July 28, 2010

SET Math

The card game known as SET® is deceptively simple. Its object is to identify, as quickly as possible, a grouping (SET) of three cards, selected from 12 cards laid out face up on a table.

A SET deck has 81 (34) cards. Each card displays a design with four attributes: shape, number, shading, and color. Each attribute has three possible values.

Attribute
Values
Shape
{oval, diamond, squiggle}
Number
{one, two, three}
Shading
{striped, solid, open}
Color
{red, green, purple}

For the purposes of the game, three cards are called a SET if, with respect to each of the four attributes, the cards are either all the same or all different. For example, the following three cards would constitute a SET because the cards are different for all attributes.


Invented in 1974 by population geneticist Marsha Jean Falco, the game has become a popular, even addictive pastime for both children and adults. It has also attracted mathematical attention.

"Although children often beat adults, the game has a rich mathematical structure linking it to the combinatorics of finite affine and projective spaces and the theory of error-correcting codes," Diane Maclagan and Benjamin Lent Davis remarked in a paper published in the September 2003 Mathematical Intelligencer. In 2002, “an unexpected connection to Fourier analysis was used to settle a basic question directly related to the game of SET, and many related questions remain open."

The cards that make up SET have several important combinatorial properties. Given any two cards, there exists one and only one card that forms a set with those cards. Hence, the probability of producing a SET from three randomly drawn cards is 1/79. Overall, the deck has 1080 unique winning combinations.

One obvious mathematical question about the game concerns the number of cards that must be dealt to guarantee the presence of a SET.

Anyone who has played the game knows that 12 cards are sometimes not enough to find a SET. Indeed, the rules specify that, if no SET is found, three additional cards must be dealt for the game to continue. This is repeated until a SET makes an appearance.

One way to find out how many cards are needed is to do an exhaustive computer search of all the possible combinations. Such a search would reveal a collection of 20 cards that has no SET. Every collection of 21 cards contains a SET.

It's also possible to picture each card as a point in four-dimensional space, where each of a point's four coordinates assumes one of three possible values. Three cards form a SET if and only if the three associated points are all on the same straight line in this finite four-dimensional space.

In effect, Maclagan and Davis noted, players of SET are searching for lines contained in a subset of this finite affine space. They then defined a cap to be a subset of the space not containing any lines and asked the maximum possible size of a cap in the given space, which is equivalent to asking for the maximum number of cards that would have no SET among them.

Interestingly, this question was answered in a mathematical context in 1971, 3 years before SET was invented.

It's also possible to extend the problem in various ways. "Although SET cards are described by four attributes, from a mathematical perspective there is nothing sacred about the number four," Maclagan and Davis wrote. "We can play a three-attribute version of SET, for example, by playing only with the red cards. Or we can play a five-attribute version of set by using scratch-and-sniff SET cards with three different odors."

You can then ask how the size of the maximal cap, a, depends on the number of attributes (dimension, d). The answer for five dimensions was worked out some years ago, and the value for six dimensions isn't yet known exactly.

Dimension
1
2
3
4
5
6
7
Maximal cap
2
4
9
20
45
112
?

There are many more possible generalizations. For example, you could add another color, shape, form of shading, and number to the cards. Such generalizations suggest a host of new mathematical questions.

SET cards can also be used for other pursuits in recreational mathematics. For example, you could look for SET magic squares. The idea is to arrange selected cards in a three-by-three array so that any line on the square yields a SET. In fact, you can start with any three cards, and there will always be a way to fill in the rest of the blanks to make a SET magic square.

In the meantime, you can visit the official SET website and try the SET daily puzzle to get yourself warmed up for deeper mathematical challenges.

Originally posted Aug. 25, 2003.
Updated July 28, 2010.

References:

Brink, D.V. 1997. The search for SET (June).

Davis, B.L., and D. Maclagan. 2003. The card game SET. Mathematical Intelligencer 25(No. 3):33-40.

Magliery, T. 1999. The SET® Home Page.

Zabrocki, M. 2001. The Joy of SET.

