Pinball Chaos

Sending a steel ball speeding across a tilted board studded with bumpers can be an addictive pastime—a tantalizing blend of skill to keep the ball in play and unpredictability in the ball's erratic path, rebound by rebound.

The pinball machine can serve as a model of deterministic chaos—a system that embodies a sensitive dependence on initial conditions. Balls with slightly different starting points end up following very different paths when they ricochet through the array of bumpers. Moreover, any uncertainty in a ball's initial position makes it difficult to predict where the ball will be even after just a few bounces.

Mathematician Henri Poincaré introduced this notion of "sensitive dependence on initial conditions" in the early part of the 20^th century, when he tangled with the intricacies of predicting planetary motion.

In his 1908 essay "Science and Method," Poincaré wrote, ". . . it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon."

Decades later, mathematician and meteorologist Edward N. Lorenz discovered the same effect embodied in equations used to model weather systems. At a meeting in 1972 he presented a paper with the provocative title "Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?"

"I avoided answering the question," he writes in his 1993 autobiography, The Essence of Chaos, "but noted that if a single flap could lead to a tornado that would not otherwise have formed, it could equally well prevent a tornado that would otherwise have formed."

Nonetheless, the term "butterfly effect" soon entered the lexicon to describe the unpredictable, even potentially drastic, consequences of a small change.

Lorenz also tells a story about pinball machines—the earlier kind without flippers or flashing lights, with nothing but arrays of pins to disturb the motion of the ball.

One spring in the 1930s during his undergraduate years at Dartmouth College, Lorenz recounts, a few pinball machines appeared in local drugstores and eateries.

"Soon many students were occasionally winning, but more often losing, considerable numbers of nickels," he writes. "Before long the town authorities decided that the machines violated the gambling laws and would have to be removed, but they were eventually persuaded, temporarily at least, that the machines were contests of skill rather than games of chance, and were therefore perfectly legal."

If that were true, however, students should have been able to perfect their skills and become regular winners, Lorenz notes. That didn't happen because of the deterministic chaos inherent in the pinball machine.

The Exploratorium in San Francisco gives you two chances to study pinball chaos. The Tinkering Studio includes a do-it-yourself model, where you can place various objects in different positions on a sloping table to investigate their effect on the trajectories of spring-projected balls.

Controlling just the initial speed of the ball, it is remarkably difficult to get a ball to follow the same path and come to rest at a given location at the bottom, even with just a few circular bumpers on the table.

Not too far away, you can also try your skill on arcade-style pinball, but with an Exploratorium twist. You don't need any coins to play, and the sides of the machine are transparent so that you can see what goes on inside.

References:

Lorenz, E.N. 1993. The Essence of Chaos. University of Washington Press.

Peterson, I. 1993. Newton’s Clock: Chaos in the Solar System. W.H. Freeman.

______. 1990. Islands of Truth: A Mathematical Mystery Cruise. W.H. Freeman.

Photos by I. Peterson

Restroom Illusions

There's something to tickle the eye just about anywhere you go at San Francisco's Exploratorium—even the restrooms.

The entryway to the main pair of restrooms features a dramatic array of black and white ceramic tiles, carefully arranged to recreate an optical effect known as the Café Wall Illusion. Described and popularized by psychologist Richard Gregory, the illusion makes the parallel lines defining each row of tiles appear sloped at different angles to each other.

The tiles, alternating black and white, are in parallel rows, but because the rows are staggered, the dividing lines between the rows look slanted.

The effect is most dramatic when the grout between the tiles is an intermediate gray in color, rather than either black or white. In the Exploratorium installation, each of three walls features grout of a different color (gray in the middle, white on the men's side, and black on the women's side) so that viewers can judge the effect of grout color on the strength of the illusion.

The eye-dazzling show doesn't end there. Within the restrooms throughout the building, individual tiles feature other classic optical illusions—all shown in black against white.

The patterned tiles were custom manufactured, then scattered among the white tiles covering the walls.

That's quite a show for a simple rest stop.

