other recorded Elvis's choice.

Elvis was certainly smart enough to consistently take the swimming-only route for shorter distances and the swim-run-swim path for longer distances. For Elvis, the bifurcation point appeared to be somewhere between 14 and 15 meters. However, the formula predicts it should have been 10.24 meters. So Elvis missed the mark in three of the nine trials, choosing to swim rather than to take the longer but (in principle) quicker route.

"Thus," Pennings and Minton remark, "Elvis knows bifurcations qualitatively, but not quantitatively."

References:

Minton, R., and T.J. Pennings. 2007. Do dogs know bifurcations? College Mathematics Journal 38(November):356-361.

Pennings, T.J. 2003. Do dogs know calculus? College Mathematics Journal 34(May):178-182.

Peterson, I. 2006. Calculating dogs. MAA Online (Feb. 20).

______. 2003. A dog, a ball, and calculus. MAA Online (June 9).

October 18, 2007

Norwegian Sol

In the forested mountains near Bergen, Norway, a glowing orb brightens the long nights of winter. The source of light is an illuminated artwork called "Sol," created by Norwegian artist Finn Eirik Modahl.


Created by sculptor Finn Eirik Modahl, "Sol" illuminates a forested landscape near Bergen, Norway. Photos courtesy of Finn Eirik Modahl.

Resting in the hollow of a concrete base that resembles tiers of frozen lava, the central structure is a geodesic truncated icosahedron constructed from steel and glass. About 6 feet tall, it shines with a warming glow—and invites visitors to come near and touch the sun.


The sculpture "Sol" is based on a geodesic truncated icosahedron.

"The concept is meant to appeal to heart and mind," Modahl says. "I want contemporary art to create a dynamic, giving the people living around the sculpture a chance to take part in it."


Children touch and clamber about on the geometric form at the heart of Modahl's "Sol."

"Sol" is one of several installations at the Pulse Sculpture Complex, which integrates artworks with innovative dwelling designs.


Glowing mysteriously in the dark, Modahl's "Sol" invites exploration and wonder.

In an earlier effort, unveiled in 2000, Modahl created a dramatic obelisk, set at the heart of grassy expanse threaded by a spiral walkway—surrounded by mountains and overlooking a fjord. Called "Elektra" and located in Tyssedal, Norway, the towering monument is made of granite and glass. The stone pillars were "cracked" rather than carved into their final form. The glass inclusion—a thick vertical vein in the stone—was made from hundreds of hand-cut glass plates, which glow with an eerie green light. Modahl describes the hue as the color of a mountain lake.

Earlier in his career, Modahl had devoted his time to installation and performance art, particularly in exploring interactions between art and commerce. For two years, he was director of the Bomuldsfabriken Kunsthall, one of the leading venues for contemporary art in Norway. Lately, in developing the concepts for his monumental sculptures, Modahl has turned more and more to mathematics for inspiration and structure.

October 1, 2007

Spontaneous Knotting

Coil a long rope into a box, then shake the box. It's very likely that the rope will emerge bearing a knot.

The same perverse tendency of spontaneous knot formation also plagues necklaces loosely stored in drawers or headphone cords stuffed into pockets or backpacks.


A coiled headphone cord, once unraveled, is likely to contain a knot.

Now, physicists have investigated this pervasive phenomenon, performing experiments to identify the physical factors that lead to spontaneous knot formation and applying mathematical knot theory to analyze the resulting knots. Dorian M. Raymer and Douglas E. Smith of the University of California, San Diego describe their findings in a paper to be published in the Proceedings of the National Academy of Sciences.

The researchers discovered that, when a string is tumbled inside a box, complex knots often form within seconds—if the string is long and flexible enough. For example, for a string of diameter 3.2 millimeters, coiled inside a cubic box 0.3 meter wide, which was spun at 1 revolution per second for 10 seconds, no knots formed when the string was shorter than 0.46 meter. When a tumbled string had a length between 0.46 and 1.5 meter, the probability of knot formation went up sharply.


This illustration represents the knot experiment, in which a knot forms in a tumbled string. Dorian Raymer, UCSD.

Overall, the experiments suggest that the probability of knot formation increases to nearly 100 percent for long agitation times, long length, and high flexibility.

To determine what knots had formed after a string was tumbled, the physicists lifted the ends of the string directly upward from the box, then joined the ends to form a closed loop. They photographed the loop, creating a two-dimensional knot diagram showing where the loop crossed over or under itself. The researchers then applied mathematical knot theory—calculating each tangle's Jones polynomial—to identify the knot.

The analysis produced a striking result. In 3,415 trials, Raymer and Smith observed knot formation 1,127 times, identifying 120 different types of knots, having a minimum crossing number up to 11. Moreover, the researchers found instances of every prime knot with up to seven crossings. The simplest possible knot—a trefoil knot—has three crossings. Composite knots—for example, a pair of trefoil knots—formed far less frequently than did prime knots.


Digital photos of knots in strings are combined with computer-generated knot diagrams. Dorian Raymer, UCSD.

The fact that the majority of the observed knots were prime suggests that knotting tends to occur from one end. Once a knot starts at one end, it's highly unlikely (though occasionally still possible) for a knot to start at the other end. Indeed, knot theory dictates that, if a separate knot is formed at each end of a string, the knots can slide together at the center of the string but can't merge to form a single prime knot.

The researchers also developed a simple model to describe knot formation on the basis of random "braid moves" of a string's free end. With multiple parallel strands lying near a string end, knots form when the end weaves under and over adjacent segments. In effect, the string end traces a path that corresponds to the knot topology.

