Defending the Roman Empire

About 1700 years ago, the Roman Empire was under attack, and Emperor Constantine had to decide where to station his diminished forces. Constantine organized his legions into four field armies. He needed to protect eight regions with these forces. The trick was to place the armies so that each region was either occupied by an army or was only one step away from an army. But an army could be sent onward only if there were another army to stay behind and defend its original position.

Take a look at this map (below) showing the regions and the steps between the regions. Where would you put the four armies?


Constantine chose to place two armies in Rome and two at his new capital, Constantinople. This meant only Britain could not be reached in one step. Defending Britain would require moving an army from Constantinople to Rome, then from Rome to Gaul, and finally to Britain—a total of four steps.

Can you do better than Constantine—either by reducing the number of regions that can’t be reached in one step or by cutting the number of steps it would take to get to the worst-off region? Try it and see.

Charles S. ReVelle, an environmental engineering professor at Johns Hopkins University in Baltimore, used a computer to test various possibilities. He came up with several alternatives. Each beats Constantine, but has its own flaws.

One possibility is to put two armies in Rome, one in Britain, and one in Asia Minor. Then every unoccupied region can be reached in one step from Rome. But the emperor would have trouble responding if a second was should suddenly erupt elsewhere in the empire.

Another solution is to put two armies in Iberia and two in Egypt. Again every unoccupied region could be reached in one step from one of these power centers. There is a political problem with this solution, however: Rome itself does not have an army.

Mathematics can help you figure out the best places to put military units, especially when you have a limited number of units and a lot of territory to defend. The same sort of math is also useful when people want to know the best place in town to put a new hospital, fire station, or fast-food restaurant.

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Muse, October 2001, p. 34.

Twisted Security

You're at the Yahoo! or Google Web site signing up for a new email account. You fill out a form, typing in your name and other information. When you come to the end, you find you have to enter a code, which you get from a string of letters and numbers shown just below the entry form. The weird thing is that the letters and numbers are all twisted and angled, with squiggly lines running through them. You have to figure out what those characters are and type them in correctly before you can complete your account registration.


You'll often run into the same thing at other Web sites—when you're signing up to play a game, voting in an online poll, buying concert tickets, joining a group, and more.

So, what's that all about?

Computers can do all sorts of things, from searching the Web at lightning speed to playing championship chess. But they're not very good at, for example, figuring out what's in a photo or identifying a person's face—things that people, even little children, can do easily. So, one way to make sure that a person rather than a computer is performing some task to to ask for something that people do much better than computers do—like trying to decipher distorted numbers and letters.


Such test images are examples of little puzzles called captchas. The word captcha was invented by computer scientist Manuel Blum of Carnegie Mellon University. It stands for "Completely Automated Turing Test to Tell Computers and Humans Apart." Alan Turing was a famous mathematician who invented a test in which you ask questions of a hidden participant and see if you can tell whether the responses are coming from a person or an "intelligent" computer. In the case of captchas, computers can automatically generate the tests and judge whether a person types in the correct answer, but they can't themselves pass the test.


Blum and his coworkers invented captchas because companies like Yahoo! wanted to keep people from writing computer programs that would seek out Web sites and automatically sign up for large numbers of free email accounts in a very short time. They could then use these addresses to send out mass mailings, or spam.

Yahoo! started using captchas a few years ago to prevent computer programs from being written that would abuse their services. Now, you'll find captches all over the Internet. But there are some problems. Computers are getting better at figuring out what's in images, so some captchas can be solved by computers. And image captchas don't work for people who are blind, for example. As an alternative, Blum and others have come up with captchas that depend on unscrambling sounds, something else that computers have trouble doing. They've also created more complex captchas, for example, comnputer-generated images that contain several, overlapping words selected randomly from a dictionary and displayed against a colored pattern.


While some researchers work on creating tougher puzzles, others put a lot of effort into programming computers to solve those puzzles. The result is the computers keep getting better at all the different things that they do.

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Muse, May/June 2007, p. 36.

You can learn more about captchas at http://www.captcha.net/.