Photos by I. Peterson

Counting on Success

The setup looks simple, but people attracted to the long row of nine-sided, wooden rings at the Exploratorium in San Francisco can't resist trying it—again and again.

Each ring bears the numbers from 1 to 9. You "shuffle" the rings, rotating them so that you can read a scrambled string of digits along the top of the row.

Starting at one end, you think of a number from 1 to 9, and you move along the top row from ring to ring according to the designated number. For example, if your chosen number is 4, you move over four rings along the row. The number on top of the ring on which you land gives you the number of steps for your next move, and so on. You continue in this way as far as you can go down the line and note the number on the ring where you end up.

Amazingly, when you do it again from the same end and with the same sequence of numbered rings but with a different starting number, you're almost sure to end up at the same ring.

Aptly titled "Sooner or Later," the Exploratorium exhibit is an ingenious variant of a card-guessing trick that magicians can use to show off their mind-reading prowess, even though its success is rooted in mathematical principle.

This particular prediction trick is known as the Kruskal count, named for physicist and mathematician Martin D. Kruskal.

The magician invites a subject to shuffle a standard deck of 52 playing cards and secretly pick a number between 1 and 10. The subject slowly and steadily deals out the cards one by one, face up, to form a pile. As she does so, she goes through the following counting routine: Suppose her secret number is 6. The sixth card that she deals becomes a "key" card, and its face value tells her how many more cards must be dealt to get to the next key card. For example, if the key card happens to be 7, she starts counting again, this time from 1 up to 7, to find the next key card. She repeats the procedure until all 52 cards are dealt. An ace counts as 1, and a king, queen, or jack count as 5.

At some point, she reaches a key card (perhaps the last card in the deck) that is not followed by enough cards for her to complete the count. This final key card becomes the subject's "chosen" card and the magician's mind-reading target.

No matter how steadily the subject deals out the entire deck (with no pauses at crucial moments or other hints of which cards are key), the magician still has a high probability of identifying the mystery card.

The reason is that, for many arrangements of the deck, every starting point leads to the same final key card. So, all a magician has to do is to select his own starting point, then count his way to the end while the subject is dealing the cards. He is very likely to end up at the same "chosen" card as the subject.

One way to see what is going on is to deal out a shuffled deck so that the cards form an array. I used colored chips to identify the key cards associated with each of the 10 possible starting points. Suppose the first card is a 10. The second chip goes on the 11th card in the array—an ace; the next chip goes on the adjacent card—a 7, and so on. Using chips of a different color, I started with the second card—a 5—and laid down a new trail. At some stage, the second trail happened to hit a card already marked by a chip. From that point on, the trails coincided.

The same behavior underlies the convergence of paths along the row of nine-sided rings in the Exploratorium exhibit. For typical "random" arrangements of the rings, you'll end up at the same ring, no matter what your starting number.

The trick doesn't work every time. For certain arrangements of the deck (or rings), one or two starting points may generate trails that never coincide with the others. In the case of a deck of 52 cards, however, the probability is about 5/6 that the magician ends up at the same card as the subject—assuming both magician and subject count accurately and correctly.

A magician can increase his chances of "guessing" correctly by starting with a card near the beginning of the deck. Giving royal cards a lower value (say, 2 instead of 5) and using two decks shuffled together also improve the odds of having trails intersect.

In 2001, Jeffrey C. Lagarias, Eric Rains, and Robert J. Vanderbei took a closer look at the Kruskal count. They used computer simulations and mathematical arguments to determine the effect of two parameters on the probability of success for the magician: the freedom to choose the initial key card and the freedom to assign values to the face cards. The researchers described their results in a paper titled "The Kruskal Count."

Lagarias, Rains, and Vanderbei compared three possibilities for assigning values to face cards. Suppose the values 11, 12, and 13 are assigned to the jack, queen, and king, respectively. In this case, the magician's probability of failure is about 0.34. Assigning the value 10 to all face cards lowers the failure probability to 0.29. Going with the usual rule of assigning the value 5 to face cards slashes the failure probability to 0.16. If the magician chooses the first card as his starting point, the failure probability drops further to 0.14.