According to a basic theorem of knot theory, such braid moves can generate all possible prime knots. This is consistent with the observation that the physicists found all prime knots with up to seven crossings among their tumbled strings.

So, what are the chances of ending up with a knot when you unravel your carefully coiled headphone cord? They're annoyingly high, especially if the cord is long, thin, and flexible.

References:

2007. Unraveled: The mystery of why strings tangle. New Scientist (Sept. 30).

Peterson, I. 1997. Knotted walks. MAA Online (Nov. 3).

Raymer, D.M., and D.E. Smith. In press. Spontaneous knotting of an agitated string. Proceedings of the National Academy of Sciences.

Seethaler, S. 2007. UC San Diego physicists tackle knotty puzzle. UCSD News Center (Oct. 1).

September 25, 2007

Folding a Klein Bottle

The intriguing mathematical object known as a Klein bottle has only one surface. In effect, it has an outside, but no inside.

In abstract terms, you can make a Klein bottle by starting with a rectangular piece of the plane. You join two opposite edges to create a tube. You then join the remaining two edges, but with the same sort of twist (so that the joined edges have opposite orientation) that you need to make a Möbius strip. The second step can't be carried out without the tube passing through itself.


A true Klein bottle exists only in a topologist's imagination. It has to intersect itself without the presence of hole—a physical impossibility. Nonetheless, that hasn't stopped illustrators and glassblowers from attempting to represent this remarkable object in some physical form. Movies.

To make one from glass, for instance, you could start with a stretchable glass tube. One end of the tube narrows into a long neck; the other end widens into a base. The neck passes through the side of the bottle, and the neck and the base join to make the neck's inside continuous with the base's outside.

Astronomer Cliff Stoll is among those who have succeeded in crafting glass Klein bottles. He now offers them for sale, in a wide variety of sizes and manifestations, at Acme Klein Bottle.


A genuine Acme Klein Bottle. Photo by I. Peterson.

Several years, Stoll invited Robert J. Lang, creator of intricate, highly original origami constructions—insects, lobsters, birds, mammals, cuckoo clocks, and more—to construct a Klein bottle from a sheet of paper. It wasn't easy, but Lang met the challenge, folding a 13-inch-by-30 inch rectangle into a 7-inch-tall paper model.



Lang later made available the crease pattern that he used to create his paper Klein bottle. Lang displayed his Klein bottle and various other objects at MathFest in August when he appeared at the A K Peters booth to sign copies of his book Origami Design Secrets: Mathematical Methods for an Ancient Art.


There are other frontiers in the realm of Klein bottle visualization. The bottle form is just one of several ways to represent this mathematical object. Indeed, the Klein surface can take on many shapes that don't look at all like bottles. One version looks like a ring pinched so that it has a figure-eight cross section. But the orientation of the figure eight changes in going around the ring.


On the construction side, Andrew Lipson specializes in building mathematical objects out of LEGO bricks. He has made a hinged Klein bottle out of bricks and other pieces. His hinged construction shows how a Klein bottle, cut in half lengthwise, becomes two Möbius strips.

References:

Banchoff, T.F. 1998. Surfaces Beyond the Third Dimension: The Klein bottle. In Communications in Visual Mathematics.

Lang, R. 2003. Origami Design Secrets: Mathematical Methods for an Ancient Art. A K Peters.

Peterson, I. 2001. Immersed in Klein bottles. MAA Online (Feb. 19).

September 12, 2007

The CMJ Hit Parade

Among the three mathematics journals published by the Mathematical Association of America, The College Mathematics Journal (CMJ) has the shortest history. It started in 1970 as The Two-Year College Mathematics Journal, published by Prindle, Weber & Schmidt in collaboration with the MAA. Issues initially appeared twice a year. As the title suggests, its focus was on the teaching of mathematics in two-year colleges. It was meant to be a forum for mathematicians interested in the special curricular and pedagogical challenges presented by such institutions.

In 1984, the publication became The College Mathematics Journal, published five times per year. Instead of focusing solely on the concerns of two-year colleges, the renamed journal was to serve all who were interested in the earlier years of college-level mathematics, especially the first two years. Editors looked for lively articles to enrich instruction and enhance classroom learning. Articles highlighted history, people, philosophy, problem-solving, applications, and more. And there was room for proofs without words, poetry, quotations, cartoons, and other such items.

All CMJ issues, as long as they are more than three years old, are now available on the Web through the JSTOR archive, a database to which many college libraries and other institutions subscribe.