July 21, 2010

Knight Moves in 3D

The crazily crinkled structure, framed within a cube, is an impressive sight. Suspended above the stairs on the sixth floor of the mathematics building at St. Olaf College in Northfield, Minn., it is also the solution to a mathematical puzzle.


Painstakingly crafted from wood by retired St. Olaf mathematician Loren Larson, the sculpture embodies a path that a knight could take on a three-dimensional chessboard to complete a tour of the 512 (83) positions on the grid, visiting each position just once before returning to the starting point.

According to the rules of chess, a knight makes an L-shaped move that shifts its position by a single square in one direction and two squares in a perpendicular direction. Indeed, the knight is the only chess piece that covers an asymmetrical pattern of squares.

Finding a sequence of 64 knight moves that visit each square of a standard chessboard—a knight's tour—is a classic problem with a long history. Leonhard Euler (1707-1783) described a method for constructing such a tour in 1759, and mathematicians have investigated many variants of the problem since (see, for example, "A Magic Knight's Tour"), including knight's tours in three dimensions.

In Larson's sculpted solution, the sticks representing the 512 moves gradually change in color from pale yellow to deep red, with the final move linking the lightest yellow piece to the darkest red piece.


The color changes give insights into the algorithm used to find the route. If you start at or near a face of the 8 x 8 x 8 cube, you first visit other positions that have relatively few outlets, such as corners or edges. In effect, you knock off the hardest cases first. The result is that the tangle of moves seems to get darker and darker as you move toward the cube's center.

Photos by I. Peterson

July 19, 2010

Fractal Drum

A drum vibrates at characteristic frequencies, depending mainly on the size, shape, tension, and composition of its sound-generating drumhead. This spectrum of frequencies—the set of pure tones produced by a vibrating membrane stretched across a frame—gives a drum its particular voice.

Physicists and mathematicians have long recognized that the shape of the boundary enclosing a membrane plays a crucial role in determining the membrane's spectrum of vibrations. So, what happens when a drum has an intricately indented rim—one so wiggly that the boundary consists of crinkles atop crinkles?

To find out, Michel L. Lapidus and his colleagues set out some years ago to study mathematically the vibrations of a drum with a fractal boundary—a fractal known as the Koch snowflake.


Starting with a large equilateral triangle, you can add smaller triangles to the middle of each side to create a six-pointed star. Adding even smaller triangles to each of the star's twelve sides generates a crinkly shape. Continuing the process by adding increasingly small triangles produces the intricately frilled Koch snowflake.

Idealized to its mathematical essence, a drum is a flat, two-dimensional surface held fixed along its rim. Only the interior moves, which greatly restricts the surface's possible motions.

The resulting vibrations, or normal modes, represent the solutions of a mathematical expression known as the wave equation. In the drum's case, the solutions specify the vertical displacement of each point on a surface bounded by a closed curve, such as a circle, rectangle, or fractal.

In the 1990s, working with Lapidus, Cheryl A. Griffith, Robert Renka, and John W. Neuberger of the University of North Texas sought to display normal-mode vibrations of a drum with a fractal boundary on the computer screen. They computed the first 40 eigenfunctions of the Koch snowflake. Their colorful images vividly showed the dramatically frilled edges of the waveforms created on such membranes.

Entranced by the beautiful forms, Neuberger commissioned mathematician and sculptor Helaman Ferguson to create a bronze rendering of one of the modes that the team studied. Ferguson called his creation Texas Snowflake 13th Eigenfunction.


Helaman Ferguson's sculpture depicts the thirteenth eigenfunction, or normal mode of vibration, of the Koch snowflake. This mode has seven bumps and sixfold symmetry.

At Neuberger's request, Ferguson then created another sculpture, this time depicting the sixth eigenfunction of the Koch snowflake, with so-called Neumann boundary conditions. This mode has threefold symmetry.


Reference:

Peterson, I. 1994. Beating a fractal drum. Science News 146(Sept. 17):184-185.

Photos by I. Peterson

July 17, 2010

Cloud Gate

The law of reflection, combined with a simple geometry, can lead to visual wonders of surprising intricacy and complexity. Cloud Gate, a sculpture by Anish Kapoor, is a particularly striking example of such a spectacle.