Weird Bottles

An ordinary bottle has an inside and an outside. If an adventurous ant with sticky feet were to walk along that bottle's surface to get from the outside to the inside, it would have to cross the lip that forms the bottle's mouth. Mathematicians have come up with a bizarre, mind-bending object they call a Klein bottle that has no such lip, or edge. What appears to be the bottle's inside is smoothly connected with its outside!


To make a model of a Klein bottle, you could start with a long tube. You would have to stretch and bend one of the tube's ends so that it twisted around and plunged through the tube's side, then met the tube's other end from the inside.

People have tried doing this with glass, but it isn't easy. Astronomer Cliff Stoll of Oakland, California, first heard about Klein bottles when he was in high school more than 30 years ago. He went to his chemistry lab, set up a Bunsen burner, and tried to make one. "After burning my fingers and cracking a dozen tubes, I gave up," he says.

A few years later, while in college, Stoll tried again, this time with gloves and better equipment. He failed once more. "Without enough heat, you can't bend the glass," he explains. "Heat it too much and the glass melts into a glob. And at the right temperature, it's practically impossible to stretch glass around a curve and down into itself."

Stoll finally found the answer several years ago when he met some expert glassblowers who had experience making intricate glassware for scientific experiments. They showed him how to make a glass Klein bottle. Stoll crafted his first bottle from a piece of lab glassware—a Pyrex 500-milliliter Erlenmeyer flask with welded glass connections. He's been making Klein bottles ever since, even selling them to people who want an example of an object that has no edges and only one surface.


When German mathematician Felix Klein discovered the mathematical object now named for him, he probably had no idea that it would become an intriguing challenge for glassblowers—and for anyone else who sees one of these weird bottles and tries to figure out whether it will hold anything.

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Muse, September 2001, p. 45.

You can see examples of Cliff Stoll's Klein bottles at http://www.kleinbottle.com/.

John Sullivan has computer-generated images of Klein bottles at http://torus.math.uiuc.edu/jms/Images/klein.html.

Dots and Boxes

The familiar game of Dots and Boxes seems simple. But it's really a lot trickier than it looks.

Mathematician Elwyn Berlekamp of the University of California at Berkeley first learned to play the game when he was in grade school. He has been studying it ever since and has even written a book about strategies for playing it.

The playing field is a rectangular or square grid of dots. You and your opponent take turns joining two dots with a line. When a player adds the fourth line that completes a box, he or she claims the box by marking it with an initial, then takes an extra turn. If the same move happens to close two boxes, the player claims both boxes but still gets only one bonus move. The game ends when all boxes are taken. The player who closed more boxes wins.


Beginners tend to connect dots at random to see what happens. It usually doesn't take long to figure out that it's a good idea to avoid adding the third side to a square. If both players avoid third sides, they end up with a kind of maze made up of several chains of connected boxes.

That's when things get interesting. Your aim is to force your opponent to be the first to add a third line to a square that belongs to a chain. Then you can claim all the boxes in the chain. But it pays not to be greedy. It's better to make a final move that leaves two boxes of a long chain unclaimed. The other player can claim the "gift" boxes, but then must add a bonus line somewhere else—and if all that is left are other chains, this lets you capture another one.


Berlekamp has worked out strategies that allow an alert player starting second to always win a game played on a three-by-three grid of squares. No such guarantee is possible if you play on a five-by-five grid of squares. Amazingly, there are so many possible moves that no person or computer can try all of them to find the perfect set of moves.

So if you want to win, try the three-by-three grid. But if you want to experiment, the five-by-five grid is the ideal playing field.

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Muse, July/August 2001, p. 36.

Tilt-A-Whirl Chaos

Much of the fun of an amusement park ride is its stomach-churning, mind-jangling unpredictability. The Tilt-A-Whirl, for example, spins its passengers in one direction, then another. The ride's cars sometimes hesitate between moves and at other times swing suddenly from one motion to another. You never know what to expect next.


These surprising movements arise from a simple geometry. A rider sits in one of seven cars, each mounted on a circular platform and free to pivot about the center of the platform. The platforms, in turn, move at a constant speed along a track with three identical hills, which tilt the platforms. So the platform movements are perfectly regular, but the cars whirl around on their own quite unpredictably.