Use of the JSTOR archive has been increasing steadily over the years since CMJ first became available in this form in 2002, and it's now possible to see which of its hundreds of articles have been particularly appealing or useful. Here are the fifteen most frequently viewed articles to date:
  1. "Learning Mathematics Through Writing: Some Guidelines" by J.J. Price. November 1989, 20:393-401.
  2. "Misconceptions about the Golden Ratio" by George Markowsky. January 1992, 23:2-19.
  3. "The History of the Calculus" by Carl B. Boyer. Spring 1970, 1:60-86.
  4. "Isaac Newton: Man, Myth, and Mathematics" by V. Frederick Rickey. November 1987, 18:362-389.
  5. "Studying Students Studying Calculus: A Look at the Lives of Minority Mathematics Students in College" by Uri Treisman. November 1992, 23:362-372.
  6. "Math Anxiety: Some Suggested Causes and Cures: Part 1" by Peter Hilton. June 1980, 11:174-188.
  7. "Evolution of the Function Concept: A Brief Survey" by Israel Kleiner. September 1989, 20:282-300.
  8. "The Fractal Geometry of Mandelbrot" by Anthony Barcellos. March 1984, 15:98-114.
  9. "Cryptology: From Caesar Ciphers to Public-Key Cryptosystems" by Dennis Luciano and Gordon Prichett. January 1987, 18:2-17.
  10. "Collegiate Mathematics Education Research: What Would That Be Like?" by Annie Selden and John Selden. November 1993, 24:431-445.
  11. "The Volume and Centroid of the Step Pyramid of Zoser" by Anthony Lo Bello. September 1991, 22:318-322.
  12. "Women Mathematicians" by Debra Charpentier. March 1977, 8:73-79.
  13. "The Growing Importance of Linear Algebra in Undergraduate Mathematics" by Alan Tucker. January 1993, 24:3-9.
  14. "Pascal's Triangle" by Karl J. Smith. Winter 1973, 4:1-13.
  15. "The Derivative of Arctan x" by Norman Schaumberger. September 1982, 13:274-276.

I was very pleased to see one of my favorite articles on the list: George Markowsky's debunking of various myths commonly associated with the golden ratio. Like the list of top articles for Mathematics Magazine, the CMJ list also suggests a strong interest in history. Authors Israel Kleiner and Carl B. Boyer have articles on both lists. Boyer's account of the history of calculus, published in the first issue of The Two-Year College Mathematics Journal, is a reprint of an article that originally appeared in 1969 in Historical Topics for the Classroom (Thirty-First Yearbook of the National Council of Teachers of Mathematics).

August 30, 2007

Rectangular States and Kinky Borders

On many maps of the United States, the states of Colorado and Wyoming appear to be rectangles.


Indeed, the enabling legislation creating the two states specifies their extent strictly in terms of lines of latitude and longitude, rather than rivers, mountain divides, or other geographical features. Wyoming stretches from 41°N to 45°N latitude and from 104° 3'W to 111° 3'W longitude. Colorado's borders lie between 37°N and 41°N latitude and between 102° 3'W and 109° 3'W longitude. Originally, the lines of longitude were measured west from Washington, D.C.

However, on the surface of a sphere, although lines of latitude are parallel, lines of longitude converge as you go north or south away from the equator. So, the northern border of each state is a little shorter than its southern border. The difference for Colorado is about 21 miles.

Hence, to a first approximation, both states appear to be trapezoids rather than rectangles, though on a curved surface. A trapezoid has one set of parallel sides.

But there's an additional wrinkle that complicates the picture.

When the states were created, surveyors mapped their boundaries using transit and compass, chronometer, and astronomical readings. They also relied on data from previous surveys and interviews with residents of the affected areas. Following the appropriate lines of latitude and longitude as best as they could, the surveyors established the borders, marking them from milepost to milepost over distances stretching hundreds of miles.

The boundary between Utah and Colorado runs 276 miles from Four Corners (the only place in the United States where four states share a point) to the Wyoming border, for example. On its northward trek, the original survey ended up about 1 mile west of where the surveyors had expected to intersect the Wyoming line, indicating that the surveyed border had at least one kink in it. Indeed, subsequent surveys revealed a discrepancy between mileposts 81 and 89 (northward from Four Corners) and another between mileposts 100 and 110. The errors put kinks into what should have been a straight line.

There were similar surveying errors along other borders, including those that define Wyoming. Interestingly, once a border is defined on the ground and accepted by the interested parties, it becomes official, even if it doesn't follow the written description.

So, perhaps it's best to describe Colorado and Wyoming as polygons. And Utah can then join the group of states that are polygons. The trickier question is determining exactly how many sides these polygons have.

Colorado's legal border, for example, "is a polygon formed by a series of line segments that run between physical monuments that were put in place by . . . survey parties," Stan Wagon and John J. Watkins comment in the September College Mathematics Journal. "This polygon has hundreds of sides."

It's time to take a closer look at the borders and start counting. The main kink in the Utah-Colorado border is visible with a few clicks on a Google map, as is an abrupt jog in the border between Colorado and New Mexico.

References:

2007. Which states are polygons? College Mathematics Journal 38(September):259.

Case, W.F. Why does the eastern border of Utah have a kink in it? Utah Geological Survey.

Van Zandt, F.K. 1976. Boundaries of the United States and the several states. U.S. Geological Survey Professional Paper 909.

August 23, 2007

The Number Pad Game

The number pad of a computer keyboard or calculator presents a three-by-three array of the digits from 1 to 9. (We exclude the digit 0, which is usually in a row by itself.) This array of numbers can serve as the basis of a simple, two-player game.


In the game, the first player turns on (or clears) the calculator, presses a digit key, then presses the + key. The second player responds by pressing a digit key in the same row or column as the last digit key pressed by the first player, except the key pressed by the first player. So, if player A presses 5, player B can respond with 4, 8, 6, or 2, but not with 5.

The first player responds to the second player's move in the same way. The two players take turns alternately until a player reaches a sum greater than a specified amount, say 30. That player loses.

Is there a winning strategy for this game? How does it depend on the specified "fatal" total?

"You can visualize the game as being played by stacking wooden blocks on top of one another," Alex Fink and Richard Guy write in the September College Mathematics Journal. "When the height exceeds 30 blocks, the stack topples and the player who was responsible loses."

For a tower of "tolerable" height 30, player A can win by touching 9 on the first move. No matter how B moves, A can always ensure that B exceeds 30. A can also win by starting with 3, but B can prolong the game by touching 1 at each turn.