"The magician should choose his key card value to be 1," the researchers concluded. "Assuming this strategy for the magician, the success probability of the original Kruskal count trick is just over 85 percent."

In general, "the effect of the choice of the magician's key card on the failure probability is small, at most 2.5 percent," they noted. "The choice to have face cards take the value 5 rather than 10 has a much larger effect on the failure probability than the magician's choice of first key card position."

The underlying mechanics of the Kruskal count also highlights how seemingly unrelated chains of events can lock together in sync after a while—a phenomenon worth watching for in other contexts.

It also calls to mind a comment by Sherlock Holmes in the story "The Disappearance of Lady Frances Carfax" by Arthur Conan Doyle: "When you follow two separate chains of thought, Watson, you will find some point of intersection which should approximate to the truth."

References:

Gardner, M. 2000. Modeling mathematics with playing cards. College Mathematics Journal 31(May):173-177.

______. 1997. Sicherman dice, the Kruskal count and other curiosities. In Penrose Tiles to Trapdoor Ciphers. Mathematical Association of America.

______. 1956. Mathematics, Magic and Mystery. Dover.

Knopfmacher, A., and H. Prodinger. 2001. A simple card guessing game revisited. Electronic Journal of Combinatorics 8(No. 2):R13.

Mulcahy, C. 2000. Mathematical Card Tricks. American Mathematical Society. Oct. 1.

Photos by I. Peterson

Martin Gardner and Mathematics, Magic, and Mystery

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cross-Stitch Symmetry

The craft of counted cross-stitch lends itself to the creation of elegant patterns on fabric, and mathematician Mary D. Shepherd has taken advantage of this form of needlework to vividly illustrate a wide variety of symmetry patterns.

Mary Shepherd with her cross-stitch symmetries sampler.

Photo by I. Peterson

The fabric that Shepherd uses is a grid of squares, and the basic stitch appears as an X on the fabric. In other words, one cross-stitch covers one square of the fabric.

Stitching over squares constrains the number of symmetry patterns that you can illustrate using this technique. The reason for this constraint is that the only possible subdivision of a square is with a stitch that "covers" half a square on the diagonal. In effect, a half cross-stitch splits a square into two isosceles triangles, covering only one of the triangles.

This means that the only angles you can create in a counted cross-stitch pattern are multiples of 45 degrees.

Wallpaper patterns have translations in each direction along two intersecting lines. Of the 17 possible wallpaper patterns, only 12 can be done with a combination of cross-stitches and half cross-stitches. The other five patterns involve angles of 60 and 120 degrees, and so are not possible in counted cross-stitch.

Six of the 12 wallpaper patterns that can be done in counted cross-stitch needlework. Top row, left to right: p1 (translation only), pg (glide reflection), pm (glide reflection axis along line of reflection). Bottom row, left to right: cm (glide reflection axis not along line of reflection), p2 (180-degree rotation), pmm (reflection and 180-degree rotation).

Courtesy of Mary D. Shepherd

Shepherd has also worked on both frieze and rosette symmetry patterns. Frieze patterns, often used for borders, have translations in two directions. A rosette pattern has at least one point that is not moved by any of the symmetry transformations (translation, rotation, reflection, and glide reflection), Shepherd notes. Hence, the only transformations that can occur in rosette patterns are reflections and rotations.

Rosette patterns, for example, give a nice visualization of the symmetries of a square (technically, the group D₄ and all its subgroups), she says.

Rosette patterns for visualizing the symmetries of a square (the dihedral group of the square).

Courtesy of Mary D. Shepherd

Shepherd provides instructions for crafting a "symmetries sampler" in the book Making Mathematics with Needlework: Ten Papers and Ten Projects (A K Peters).

She has also used counted cross-stitch examples in the classroom to illustrate and explore ideas about symmetry groups and subgroups. 