The only thing that a Tilt-A-Whirl operator can adjust is the speed at which the platforms travel around the track. When the platforms move at very low speeds, each car completes one backward turn as its platform goes over a hill. At high speeds, a car gets slammed to its platform's outer edge and stays in that position. In either case, the motion is predictable. It's only at speeds somewhere in between these extremes that a car's motion becomes complicated and unpredictable. So it's important for a Tilt-A-Whirl operator to make sure the ride runs at the proper speed, about 6.5 revolutions per minute.


At just the right speed, it becomes nearly impossible to predict exactly what will happen from one moment to the next during a ride, or from one ride to the next. Because a car's motion depends on the weight of its passengers and where they are sitting, the thrills and chills are different each trip.


Do you like the Tilt-A-Whirl? You can find similar thrills at the amusement park by looking for rides with cars that are free to rotate or shift back and forth as they follow a fixed track.

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Muse, May/June 2001, p. 34.

Lively Tiles

The tiles you see in your bathroom or on a kitchen floor are usually square, though you may also come across tiles that are shaped like hexagons or octagons.

But tiles don't have to be polygons. Many of the drawings by M.C. Escher, a Dutch artist who lived from 1898 to 1972, contain interlocking tiles in the shape of birds, fish, reptiles, butterflies, and other living things.

As a young boy, Escher was intrigued by the different ways in which you can cover areas neatly with small, identical pieces. Drawing was his favorite subject in school, and after graduating, he became an artist. At age 24, he visited Spain and discovered the intricate mosaics (designs made with tiles) in the Alhambra, a 13th-century Moorish palace in Granada. Those designs inspired him to create the amazing tile patterns that appear in his art.

Escher used a number of different strategies to draw intriguing tiling patterns. Suppose you had a stack of square tiles, some red and some black. You could lay them out in neat rows of alternating red and black tiles to create a checkerboard pattern. In this case, the repeating unit would be a block of four tiles, two red and two black. Starting with one such unit, you can generate the entire pattern by using it like a stamp to fill in the rows and columns of the design. If all the tiles were the same color, the repeating unit would be a single tile.


To make his artworks more interesting, Escher often disguised the underlying geometric pattern by using tiles of different shapes and colors. When you look at one of these patterns, you might think that each tile is shaped like, say, a fish. But if you imagined you had a stack of fish tiles and tried to lay out the pattern with them, you'd quickly find you couldn't do it. This is because the true tile, the repeating unit that can be used to lay out the entire pattern, is not a fish. Instead it might be several fish of different colors joined together. So each of Escher's tiling patterns is a puzzle; what is the basic tile?

One person who has studied Escher's art is Doris Schattschneider, a math professor at Moravian College in Bethlehem, Pennsylvania. While examining Escher's notebooks, Schattschneider found that Escher had worked out his own mathematical system for classifying the tiles. He used special symbols to describe how portions of the edges of tiles related to each other and to edges of adjacent tiles. The system allowed him to find all the different ways in which he could interlock and color various shapes of identical tiles to create pleasing patterns.

Escher's study of geometric shapes, combined with his artistry and imagination, led to all sorts of fabulous drawings.

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Muse, April 2001, p. 26.

Doris Schattschneider. Visions of Symmetry: Notebooks, Periodic Drawings, and Related Works of M.C. Escher (New York: W.H. Freeman, 1990).

Take a closer look at some of Escher's designs at http://library.thinkquest.org/16661/ and http://www.mcescher.com/.

You can find activities and investigations related to tilings (tessellations) and symmetry at http://britton.disted.camosun.bc.ca/home.htm.

Fancy Folding

The amazing thing about origami is the enormous number of different objects you can make by folding a square sheet of paper. No glue or scissors allowed! You can make airplanes, flowers, butterflies, and noisemakers, or flapping birds, fierce devils, and fully equipped lobsters.


Robert Lang's incredible origami lobster. Courtesy of Robert Lang.