Fink and Guy also work out the winning moves for any "tallest tolerable tower." For example, as noted above, the winning initial moves for 30 are 3 and 9. Interestingly, there is always a winning first move for the first player unless the tallest tolerable towers are 27, 43, or 64 in height. For 27, 43, or 64, there is no such strategy for the first player.

For shorter towers, you have to pay attention and pick your opening move carefully. However, that changes for heights greater than 107. "For towers taller than 107, you can always win by playing 3, 5, or 7—your opponent can never reply with one of these numbers, and whatever is played, you can reply with either of two of them," Fink and Guy report.

You can also reverse the rules and play the game so that the first player to exceed the total wins (rather than loses). Under this rule, there are no winning moves for towers of 12, 42, 76, 97, and 40k + 114. Curiously, beyond 124, the set of winning moves repeats itself with a period of 80. So, the same winning initial moves work for towers of 125 and 205, and so on!

Other variants of the game are possible. Fink and Guy consider configurations that include the 0 key in a separate row and look at how what happens depends on whether the key is in the middle, left, or right column—or even straddles two columns.


Any of these game variants can be a pleasant pastime. In some cases, the patterns of winning moves are simple enough to master that you could have a distinct advantage over any friends who dare to take on the challenge.

References:

Fink, A., and R. Guy. 2007. The number-pad game. College Mathematics Journal 38(September):260-264.

August 16, 2007

Symmetries of Beaded Beads

To create a beaded bead, you need a supply of beads, a needle with plenty of thread, and a lot of patience and care. It also helps to consult the innovative designs and patterns of Gwen Fisher and Florence Turnour. The result is an elegant form of beadwork—sparkling, colorful clusters of beads typically woven around one or more large holes.


Fisher is a mathematician at California Polytechnic State University in San Luis Obispo, so it's natural for her to look at these creations mathematically. Indeed, many beaded beads can be viewed as polyhedra, where the hole through the middle of each bead corresponds to a polyhedron's edge.

Different weaving patterns bring different numbers of these "edges" together to form the vertices of a polyhedron, Fisher says. The holes of 12 beads, for example, can be strung together to form the 12 edges, eight vertices, and six faces of a cubic bead cluster.

Such polyhedral beaded beads also have particular symmetries, classified by the three-dimensional finite point groups. Earlier this month at MathFest 2007 in San Jose, Calif., Fisher described weaving techniques that allow the realization of beaded beads with all the possible symmetries of polyhedra.


"Any polyhedron can be modeled as a beaded bead," Fisher and Blake Mellor of Loyola Marymount University write in a recent paper published in the Journal of Mathematics and the Arts. In other words, given any polyhedron, it's possible to weave a beaded bead with the same set of symmetries.

"The challenge is to create the patterns to accomplish this so that we are also creating beaded bead that are objects of beauty in their own right," the authors note. "In meeting this challenge, we developed many new designs that may not have been created otherwise." Fisher displayed many of the resulting bead clusters at MathFest.

A symmetry of a geometric object is a rigid transformation, such as a reflection or rotation, that leaves the appearance of the object unchanged. Any rotation or reflection of a sphere leaves the object unchanged, so the symmetry group of a sphere is the infinite collection of these motions, commonly designated O(3). Any given polyhedron shares some finite collection of these symmetries, so its symmetry group is a finite subgroup of O(3).

These infinitely many subgroups divide naturally into 14 classes: the prismatic groups, which correspond to the seven infinite frieze groups (border patterns); and seven additional groups related to the symmetric groups of the Platonic solids.

Fisher notes that beaded beads with any of the prismatic symmetries can be created using a weaving technique, known as the fringe method, that she and Turnour developed.


Other techniques allow the realization of such exotic objects as the beaded Sierpinski tetrahedron.

"Bead weaving is a phenomenally rich medium for creating mathematical art in three dimensions," Fisher and Mellor conclude. And the construction of beaded beads leads to additional mathematical questions. For example, what is the minimum number of beads required to realize a given point group? What is the minimal length of thread required?

References:

Fisher, G.L., and B. Mellor. 2007. Three-dimensional finite point groups and the symmetry of beaded beads. Journal of Mathematics and the Arts 1(No. 2):85-96.

Peterson, I. 2007. Knitting network. MAA Online (Jan. 29).

July 19, 2007

Tom Lehrer's Derivative Ditties

Way back in the 1960s, one of my favorite TV shows was a satire program called That Was The Week That Was (also known as TW3), which aired weekly on NBC. It was the U.S. version of a late-night British series that was the talk of the country in the early 60s. The U.S. edition featured Elliot Reed, Alan Alda, Phyllis Newman, Henry Morgan, singer Nancy Ames, and, later, David Frost.

One of the program's highlights was its opening song, with lyrics that changed every week to reflect current news events. The show's resident songwriter was Tom Lehrer, who tackled and parodied such subjects as pollution ("Pollution"), Vatican II ("The Vatican Rag"), race relations ("National Brotherhood Week"), education ("New Math"), and nuclear proliferation ("Who's Next?"). A selection of these songs (uncensored) subsequently appeared on a record album titled That Was The Year That Was.

The fact that some of Lehrer's songs have mathematical elements reflects his background. Lehrer earned B.A. and M.A. degrees in mathematics from Harvard University and even entered Harvard's doctoral program, though he never completed his degree. In 1972, he came to the University of California, Santa Cruz, often teaching an introductory course titled "The Nature of Mathematics." He also taught a class in musical theater and would, on occasion, perform songs in his lectures.