The large structure—66 feet long, 42 feet wide, 33 feet high—seems barely tethered to the ground of AT&T Plaza in Chicago's Millennium Park. It has no color of its own; it vaguely resembles a giant jelly bean.


Inspired by globules of liquid mercury, the sculpture was crafted from sheets of stainless steel, seamlessly pieced together into an elliptical form to offer an enthralling lesson in optics. Its highly polished, gracefully curved surface mirrors the surrounding cityscape of tall buildings and the clouds above.


The inwardly curved underside—a 12-foot-high arch gateway—creates a hollow that invites exploration and offers additional visual surprises. Captured in startling, perplexing reflections, the viewer becomes part of the artwork—a part of the visual experience.


Photos by I. Peterson

July 15, 2010

Puzzling Lines

Artist Sol LeWitt (1928-2007) often featured geometric and combinatorial themes in his numerous creations (see "LeWitt's Pyramid"). Indeed, many of his earlier artworks were explicitly combinatorial.

In 1973, for example, he composed Straight Lines in Four Directions and All Their Possible Combinations. This study consisted of 15 square etchings, each inscribed with one or more horizontal, vertical, and diagonal lines in different orientations.

When Barry Cipra, a mathematician and writer in Northfield, Minn., first saw this set of etchings, arranged in a grid, he was intrigued by how the eye automatically tried to connect the lines from one square to the next. In the arrangement that he saw, however, none of the horizontal or vertical lines went completely from one side of the grid to the other.

Cipra asked himself if it would be possible to rearrange 16 squares (one of them blank), without rotating any of the squares, so that all horizontal, vertical, or diagonal lines are unbroken within a four-by-four grid. That was the birth of a challenging mathematical puzzle.


Notice that some of the dark lines in the figure (adapted from LeWitt's design) continue from one square to the next. Is it possible to arrange the sixteen squares, keeping them in a four-by-four grid and not rotating any of them, so that all the lines go all the way from one edge of the grid to the other? If so, how many different solutions are possible?

Playing with square pieces cut from stiff cardboard, Cipra quickly discovered that the puzzle has a solution. Indeed, it has three distinct solutions.

Rotating any one of the three distinct solutions through 90 degrees, reflecting it, or performing a combination of these two operations generates another, related solution. So does taking the topmost row (or the leftmost column) and moving it all the way to the bottom (or to the right).

"In other words, each solution could be drawn on the surface of a torus," Cipra notes. He went on to prove that there are no other possibilities by showing that all solutions must have the toroidal property.

Cipra's LeWitt puzzle resembles the famous "15" sliding-tile game. A square tray holds fifteen tiles, with one vacant space. The player slides the pieces around in such a way as to rearrange the initial configuration into the desired one. Using such moves, is it possible to go from an arbitrary arrangement of LeWitt's squares to each of Cipra's solutions?

Cipra hasn't tried that yet. However, "it should be straightforward to check whether the three solutions are related by an even or odd number of pairwise interchanges and also how they relate to the initial configuration," he remarks.

At the same time, Cipra adds, "the LeWitt puzzle is hard enough to solve when you have complete freedom to move the pieces."

Cipra's puzzle also serves as a reminder of the essential playfulness and simplicity of LeWitt's art, where logic, clarity, and beauty interact to evoke visual delight and deep thought.


Original version posted June 12, 2000.
Updated July 15, 2010.

References:

Cipra, B. 2002. The Sol LeWitt puzzle: A problem in 16 squares. In Puzzlers’ Tribute: A Feast for the Mind, D. Wolfe and T. Rodgers, ed. A K Peters.

______. 1999. What’s Happening in the Mathematical Sciences 1998-1999 (Volume 4). American Mathematical Society.

July 13, 2010

Geometreks in Pittsburgh

The summer meeting of the Mathematical Association of America, MathFest 2010, will be held Aug. 5-7 in Pittsburgh. The city has a wonderful array of public art and some spectacular architecture, making it an apt venue for excursions into mathematical art and more.


Four rectangular blocks add a colorful touch to the stark entrance to Wean Hall, Carnegie Mellon University, Pittsburgh.