Tom Hull, a mathematician at Merrimack College in North Andover, Massachusetts, has been making origami models since he was eight years old. He got started when his uncle gave him a book about origami. When he got to college, Hull found a way to combine his interest in origami with a career in mathematics. He even contributed to a book for beginners, called Origami, Plain and Simple, while he was a student.

Hull is now inventing new types of origami designs based on mathematics. Some of these designs are flat. They look like tiles—the sorts of repeating patterns that you might see in fancy bathrooms, for example. Others are made from identical simple units, each one folded from a square sheet of paper, that interlock to form three-dimensional structures that look like sticky burrs or crazy crystals.


Tom Hull's amazing five intersecting tetrahedron. Courtesy of Tom Hull.

Physicist Robert J. Lang likes to work out rules that, when given to a computer, show what creases to make to end up with a desired origami figure. He is famous for highly complex designs, such as his paper lobster, which comes fully equipped with legs, pincers, feelers, and tail.

There's a lot of math in origami, from the patterns the creases make to the sets of precise instructions people follow to create certain objects. Hull himself uses origami to help explain angles and other geometric concepts to students. He also finds origami relaxing after a hard day in the classroom.

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Muse, March 2001, p. 24.

Tom Hull has a web page about origami mathematics at http://www.merrimack.edu/~thull/OrigamiMath.html.

The Exploratorium in San Francisco features an illustrated article about origami designs on its Web page at http://www.exploratorium.edu/exploring/paper/index.html.

Robert Lang has a Web site at http://www.langorigami.com/.

Mathematician Helena Verrill illustrates a variety of her origami tiling designs and provides instructions for making them at http://www.math.lsu.edu/~verrill/origami/.

You'll find diagrams and instructions for making all sorts of origami figures and patterns at http://www.paperfolding.com/diagrams/.

Puzzling Art

An art gallery isn't the first place you would think to look for math, let alone a mathematical puzzle. But Barry Cipra, a mathematician and writer in Northfield, Minnesota, found a puzzle in a set of large drawings by American artist Sol LeWitt.

LeWitt is famous for the use of simple patterns and basic geometric shapes in his drawings, sculptures, and paintings. In 1973, he composed an artwork that consisted of 15 squares. On each square, he drew a horizontal, vertical, or diagonal line, or some combination of these lines. The 15 squares, arranged into a four-by-four grid with one blank space, represented all possible combinations of such lines.

When Cipra first saw this set of drawings, he was intrigued by how his eye automatically tried to connect the lines from one square to the next. In LeWitt's arrangement, however, none of the horizontal, vertical, or diagonal lines actually went from one side of the grid to the other.

Cipra asked himself if it would be possible to rearrange the 16 squares (including the blank one), without rotating any of them, so that the horizontal, vertical, and diagonal lines did go all the way across the grid.


Straight Lines in Four Directions and All Their Possible Combinations: The math problem. Make a copy of these squares on a sheet of paper, then cut out the squares to see if you can solve the puzzle. The rule is, you aren't allowed to rotate the pieces to find a solution. Be careful!

Using square pieces cut from stiff cardboard, Cipra came up with a solution. He later used math to prove that there are exactly three completely different solutions to the puzzle. Can you find one? Or all three?

The question is, could LeWitt himself find a solution to the LeWitt puzzle? Did the artist even know he made a puzzle?

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Muse, February 2001, p. 34.

Answers:

Data in Hiding

When Viviana Risca was a high school student in Port Washington, New York, she found a way to hide secret messages among DNA molecules. Heaps of DNA strands sit like microscopic spaghetti inside plant and animal cells. They are a kind of secret code already, only the secret message they normally carry is instructions for making a living creature, such as a clam or a peacock.

How did Risca use DNA to send a message? DNA is a big molecule that looks like a twisted ladder. The rungs on the ladder are chemical units called bases. Working with researchers at the Mount Sinai School of Medicine in New York City, Risca created a single strand of DNA made up of a sequence of bases corresponding to the letters of a message. She mixed this message strand with many other, different DNA molecules.