In 1974, The American Mathematical Monthly published three Lehrer songs devoted to higher mathematics (mainly calculus). The May issue featured "The Derivative Song," sung to the tune of "There'll Be Some Changes Made," composed in 1922 by Billy Higgins and W. Benton Overstreet. It was originally performed as part of The Physical Revue, a show presented by the Harvard Physics Department in 1951 and 1952.
You take a function of x and you call it y,
Take any x-nought that you care to try,
You make a little change and call it delta x,
The corresponding change in y is what you find nex',
And then you take the quotient and now carefully
Send delta x to zero, and I think you'll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It's just dy/dx.

The June-July issue featured "There's a Delta for Every Epsilon (Calypso)." It was also performed earlier in The Physical Revue.
There's a delta for every epsilon,
It's a fact that you can always count upon.
There's a delta for every epsilon
   And now and again,
   There's also an N.

But one condition I must give:
The epsilon must be positive
A lonely life all the others live,
   In no theorem
   A delta for them.

How sad, how cruel, how tragic,
How pitiful, and other adjec-
Tives that I might mention.
The matter merits our attention.
If an epsilon is a hero,
Just because it is greater than zero,
It must be might discouragin'
To lie to the left of the origin.

This rank discrimination is not for us,
We must fight for an enlightened calculus,
Where epsilons all, both minus and plus,
   Have deltas
   To call their own.

"The Professor's Song" appeared in the August-September issue. It is sung to the tune of "If You Give Me Your Attention" from the Gilbert and Sullivan operetta Princess Ida.
If you give me your attention, I will tell you what I am.
I'm a brilliant math'matician — also something of a ham.
I have tried for numerous degrees, in fact I've one of each;
Of course that makes me eminently qualified to teach.
I understand the subject matter thoroughly, it's true,
And I can't see why it isn't all as obvious to you.
Each lecture is a masterpiece, meticulously planned,
Yet everybody tells me that I'm hard to understand,
   And I can't think why.

My diagrams are models of true art, you must agree,
And my handwriting is famous for its legibility.
Take a word like "minimum" (to choose a random word), (*)
For anyone to say he cannot read that, is absurd.
The anecdotes I tell get more amusing every year,
Though frankly, what they go to prove is sometimes less than clear,
And all my explanations are quite lucid, I am sure,
Yet everybody tells me that my lectures are obscure,
   And I can't think why.

Consider, for example, just the force of gravity:
It's inversely proportional to something — let me see —
It's r3 — no, r2 — no, it's just r, I'll bet —
The sign in front is plus — or is it minus, I forget —
Well, anyway, there is a force, of that there is no doubt.
All these formulas are trivial if you only think them out.
Yet students tell me, "I have memorized the whole year through
Ev-rything you've told us, but the problems I can't do."
   And I can't think why!

(*) This was performed at a blackboard and the professor wrote: /VVVVVVVVVVVVVVV.

An earlier version of "The Professor Song," performed in The Physical Revue, has somewhat different lyrics.

Lehrer continued to return to mathematical themes in his songwriting. In 1993, for example, he composed the song "That's Mathematics!" for a special program, sponsored by the Mathematical Sciences Research Institute in Berkeley, to commemorate the proof by Andrew Wiles of Fermat's Last Theorem. At the Fermat Fest, Morris Bobrow performed not only "That's Mathematics!" but also "The Derivative Song" and "There's a Delta for Every Epsilon." And he rounded out the musical part of the program with a Danny Kaye classic, "The Square of the Hypotenuse," written by Saul Chapin and Johnny Mercer for the movie Merry Andrew. The Fermat Fest program is available as a video from MSRI.

References:

2003. Stop clapping, this is serious. Sydney Morning Herald (March 1).

Lehrer, T. 1974. The professor's song. The American Mathematical Monthly 81(August-September):745.

______. 1974. There's a delta for every epsilon (calypso). The American Mathematical Monthly 81(June-July):612.

______. 1974. The derivative song. The American Mathematical Monthly 81(May):490.

Peterson, I. 2000. Square of the hypotenuse. MAA Online (Nov. 27).

Purdom, T.S. 2000. Still a sly wit, now mostly for himself. New York Times (July 16).

July 6, 2007

Binary Frieze

Like a surreal piano keyboard, it stretches in a narrow band, high along the walls of the cavernous lobby of the Ernst & Young office building in downtown Toronto. Created in 1993 by artist Arlene Stamp, this artwork is based on a visual representation of binary numbers.


A base 2, or binary, number system has only two digits: 0 and 1. In place of the 1s, 10s, 100s, and 1,000s columns (representing powers of 10), the columns of a binary number represent powers of 2. The right-hand binary digit is in the 1s column, the next digit to the left is the 2s column, the next digit farther to the left is in the 4s column, and so on. So the base-10 number 7 would be 111 in base 2; 20 would be 10100; and 255 would be 11111111.

Using black squares to represent 0s and white squares to represent 1s, Stamp filled in a strip of graph paper that was 8 squares wide and 256 squares long with the sequence of binary numbers from 00000000 to 11111111. To her amazement, the resulting pattern resembled a fractal, with each section branching into smaller and smaller fingers.


"Suddenly, I could see the inextricable link between pictures of fractals generated by computers and simple binary numbers, which underlie the structure of the computer itself," Stamp says.