With a distinct mathematical bias, here's my list of sites that I hope to visit and photograph while I'm in Pittsburgh for the meeting.

David L. Lawrence Convention Center: cable-suspended roof. The convention center itself is home to 25 artworks.
1000 Fort Duquesne Boulevard.

Thirteen Geometric Figures by Sol LeWitt.
Wood Street T Station, Mezzanine Level.

Alcoa Corporate Center: undulating glass fa├žade.
201 Isabella Street, North Shore between the Rachel Carson and Andy Warhol Bridges.

Atop Penn Avenue Place facing the Allegheny River (visible from the North Side).

Cubed Tension by Sylvester Damianos. A rectangular prism bent to form a cube.
Allegheny Center, near the entrance to the Carnegie Library of Pittsburgh Allegheny Regional Branch.

Children's Museum of Pittsburgh: Articulated Cloud wind sculpture by Ned Kahn.
Allegheny Center, 10 Children's Way, Allegheny Square.

Aerial Scape, Skyscape by Virgil Cantini.
One Oliver Plaza, Rear Lobby, 210 Sixth Avenue.

Up & Away by Clement Meadmore.
PNC Bank Plaza, Fifth Avenue and Wood Street.

L's-One Up One Down by George Rickey.
National City Center, 20 Stanwix Street.

PPG Place: glass spires; 44-foot-tall obelisk.
Fourth Avenue and Market Street.



Mellon Square.

Pennsylvanian: central dome skylight.
1100 Liberty Avenue, Liberty Avenue and Grant Street.

501 Grant Street.

North Light by David Von Schlegell.
One Oxford Centre, Grant Street and Fourth Avenue.

Carnegie by Richard Serra.
Entrance to Carnegie Museum of Art, Forbes Avenue.

Kraus Campo by Mel Bochner and Michael Van Valkenburgh. Garden featuring a French curve piece covered in tile with number sequences.
Carnegie Mellon University, on top of the Posner Center.

Light Up! by Tony Smith.
University of Pittsburgh, Hillman Library Courtyard.




Hamerschlag Hall at Carnegie Mellon University, with the University of Pittsburgh's Cathedral of Learning in the background (right).

The Greater Pittsburgh Arts Council has excellent walking-tour guides that highlight a variety of artworks and structures downtown and in the Oakland neighborhood. Much of the information in this article comes from these guides.

Photos by I. Peterson

July 11, 2010

Hexagon Spire

Looking up can present you with remarkable geometric vistas in a variety of settings. One of my most vivid memories is of the interior of a spire that consisted of a sequence of stacked, rotated hexagons of diminishing size, producing a convincing illusion of infinite extent.




The spectacular, soaring spire of the Man in the Community pavilion at Expo '67 in Montreal consisted of a sequence of nested hexagons (video tour of pavilion).


The sequence of inscribed hexagons suggests a variety of math problems. What is the ratio of the areas of the circumscribed regular hexagon to the inscribed regular hexagon? Solution.




Photo by I. Peterson

July 10, 2010

Bending a Square Prism

The sculptures of Clement Meadmore (1929-2005) are based on simple geometric elements—often a square prism, wrought into an evocative form.




Meadmore's Slew (1971) is in front of the Esther Raushenbush Library, Sarah Lawrence College, Bronxville, N.Y.

In a typical sculpture, a single rectangular volume twists and turns upon itself before lunging into space, suggesting a forceful release of tension or a mood of exhilaration.



Out of There (1974) Hale Boggs Federal Building Plaza, New Orleans.

"I believe that we have a sense of three-dimensional form that, along with our sense of balance, enables us to respond directly and naturally to the considered arrangement of clearly defined forms in space," Meadmore once noted. "In my work the forms are geometric; my goal is to make geometry yield an expressive result. I also search for configurations that enable the viewer to see and understand the whole sculpture from any single viewpoint."




Split Ring (1969) Portland Art Museum, Oregon.

Meadmore's emphasis on permutations of simple shapes arose from his belief that a sculpture should maintain a constant visual identity from all viewpoints.