The researchers then dripped a tiny amount of the DNA onto a small dot printed on filter paper. They cut out the dot, taped it over the period in a typed note, and mailed the letter.

The person who got the secret message had to know three things to decode it. He had to know that one of the periods in the letter was a phony. He had to know a special marker DNA sequence that would allow him to chemically pick the message strand out from all the other DNA strands. And he had to know which series of bases corresponded to which letters of the alphabet.

The letter's recipient lifted the DNA dot, ran it through some chemical steps that picked out the right DNA strand and read out its sequence of bases, and decoded the message using Risca's key. Risca won the top prize in the 2000 Intel Science Talent Search for this project.

When spies of the future want to pass secret messages to one another, maybe they'll bone up on molecular biology.

Decoding a DNA Message

The information in a DNA molecule is in the rungs of this ladderlike molecule. These are chemical units called bases, and there are only four of them: adenine, thymine, guanine, and cytosine—abbreviated A, T, G, and C. Normally, sets of three bases specify part of a protein, the molecules that do most of the work in a living creature. In Risca's code, however, sets of three bases stood for letters of the alphabet. For example, she made the letter L by adding the bases TGC to the message strand she was building.

Key to Risca's Code:


Try decoding Risca's secret message:

AGTCTGTCTGGCTTAATAATGTCTCCTCGAACGATGGGATCTGCTTC
TGGATCATCCCGATCTTTGAAA.

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Muse, January 2001, p. 22.

Ant Math

Imagine stepping into a pitch-black room. How might you figure out the room's size in the dark? You could, for example, carefully follow the walls and count how many steps you took going from one corner to the next all the way around. That would give you a rough idea of how big the room was.

Certain ants have their own method for estimating size in the dark. Ants of the species Leptothorax albipennis live in small, flat cracks in rocks. A typical colony consists of a single queen, her brood, and between 50 and 100 workers. When a nest happens to get wrecked, the colony sends out scouts to find places to start a new nest.

How does a scout estimate a crack's size so it can tell if it would make a good nest?

Experiments by mathematical biologists Eamonn Mallon and Nigel Franks of the University of Bath in England show that a scout usually visits a possible home twice. On its first visit, the scout spends about two minutes in the crack. As it scurries about, it lays down a trail of smell molecules, or pheromones. On its second visit, it follows a different path, repeatedly crossing its first track, slowing down each time it does.

The scout ant apparently smells each crossing. It can then estimate the crack's size from the number of crossings: the more crossings, the smaller the floor area.

As long as a scout explores the whole crack, the method works for different cracks and for cracks of different shapes. It also works in complete darkness. It's a neat example of how animals can use very simple rules of thumb to make good decisions.

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Muse, December 2000, p. 23.

Morphing Art

When you're riding along on your bike or in a car, you sometimes see the word "ONLY" painted on the roadway just before an intersection. The white letters look normal from where you're sitting. But if you were standing beside the "ONLY" instead of riding toward it, the letters would look stretched out. It's only when you look at them at the proper angle that they don't look distorted.

Artists have long used the same idea to create visual puzzles. A viewer sees an object in a picture correctly only if he or she finds the right angle at which to look at it. Such distorted pictures are called anamorphic images.

One of the most famous examples is a painting called The Ambassadors by the German artist Hans Holbein the Younger. It shows two young men standing in front of tables overflowing with books, instruments, and globes. At their feet, the artist painted a weird shape that turns out to be a grinning skull when you hold the picture at a slant (see http://www.abcgallery.com/H/holbein/holbein16.html).

You can create your own slant picture. You start with a piece of paper ruled into square cells and another ruled with the same number of trapezoids—squares that are stretched out in a special way. Draw your picture on the square grid. Then carefully copy the contents of each square of the original grid to the corresponding trapezoid of the other grid. You'll find that you'll need to stretch the lines of your drawing to make sure everything fits together. You end up with a stretched-out version of your original picture, but if you look at it from the right angle, it'll look undistorted again.


Some artists have tried more elaborate schemes. It's possible, for example, to draw or paint a picture you can understand only if you look at its reflection in a mirror shaped like a cylinder or a cone. The mirror takes the distortion out of the image.