For the Ernst & Young building, Stamp constructed her "Binary Frieze" out of vinyl floor tiles, which were cut by hand, then assembled and glued onto aluminum supports. To perturb the relentless linearity of the fundamental binary number pattern, she also introduced a wave effect and an element of unpredictability by tilting and overlapping adjacent rectangles in certain sections of the mosaic.

The wave effect results from moving a rectangle through two functions, the first determining the width of overlap between adjacent positions of a rectangle and the second determining the degree of tilt. "The interesting thing about these interrelated functions is that they never fall into synchronization, so you cannot predict a particular position ahead," Stamp says. "Each new position of the rectangle is dependent on the previous step and the system flips into reverse when it reaches a particular limit, causing an overall wave pattern."


At the time, Stamp says, "I was very interested in the possibilities for non-repeating patterns in public spaces as a way of enlivening large areas, like floors and walls, that are so often covered up with simple repeating motifs."


"The expansive walls of the Ernst & Young lobby offered me an unusual opportunity to explore nonperiodic pattern over a large area, visible from a distance—perfect for the material presentation of the unfolding fractal pattern of binary numbers," she adds.

A second nonrepeating mosaic design, created by Stamp for the Downsview subway station in Toronto the same year, is based on the digits of pi.

References:

Peterson, I. 2002. Tiling with pi. Math Horizons 10(November):11-15.

______. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

______. 2000. Sliding pi. MAA Online (June 5).

June 28, 2007

Wrapping a Perfect Sphere

The famous Austrian candy known as Mozartkugel (Mozart sphere) consists of a marzipan core that is smothered in nougat or praline cream, then coated with dark chocolate. Many companies make Mozartkugeln, but one maker, Mirabell, claims that its product is unique. It's the only Mozartkugel that's a perfect sphere.


Mozartkugeln made by Fürst. Photo by Clemens Pfeiffer, Vienna, Austria.

Each sphere comes wrapped in a square of aluminum foil. So, to minimize waste, it's natural to ask about the smallest piece of foil that would cover a sphere. It doesn't have to be a square, but it would be helpful if the unfolded shape would tile the plane. In other words, the pieces could then be cut from a large sheet of foil without any waste.

That's the tantalizing question that Erik Demaine, Martin Demaine, John Iacono, and Stefan Langerman recently bit into. They presented their findings earlier this year at the 23rd European Workshop on Computational Geometry, held in Graz, Austria.

Mathematically, the problem involves transforming a flat sheet into a positive-curvature sphere. Normally, folding (as in mathematical origami) preserves distances and curvature. Instead, the wrapping task requires infinitely many, infinitesimally small "folds" (but no stretching) to accomplish the required transformation. So, Demaine and his collaborators define a wrapping as "a continuous contractive mapping of a piece of paper into Euclidean 3-space." Contractive means that every distance either decreases or stays the same, as measured by shortest paths on the piece of paper before and after mapiing via the folding.

The analysis shows that a square with diagonal 2π and area 2π2 covers a unit sphere. No smaller square can serve as a wrapping.

It's interesting to note, Demaine and his colleagues say, that a rectangle of dimensions 2π by π has the same area. It, too, can wrap around a sphere. Indeed, Mirabell wraps its Echte Salzburger Mozartkugel using such a rectangle, expanded a bit to ensure overlap.

An equilateral triangle will also do the job. In this case, the requisite triangle's area is about 0.1 percent less than the 2π2 area of the smallest wrapping square.

Demaine and his colleagues also come up with a three-petal configuration (below) that tiles the plane. In this case, each wrapping unit takes up an area of only about 1.6π2.


"This paper initiates a new research direction in the area of computational confectionery," the researchers conclude. "We leave as open problems the study of wrapping other geometric confectioneries, or further improving our wrappings of the Mozartkugel."

References:

Demaine, E.D., M.L. Demaine, J. Iacono, and S. Langerman. 2007. Wrapping the Mozartkugel. In Abstracts of the 20th European Workshop on Computational Geometry.

Karafin, A. 2007. "Puzzles Will Save the World." Boston Globe (June 24).

June 8, 2007

Math Mag's Greatest Hits

In its various incarnations, Mathematics Magazine, which is published by the Mathematical Association of America (MAA), has been around for more than 75 years. All issues of the Magazine, as long as they are more than three years old, are available on the Web through the JSTOR archive, a database to which many college libraries and other institutions subscribe.

Use of the JSTOR archive has been increasing steadily over the years since the Magazine first became available in this form in 2002, and it's now possible to see which of its hundreds of articles have been particularly appealing or useful. Here are the fifteen most frequently viewed articles to date:
  1. "The Golden Section and the Piano Sonatas of Mozart" by John F. Putz. October 1995, 68:275-282.
  2. "Rigor and Proof in Mathematics: A Historical Perspective" by Israel Kleiner. December 1991, 64:291-314.
  3. "The Influence of Mathematics on the Philosophy of Descartes" by R.H. Moorman. April 1943, 17:296-307.
  4. "Hypatia of Alexandria" by A.W. Richeson. November 1940, 15:74-82.
  5. "Humanizing Mathematics" by R.C. Archibald. November 1932, 7:8-11.
  6. "The 2500-Year-Old Pythagorean Theorem" by Darko Veljan. October 2000, 73:259-272.
  7. "The Number System of the Mayas" by Gary D. Salyers. September 1954, 28:44-48.
  8. "The Evolution of Group Theory: A Brief Survey" by Israel Kleiner. October 1986, 59:195-215.
  9. "Zero: The Symbol, the Concept, the Number" by Carl B. Boyer. March 1945, 18:323-330.
  10. "Descartes and Problem-Solving" by Judith Grabiner. April 1995, 68:83-97.
  11. "Assigning Driver's License Numbers" by Joseph A. Gallian. February 1991, 64:13-22.
  12. "Mathematics and Literature" by D.O. Koehler. March 1982, 55:81-95.
  13. "Good-bye Descartes?" by Keith Devlin. December 1996, 69:344-349.
  14. "Another Proof of the Fundamental Theorem of Algebra" by Daniel J. Velleman. June 1997, 70:216-217.
  15. "The Evolution of Mathematics in Ancient China" by Frank Swetz. January 1979, 52:10-19.
Interestingly, the list includes articles from every decade of Mathematics Magazine's existence, except the 1920s. The topics are widely varied, though there is a distinct bias toward the history of mathematics. And the name Descartes appears in three of the titles! Curiously, the article "Humanizing Mathematics" is simply a four-page list of books (including several devoted to puzzles and recreational mathematics) and articles (selected mainly from the Mathematics Teacher), compiled by R.C. Archibald of Brown University in 1932.