Reference:

Shepherd, Mary D. 2007. “Symmetry Patterns in Cross-Stitch.” In Making Mathematics with Needlework: Ten Papers and Ten Projects, sarah-marie belcastro and Carolyn Yackel, editors. A K Peters.

Möbius Mentions I

References to Möbius strips can pop up unexpectedly in all sorts of settings, including novels. I'm especially intrigued by examples in which the term is used without further explanation. The reader is expected to understand the reference and know all about the peculiar one-sided, one-edged character of this mathematical object.

One example that I recently encountered is in an amusing, quirky mystery, Forbidden Fruit by Kerry Greenwood. Near the beginning of the book, the author offers the following passage:

"I donned a loose caftan made to a Therese Webb pattern. I had made it seven times, and each time I had a moment when I feared that the fabric would have to be folded into another dimension to fit the design. This möbius robe was made of a fine blue butterfly batik."

I'm not sure I can picture exactly what this garment looks like, but it sounds suitably exotic.

A crocheted Möbius band, by Josh Holden.

Photo by I. Peterson

Another mystery with a Möbius reference is What's the Worst That Could Happen by Donald Westlake.

"When Brandon . . . stood at the picture window, with its view out over the Battle-Lake, at the moment peaceful, with the tall Moebius shape of the hotel beyond it."

In this case, I definitely can't picture the twisty hotel, but the setting is Las Vegas, so it's probably not beyond the realm of possibility. Yet it sounds pretty weird.

A design for a building with a Möbius-strip ramp.

Courtesy of R.J. Krawczyk and J. Thulaseedas

In some novels, the Möbius metaphor or simile is straightforward and apt. Consider the following passage from Goodbye Without Leaving by Laurie Colwin.

"Laundry," I said, yawning. "I never really understood about laundry before. It's a kind of Moebius strip—no end and no beginning."

For some examples in articles and other settings see "Words with a Twist" and "Möbius Pop."

Reference:

Thulaseedas, J., and R.J. Krawczyk. 2003. Möbius concepts in architecture. ISAMA/Bridges 2003 Conference: Mathematical Connections in Art, Music, and Science. July 23-25. Granada, Spain. Available at http://www.iit.edu/~krawczyk/jtbrdg03.pdf.

Pentagons and Game Balls

The game of sepak takraw, a popular form of kick volleyball in Southeast Asia, uses a woven ball that, in its simplest form, has 12 pentagonal holes and shows a weaving pattern with 20 intersections. The twelve pentagons remind me of a regular dodecahedron, one of the Platonic solids.

Traditionally made from rattan, such a ball may be constructed from six long strips (instructions), with five strips defining the pentagonal holes and a sixth strip forming a closed loop that wraps around twice (video).

This ornamental model, with woven strands intersecting to form pentagons atop a glass sphere, shows the basic geometry underlying a woven sepak takraw ball.

More complicated weavings produce triangular patterns on the ball's surface (below).

The two examples shown above are part of a display of artistically crafted woven spheres in the lobby lounge of the AT&T Executive Education and Conference Center at the University of Texas at Austin. I couldn't help but be reminded of sepak takraw balls when I first saw the display.

Unfortunately, I couldn't find any information about these intriguing models, and would love to know more about them.

Photos by I. Peterson

Elevator Buttons and Stone Steps

Human activity can leave telltale marks on its surroundings. These marks, in turn, can provide clues about the nature of the activity that created them or about the setting itself. See, for example, "Statistical Wear" and "An Irresistible Edge."

Recently, I started paying attention to wear caused by finger contact in the vicinity of elevator buttons. Shown below is a set of buttons for a hotel elevator. What can you tell about the setting just from the scuff marks?

The wear pattern suggests that this particular set of buttons is most likely located on the lobby floor; many more people have pressed (or tried to press and missed) the "up" button than the "down" button. The curious tail toward the right indicates that people tended to come from the right, presumably making contact with the brass plate prematurely.

Indeed, these buttons are on the lobby floor of a hotel in Toronto, and the only entrance to this bank of elevators is from the right, with no exit to the left.

You'll see similar wear marks in the example below, from a hotel in Austin, though the distribution isn't quite as strikingly asymmetric.