"If from any single viewpoint, one is unable to deduce the nature of the other side, one is only seeing a half-sculpture and will continue to see half-sculptures as one walks around the work," Meadmore explained. "By using a constant square cross section, I am able to make the whole sculpture comprehensible from any single angle, never inducing the feeling that one is seeing a half-sculpture."




Wingspread (1999) Private courtyard, 400 Chambers Street, New York City.

As muscular, dynamic manifestations of geometric shape, Meadmore's massive steel artworks are instantly recognizable. They can be found outdoors in many cities throughout the United States and elsewhere in the world.


Photos by I. Peterson

July 9, 2010

Triune Twists

Standing near the southwest corner of Philadelphia's City Hall, the weathered bronze sculpture has an edge that seems to wind around into a mathematical knot. Its curved surfaces, dedined by the twisty edge, resemble patches of soap film—pieces of minimal surfaces.


Titled The Triune, this bronze is the work of Robert Engman, long a professor of sculpture at the University of Pennsylvania. It is one of several of his artworks that are on public display in Philadelphia.


Located at the center of Miller Plaza on the University of Pennsylvania's campus, Engman's Quadrature #1 flaunts sinuous contours of painted steel against the contrasting rectangular grid of a massive pyramid.

Engman inspired his students to work with mathematical forms and ideas, just as he did in many of his own creations. His work often depicted intersections of minimal forms.


Engman's mobile of cast aluminum, titled After Iyengar, hangs in the lobby of the Chemistry Building at the University of Pennsylvania. It is one of three sculptures he created to honor Indian Yoga master B.K.S. Iyengar.

The Hirshhorn Museum and Sculpture Garden in Washington, D.C., has a number of Engman works, including a version of After Iyengar.

Reference:

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

Photos by I. Peterson

July 8, 2010

Paul Halmos on Writing Mathematics

As a mathematician, Paul R. Halmos (1916-2006) made fundamental contributions to probability theory, statistics, functional analysis, mathematical logic, and other areas of mathematics. He was also known and widely recognized as a masterly mathematical expositor. And he served as editor (1981-1985) of the American Mathematical Monthly.

Halmos described his approach to writing in an essay published in the book How to Write Mathematics (American Mathematical Society, 1973). One paragraph presents the essence of the process:

"The basic problem in writing mathematics is the same as in writing biology, writing a novel, or writing directions for assembling a harpsichord: the problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order that you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation.”

Halmos adds, “That’s all there is to it.”

Halmos then expands on what he sees as the key elements of good mathematical writing.

1. Say something. To have something to say is by far the most important ingredient of good exposition.

2. Speak to someone. Ask yourself who it is that you want to reach.

3. Organize. Arrange the material so as to minimize the resistance and maximize the insight of the reader.

4. Use consistent notation. The letters (or symbols) that you use to denote the concepts that you’ll discuss are worthy of thought and careful design.

5. Write in spirals. Write the first section, write the second section, rewrite the first section, rewrite the second section, write the third section, rewrite the first section, rewrite the second section, rewrite the third section, write the fourth section, and so on.

6. Watch your language. Good English style implies correct grammar, correct choice of words, correct punctuation, and common sense.

7. Be honest. Smooth the reader’s way, anticipating difficulties and forestalling them. Aim for clarity, not pedantry; understanding, not fuss.

8. Remove the irrelevant. Irrelevant assumptions, incorrect emphasis, or even the absence of correct emphasis can wreak havoc.

9. Use words correctly. Think about and use with care the small words of common sense and intuitive logic, and the specifically mathematical words (technical terms) that can have a profound effect on mathematical meaning.

10.Resist symbols. The best notation is no notation; whenever it is possible to avoid the use of a complicated alphabetic apparatus, avoid it.

Halmos concludes: “The basic problems of all expository communication are the same. . . . Content, aim, and organization, plus the vitally important details of grammar, diction, and notation—they, not showmanship, are the essential ingredients of good lectures, as well as good books.”

The 44-minute  film I Want to Be a Mathematician: A Conversation with Paul Halmos is based on an interview with Paul Halmos, in which he discusses various aspects of writing, teaching, and research (Trailer).