About 200 years ago, anamorphic paintings for cylindrical or conical mirrors were popular toys in both Europe and Asia. You can still find examples in some museums. Nowadays, you can buy anamorphic jigsaw puzzles, which you can assemble and view with a special mirror to reveal a hidden image. And artists have created amazing pictures that must be reflected by shiny spheres, mirrored pyramids, or other mirrored shapes to show their true identity.

It's a neat game of hide-and-seek for the eye.

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Muse, November 2000, p. 26.

Learn more about anamorphic images at http://www.anamorphosis.com/ or http://www.physics.uoguelph.ca/morph/morph.html.

You can buy MorphMagic anamorphic jigsaw puzzles at http://www.mathartfun.com/shopsite_sc/store/html/AnaPuzzles.html.

Views from Flatland

What do you think the world would look like if you and everything in it were squished flatter than a pancake? Like shadows, you and your friends would freely flit about the surface. But if you couldn't rise above or below the surface, objects with any thickness would look very strange to you.

That's the idea behind a book called Flatland, written more than 100 years ago by Edwin A. Abbott. Head of a school for boys in London, Abbott imagined a world in which the inhabitants are geometric shapes: straight lines, triangles, squares, pentagons, circles, and other figures, all living on the surface of a perfectly flat world.


Curiously, though Flatlanders are two-dimensional, they appear to one another to be straight lines. To see why, place a coin on a table. From directly above, the coin looks circular. Seen at an angle, the coin looks more like an oval. If your eye is level with the table, the oval thins to little more than a straight line.

Unlike many people in Great Britain at the time he wrote Flatland, Abbott believed that girls deserve as good an education as boys. He also favored granting more rights to women, including the right to vote. To make fun of the way women were treated in British society, Abbott made Flatland males triangles and other polygons, but he made Flatland females straight lines. That made them dangerous because they can run you through like a needle. Unlike males, females can become invisible at will. Can you see how?


In one dramatic episode in Flatland, a three-dimensional sphere visits the Flatlanders' world. You can think of a sphere as made of a stack of circles. Only one of these circles would intersect Flatland at any given moment. To a Flatlander, each circle would look like a line. The length of the line would depend on the size of the circle, so to a Flatlander, a sphere rising through Flatland would look like a dot that grew longer and longer, then shorter and shorter, until it became a dot again and vanished.


A tiny water critter skimming along the surface of a still pond would get the same sort of view if you were wading nearby. It would see the nearly circular cross sections of your legs as mysteriously shifting lines.

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Muse, October 2000, p. 26.

The complete text of Flatland is available at http://www.geom.uiuc.edu/~banchoff/Flatland/.

To learn more about Flatland: The Movie, go to http://www.flatlandthemovie.com/.

Eye Reflections

Ever caught someone staring off into space and wondered what the heck they are looking at? To find out, you might follow their gaze, study their expression, or hey, just ask them, right? But now there's an even better, more precise way: Observe what is being reflected on the surface of their eyes.


Here's how to try this out: Get close and look right into a friend's eyes. You'll see a curved reflection of your face, a nearby window, a bench, a tree, or whatever else might be in view. Among the many important things they do, eyes can act like little mirrors that reflect exactly what a person is looking at. So, you can learn a lot about what a person is looking at and what his or her surroundings are like just by studying these reflections.

Interestingly, the curved panorama that you see reflected in an eye is broader than the image that falls on the retina at the back of the eye, which converts light into electrical impulses that go to the brain and tell you what you are seeing. This means that the wide-angle reflection shows more of the surroundings than the viewer (the person whose eyes are doing the looking) would actually detect at any given moment.

You can observe these reflections in a detailed, high-resolution photograph of a person's eye. If you look closely at the enlarged image, you might even catch a glimpse of the photographer who took the picture. A digital image of the reflection in the eye at that moment would also show what the viewer might have seen if he or she had gazed in another direction.