Wear marks appear in all sorts of settings. Foot traffic can be responsible for some of the more striking examples, particularly when the marks appear on stone, as seen in the photo below of a stone staircase in Wells Cathedral in Great Britain.

What can you say about the traffic patterns that these worn steps reveal?

Photos by I. Peterson

Roots of a Base Tiling

Like a strangely ordered root system, a fragment of an Archimedean tiling serves as the supporting structure for a metal pillar in an odd artwork in downtown Toronto. The sculpture, which consists of a pair of quirkily incomplete pedestrian bridges, stretches across paving bricks amid the restored, industrial-style buildings of the city's Distillery Historic District.
 
Titled Passerelle et Portance, the sculpture was created by Claude Millette and installed in 2006.

The tiling that forms the base consists of regular hexagons framed by squares and equilateral triangles. This example of a semiregular tiling is sometimes described as a rhombitrihexagonal tiling.

Rhombitrihexagonal Archimedean tiling. R.A. Nonenmacher, Wikipedia.

  
Photos by I. Peterson

Wild Beasts around the Corner

Some mathematical problems are easy to describe but turn out to be notoriously difficult to solve. In some instances, these difficulties may stem from fundamental issues of provability, especially for mathematical problems apparently poised between order and chaos.

In a provocative article titled "On Unsettleable Arithmetical Problems," John H. Conway (Princeton University) offers some remarkably simple assertions that are true yet are neither provable nor disprovable. The article appears in the March 2013 American Mathematical Monthly.

"It is usually thought that they must necessarily be complicated," Conway writes, but "these wild beasts may be just around the corner."

Conway bases his examples on the infamous Collatz 3n + 1 problem, which concerns a sequence of positive integers.

Start with any positive integer n.

If n is even, divide it by 2 to give n' = n/2.
If n is odd, multiply it by 3 and add 1 to give n' = 3n + 1.

For example, starting with 5, you get the following sequence: 5, 16, 8, 4, 2, 1, 4, 2, 1,. . . 

Starting with 11, you get the sequence: 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1,. . . 

The numbers generated by these rules are sometimes called "hailstone numbers" because their values fluctuate wildly (as if suspended in turbulent air) before they finally crash (to the ground) as the repeating string 4, 2, 1.

Computational experiments suggest that such a sequence will always eventually crash. However, for some starting integers, it takes a huge number of steps to reach the repeating cycle.

Will you always end up in a repeating cycle? No one has yet found a counterexample. Computations by Tomás Oliveira e Silva have verified that no such counterexample can start with an integer less than at least 5 x 10¹⁸.

Jeffrey C. Lagarias (University of Michigan) has categorized the 3n + 1 problem and its ilk as tantalizing but dangerous, luring mathematicians into weeks, if not years, of futile labor.

In his paper, Conway shows how simple assertions inspired by the Collatz problem cannot be settled—neither proved nor disproved.

Then, in an intriguing postscript, Conway presents an argument that has convinced him that the Collatz conjecture is itself very likely to be unsettleable, rather than, as he originally thought, having just a slight chance of being unsettleable. He uses the fact that there are arbitrarily tall "mountains" in the graph of the Collatz game.

"I don't want readers to take these words on trust but rather to encourage those who don't find them convincing to try even harder to prove the Collatz Conjecture!" Conway wryly concludes.

References:

Conway, J.H. 2013. On unsettleable arithmetical problems. American Mathematical Monthly 120(March):192-198.

Lagarias, J.C., editor. 2010. The Ultimate Challenge: The 3x + 1 Problem. American Mathematical Society.

______. 1986. The 3x + 1 problem and its generalizations. American Mathematical Monthly 92(January):3-23.

Oliveira e Silva, T. 2010. Empirical verification of the 3x + 1 and related conjectures. In The Ultimate Challenge: The 3x + 1 Problem, J.C. Lagarias, ed. American Mathematical Society.

The special, full-color March 2013 issue of the American Mathematical Monthly is available for purchase.