Several of the top articles are also available in the new MAA book The Harmony of the World: 75 Years of Mathematics Magazine, which features 38 selections from the archive.

For anyone who does not have access to the JSTOR archive at a local library, these articles will soon be available individually from JSTOR for a fee (and they can be found in a Google search). MAA members also have the option of obtaining an individual subscription to JSTOR, which provides access to the full archives of not only Mathematics Magazine but also The American Mathematical Monthly and The College Mathematics Journal.

References:

Alexanderson, G.L., ed. 2007. The Harmony of the World: 75 Years of Mathematics Magazine. Mathematical Association of America.

June 4, 2007

Unwrapping Curves

In Mathematical Snapshots, Hugo Steinhaus described the following method for generating an interesting curve. Wrap a strip of paper around a cylindrical candle. Using a sharp knife, slice the wrapped candle into two pieces, making the cut at an oblique angle. The candle's cross section at the location of the cut is an ellipse. Unwinding each of the resulting paper scraps reveals a sinusoidal edge where the paper was cut.


An elliptical cross section of a cylinder becomes sinusoidal when unwrapped.

A number of years ago, Mamikon A. Mnatsakanian showed that you can get the same result when you dip a cylindrical paint roller at an angle into a container of paint. When you roll the applicator on a flat surface, you also see a sinusoidal wave pattern.


A paint roller can be used to print sinusoidal waves on a flat surface.

Now, Mnatsakanian and Tom M. Apostol have extended Steinhaus's original demonstration further to include a wide variety of cross sections. "Imagine the elliptical cross section replaced by any curve lying on the surface of a right circular cylinder," they write in the May American Mathematical Monthly. "What happens to this curve when the cylinder is unwrapped?"

Apostol and Mnatsakanian also consider the inverse problem. What happens to curves on flat surfaces that are then rolled into cylinders? You can experiment with such transformations yourself simply by drawing curves (lines, circles, parabolas, and so on) on rectangular sheets of transparent plastic, then seeing what happens when these sheets are rolled into cylinders of different radii.

"A few trials reveal an enormous number of possibilities, even for the simple case of a circle," the mathematicians note.


Rolling a circle drawn on a sheet of transparent plastic into cylinders of various radii produces strikingly different geometric patterns. The dashed curves are on the rear half of the cylinder.

Working with just a straight line segment drawn on a transparency produces striking visual evidence of the nature of geodesics on a cylinder. A line segment on a flat sheet is the shortest path between the line's endpoints. Because distances are preserved when a cylinder is unwrapped, such a line segment, when rolled into a cylinder, becomes a geodesic (or the shortest path) on the cylinder. You see, no matter what the cylinder's radius, that such a geodesic on a circular cylinder is always part of a circular helix. Viewed from the side, the pattern looks like a sine wave.

For circular cylinders, Apostol and Mnatsakanian note, a sinusoidal influence is always present.

They go on to investigate the curves that result from drilling holes into cylinders and from unwrapping curves from a right circular cone.

A plane, for example, cuts a cone along a conic section. "We can analyze the shape of the corresponding unwrapped conic," Apostol and Mnatsakanian report. "This leads to a remarkable family of periodic plane curves that apparently have not been previously investigated."

References:

Apostol, T.M., and M.A. Mnatsakanian. 2007. Unwrapping curves from cylinders and cones. American Mathematical Monthly 114(May):388-416.

Steinhaus, H. 1969. Mathematical Snapshots, 3rd. ed. Oxford University Press.

May 29, 2007

Random-Turn Hex

In most two-person games, players take turns, proceeding from turn to turn in an orderly manner. If, instead of alternating turns, the players use a coin toss to decide who makes each move, a deterministic pastime becomes a random-turn game.

Interestingly, random-turn games can be easier to analyze than their deterministic counterparts, Yuval Peres, Oded Schramm, Scott Sheffield, and David B. Wilson report in the May American Mathematical Monthly. Such games also "exhibit surprising structure and symmetry," the researchers say.

The game of Hex, for example, is played on a diamond-shaped board made up of hexagons. Each player has pieces (or stones) of a particular color. The two players alternately place stones of their respective colors on unoccupied hexagons. A player wins by completing an unbroken chain of stones, creating a path that connects the two opposite sides of his or her color.

The game can't end in a tie. One player can block the other only by completing his or her own chain. It's possible to prove that there exists a winning strategy for the first player on a board of any size, but there is no known optimal strategy for the standard 11 by 11 board (or for larger boards).