Two computer scientists, Ko Nishino of Drexel University and Shree K. Nayar of Columbia University, have developed a system for gleaning information from these reflected images. They use a digital camera to snap closeups of people's faces. A computer then processes the resulting images, letting the researchers reconstruct the views. By studying the reflections, they can even work out exactly what a person in any photo is actually looking at. There goes the "I just didn't notice" excuse!

In one recent application, photo buffs enlarged images of eyes in black-and-white portraits taken more than 150 years ago. Though fuzzy, the resulting light-and-dark patterns provided tantalizing glimpses of the places where the pictures were taken.

And, if you're like James Boind in the movie Goldfinger, a reflection in a villain's eye might save your life when you glimpse the villain's accomplice trying to sneak up behind you.

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Muse, April 2007, p. 29.

Photo credits:
Nishino and Nayar

Tesseracts: Cubes Get Hyper

Madeleine L'Engle, who wrote these words in A Wrinkle in Time, used tesseract to mean a shortcut through space and time. In her story, space-time wrinkles, or folds onto itself, creating new paths that allow characters to tesser, or travel from one end of the galaxy to the other in an instant.


Mathematicians also use the word tesseract, but they mean something different. A tesseract is another name for a four-dimensional cube, or a hypercube.

Here's one way to picture what this strange object might look like. You can start by imagining a point floating in space. A mathematical point has no length or width. Mathematicians say the point has no dimension. Moving the point along a straight path to a new position traces out a line. That line is a one-dimensional object. Shifting the line at right angles to its length traces out a square, and a square is a two-dimensional object. Moving the square at right angles to its flat surface traces out a cube—a three-dimensional object.


Here's the mind-boggling part. You have to try to imagine what would happen if you could move the cube in a new, fourth dimension at right angles to the three you've already used. Any drawing or model you might make of the resulting object would look horribly distorted. But you can still get an idea of what a tesseract would look like and even sometimes "see" it in flashes.

How might you recognize a tesseract if you ever encountered one in your multidimensional travels? Well, in our three-dimensional world, it might look something like a large cube that seemed to be spitting out a smaller one.

Even though L'Engle's concept of a tesseract is different, her book has inspired many readers to think more deeply about time and space and mathematics.

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Muse, September 2000, p. 18.

You can learn more about dimensions, tesseracts, and even Madeleine L'Engle at http://www.mathaware.org/mam/00/918/index.html.

Tricky Tables

The shape of a billiard table has a lot to do with the types of shots you can make in a game of billiards. Which of these odd tables do you think would be your best bet for hitting another ball?


With the help of a little geometry, an expert billiards player can figure out exactly where a ball will go. Unless it's a trick shot, the ball will travel in a straight line until it hits a cushion. When it bounces, it obeys a basic law of physics: the angle between the incoming ball's path and the cushion is the same as the angle between the outgoing ball's path and the cushion.


But if one bounce is predictable, many bounces may not be—especially on a tricky table.

Suppose you have a circular table. The mathematician Charles L. Dodgson, who as Lewis Carroll wrote Alice's Adventures in Wonderland, once published a set of rules for a two-player game of circular billiards. In his game, you had to hit other balls as well as bumpers to rack up points quickly.


Suppose you had a ball in the center of the table. To figure out what a ball will do on a table, mathematicians imagine what would happen if there were no friction and the ball could travel forever. It turns out that a ball can follow a path on a circular table that never passes anywhere near the center of the table. So a ball sitting there would never get hit. Dodgson's game isn't as easy as it sounds.


What about a rectangular table? Try setting a large circular pan, hoop, or other round object in the middle of a rectangular poil table or air-hockey table. Put a weight on the object to keep it in place, then see how it affects the movements of a ball or hockey puck during a game.


You'll find that balls or pucks that start off in nearly the same direction soon are on wildly different paths. So even though each bounce is predictable, after many bounces it is hard to say where a ball or puck will end up!

What about the ellipitical table? If the rectangular table with a circular obstacle is unpredictable, the elliptical table is totally predictable. Suppose balls are placed in the spots shown below:


If a player shoots one of the balls in any direction, it will hit the edge, bounce off and collide with the other ball. The player doesn't even have to aim. One ball will always hit the other!