In random-turn Hex, the players toss a coin to decide who places the next stone. Peres and his colleagues show that the probability that the first player wins when both players play optimally is the same as the probability that the first player wins when both players play randomly.

So, winning the game is akin to creating a path that crosses from one side of the board to the other after filling in empty hexagons at random—a process known as independent Bernoulli percolation.

"In this and other games, the set of moves played during an entire game (when both players play optimally) has an intriguing fractal structure," Peres and his colleagues observe.

The researchers also prove that, in a random-turn selection game such as Hex, any optimal strategy for one of the players is also an optimal strategy for the other player.

It turns out that the best first move in random-turn Hex is the site that is most likely to be crucial for a percolation crossing. For the standard board, the best first move is near the center.

The authors note that "random-turn games are natural models for real-world conflicts, where opposing agents (political parties, lobbyists, businesses, militaries, etc.) do not alternate turns. Instead, they continually seek to improve their positions incrementally."

References:

Browne, C. 2000. Hex Strategy: Making the Right Connections. A K Peters.

Gardner, M. 1959. The game of Hex. In Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Mathematical Puzzles and Games. University of Chicago Press.

Peres, Y., O. Schramm, S. Sheffield, and D.B. Wilson. 2007. Random-turn hex and other selection games. American Mathematical Monthly 114(May):373-387.

May 10, 2007

Closet Doors

As anyone who lives in a tiny apartment or cluttered room has experienced, opening or closing a closet door requires a certain amount of floor space.

Suppose, for example, that a closet has a standard door, which can swing open so that it's at right angles to the closet interior. As it opens (or closes), the door covers a quarter of a circle, which represents the area that must be kept clear of obstacles. If the closet (or door) has width r, the door requires πr2/4 square feet of floor space.

To save floor space, some closets have so-called bifold doors. In this case, the door has hinges at one side and in the middle so that it folds as it opens. The unhinged side typically runs along a track that keeps it aligned with the closet opening. Note that "bifold" is a misnomer. There is only one fold (but two panels).


When Derek Seiple was a high school student, he wondered how much floor space is needed to accommodate the opening and closing of a bifold door. How big is the saving over a standard door? When he got to college (Penn State University at DuBois), Seiple was encouraged to investigate the problem, and the results appear in the April Mathematics Magazine.

"If you have a closet (or any doorway) covered with a bifold door there is an astroid lurking just inside and the only way you can get to it is to coax it carefully with a little bit of calculus," Seiple and his coauthors note. "If your door has more than one fold there are even more interesting objects waiting to be discovered."

Seiple's analysis shows that a bifold door traces out a path that consists of two curves. Given that each panel has a width r/2, a closing bifold door first sweeps out a circular arc of radius r/2. At 45 degrees, however, its path changes. Whereas the first part of the path was convex, the second part is concave. It now traces out part of a type of curve known as an astroid.

In this case, the area swept out by the door is 5πr2/64. That's a saving of nearly 70 percent, compared with a standard door.

"It is clear that adding 2, 3, 4, or n folds will reduce the floor space required even further," Seiple says.

At the same time, "adding more hinges has no effect on the astroidal portion of the curve," he adds. "The very same astroid appears regardless of the number of folds in the door as long as all of the panels are hinged so that they make the same angle with the front of the closet."

Interestingly, the entire path would be an astroid if you happened to have a "door" with no hinges. As one side of the door moved along a track across the closet opening, the other side would move along a track at right angles to the closet opening.


As the door closes (above), points A and B move toward points B and C respectively.

You'd see the same path traced by an initially vertical ladder that slips down and away from a wall.

References:

Seiple, D., E. Boman, and R. Brazier. 2007. Mom! There's an astroid in my closet! Mathematics Magazine 80(April):104-111.

May 4, 2007

Integral Heptagons

It isn't hard to find three points such that the distance between each pair of points is an integer. Three points defining a right triangle with sides 3, 4, and 5 represent one such example. Triangles characterized by Pythagorean triples and many other triangles exhibit such integral relationships.

It's not so clear that there are sets of four points in which the distance between each pair is an integer, but there are. It's even less clear for five, six, seven, or more points.

The problem of finding sets of points with all mutual distances integers has intrigued many mathematicians, including Abram Besicovitch (1891–1970) and Paul Erdős (1913–1996). Erdős originally asked for five points in the plane, no three on a line, no four on a circle with the distance between each pair of points an integer.

When that problem was solved, six points became the target. There proved to be infinite families of such point sets.

Now, the seven-point case has been solved. Using an exhaustive computer search, Tobias Kreisel and Sascha Kurz of the University of Bayreuth found a integral heptagon, in which no three points lie on a line and no four points lie on a circle. In fact, they came up with two examples.

The following table gives the distances between the pairs of points in the smallest possible integral heptagon.


For a diagram of this heptagon, see Ed Pegg's current Math Games column.

In each case, you can also look the smallest possible diameter, d, where the diameter is the largest occurring distance in a point set. For four points, d = 8; for five points, d = 73, and for six points, d = 174. The new results show that, for seven points, d = 22,270.

The new target? Are there eight points in the plane, no three on a line, no four on a circle with pairwise integral distances?

References:

Brass, P., W. Moser, and J. Pach. 2005. Research Problems in Discrete Geometry. New York: Springer.

Guy, R.K. 1994. Unsolved Problems in Number Theory, 2nd. ed. New York: Springer.

Kreisel, T., and S. Kurz. Preprint. There are integral heptagons, no three points on a line, no four on a circle.