So the best bet for hitting another ball is the elliptical table. Did you guess right?

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Muse, July/August 2000, p. 26.

Weird Dice

The dice game known as Piggy punishes the greedy player. In Piggy, you and your opponent take turns rolling a pair of ordinary dice. Your score is the sum of the face values of the dice, so if you roll a three and a four, you get seven points. The first player to reach 100 points wins.

Here's the fiendish part. You can roll as many times as you want in a turn, but as soon as you roll doubles, you lose all the points you have won in that turn, and your turn is over.


Now suppose that you can choose between using a standard pair of dice or a special pair of dice. Instead of having faces marked with 1, 2, 3, 4, 5, and 6 dots, one of the weird dice has faces labeled 1, 2, 2, 3, 3, and 4, and the other has faces labeled 1, 3, 4, 5, 6, and 8.

Amazingly, your chances of rolling any particular total are the same with either pair of dice.

The chart shows there are only two ways (shaded yellow-green) to roll a sum of three with the standard dice. The same is true of the weird dice. Similarly, with both pairs of dice there are six ways (shaded purple) to roll a sum of seven, and so on.


Would you use the weird dice in a game of Piggy? If you did, your chances of getting a particular total on a given roll would not change. However, if you look closely at the chart, you'll see that doubles come up more often with standard dice than with weird dice. (Doubles are shaded blue.) Since the Piggy player who rolls doubles more often usually loses, you should choose the weird dice.

If you want to try out weird dice with Piggy or a board game, such as Monopoly, you can easily convert standard dice into weird dice. Take one standard die, then cover the 5 with a small sticker and label it 2, and cover the 6 and label it 3. Take another standard die and put a sticker over the 2, labeling it 8.

In a game like Monopoly, where rolling doubles has special consequences, you might find that using the weird dice changes your strategy a little.

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Muse, May/June 2000, p. 18.

Four Corners, Four Faces

To Arthur Silverman, a sculptor in New Orleans, tetrahedrons, or triangular pyramids, are very special. He's been creating sculptures based on the tetrahedron for more than 20 years. You might see examples on display in plazas and office buildings in New Orleans, San Francisco, and other cities in the United States.

Until the age of 50, Silverman had been a successful surgeon. He gave that up, however, to return to interests that had captured his attention when he was a teenager and had visited art museums and tried carving wood.


To make a tetrahedron (above), imagine four points in space. If you join all of the points, the points become the corners of four triangles. The four triangles form the faces of a tetrahedron.

"The tetrahedron is very exciting visually," Silverman says. "It's difficult to anticipate what you are going to see." For example you can stretch several edges of a tetrahedron to create a slim, tall tower. Silverman has a pair of such towers, each 60 feet high, in the middle of a plaza fountain in New Orleans.


The Energy Centre fountain in New Orleans, a sculpture by Arthur Silverman, is made of two tetrahedrons.

You can join tetrahedrons together to create an angular wall down which water can tumble and fall. You can stack them in various ways to create a monument. Or you can balance then on edge or on a corner.

You can also slice tetrahedrons to get interesting cross sections, which can then be used as tiles to cover a wall. You can divide tetrahedrons into intriguing pieces, and then rejoin them in various ways. The possibilities seem endless.


If each face of a tetrahedron is an equilateral triangle, the result is a regular tetrahedron, one of the five Platonic solids. Here, a tetrahedron is shown inside a cube, another Platonic solid.

Silverman has produced more than 300 sculptures based on the tetrahedron. "When I get an idea, I play with it as long as I can," he notes.
Sometimes it takes an artist to reveal the many wonders of a seemingly simple geometric form like the humble tetrahedron.

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Muse, April 2000, p. 26.

You can learn more about Arthur Silverman's art at http://www.artsilverman.com/, http://www.barnettfineart.com/artistsDetail.php?ARTIST_ID=64, http://www.bilhenrygallery.com/silverman/, and http://www.lemieuxgalleries.com/artist_silverman.html.

Photo credit:
Courtesy of Arthur Silverman.