The Fabulously Odd 11-Cell

Polyhedra consist of triangles, squares, pentagons, hexagons, and other polygons that are joined together to form closed, three-dimensional objects. Different rules for linking various polygons generate different types of polyhedra.

The Platonic solids, known to the ancient Greeks, consist entirely of identical regular polygons, which are defined as having equal sides and equal angles. There are precisely five such three-dimensional objects: the tetrahedron (made up of four equilateral triangles), cube (six squares), octahedron (eight equilateral triangles), dodecahedron (12 regular pentagons), and icosahedron (20 equilateral triangles).

These objects are all convex; in other words, they have no indentations. Their extreme regularity also gives them a high degree of symmetry.

Euclid's Elements contains a lengthy mathematical description of the Platonic solids—and a proof that there are no more than five.

"It is hard to overstate how profoundly amazing this proof must have been—and remains," Jaron Lanier writes in the April Discover. "The identities of the five shapes, and the certainty that there can be no more than five, [are] absolute and universal."

You can extend the notion of a Platonic solid to four dimensions, though the results can be very difficult to visualize. There are six such analogs, having 5, 8, 16, 24, 120, and 600 cells, where a cell is the three-dimensional analog of a polygonal face, or side.

But there's another four-dimensional curiosity that can also be considered analogous to a Platonic solid: a four-dimensional shape made up of 11 identical cells, or 11-cell. This curious form was discovered in 1976 by Branko Grünbaum, then later rediscovered and analyzed by H.S.M. Coxeter.

The fact that this shape consists of 11 (a prime number) faces might suggest that the form would lack the high degree of symmetry expected of a Platonic solid. Indeed, Freeman Dyson once asked, after hearing of Coxeter's discovery, "Can you imagine a regular polyhedron, a body composed of perfectly symmetrical cells arranged in a perfectly symmetrical structure, having a total of eleven faces?"

But the 11-cell has an unusual form. Its identical cells, when separated, aren't conventional, three-dimensional objects. And their sides actual pierce or coincide with each other in the combined form.

Plato would have been delighted to know about it, Dyson commented to Coxeter.

Lanier, with the help of computer scientist Carlo Séquin of the University of California, Berkeley, set out visualize this odd object. Each component of an 11-cell is a special shape called a hemi-icosahedron. You can visualize it as half an icosahedron that is folded into an octahedron, with some missing faces and a few others that coincide or interpenetrate. Yet, in four dimensions, these strange cells fit together in a perfectly regular manner.

The following image, created by Séquin and Lanier, shows a partial 11-cell—one consisting of a symmetrical arrangement of five hemi-icosahedra.


The 11-cell is also self-dual. If you draw lines between the centers of every facet of the 11-cell, you get another 11-cell.

Lanier originally dubbed this shape a hendecatope, meaning "11-related place" in Greek. The name was later changed to hendecachoron to differentiate it from certain other objects. Lanier and Séquin have prepared a presentation on the hendecachoron for the upcoming ISAMA '07 conference in College Station, Texas.

Last year, Dimitri Leemans and Egon Schulte showed that there can be only two shapes like the 11-cell. The other is the 57-cell, which had been discovered by Coxeter. However, 57 is not a prime number, so the 11-cell retains a unique place in the pantheon of polytopes.

References:

Dyson, F. 1983. Unfashionable pursuits. Mathematical Intelligencer 5(No. 3):47-54.

Lanier, J. 2007. Jaron's world: Shapes in other dimensions. Discover 28(April):28-29.

Leemans, D., and E. Schulte. In press. Groups of type L₂(q) acting on polytopes. Advances in Geometry. Abstract.

Peterson, I. 2001. Polyhedron man. Science News 160(Dec. 22&29):396-398.

Roberts, S. 2006. King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company.

Séquin, C.H., and J. Lanier. Preprint. Hyperseeing the regular hendecachoron.

Meccano Math

When I was growing up in Canada, one of my favorite playthings was my Meccano set—a wondrous collection of metal strips, plates, wheels, gears, and other parts that could be assembled into all sorts of mechanical contraptions.

Over the years, ingenious mechanics have assembled available parts into cars, trucks, cranes, construction equipment, and much more. Enthusiast Tim Robinson has even used standard Meccano parts to create scale models of mechanical computers (two difference engines and a differential analyzer) invented by Charles Babbage (1791–1871) in the 19th century.

Theoretical physicist Gerard 't Hooft of the University of Utrecht also has fond memories of playing with a Meccano set as a child. He has recently turned this long-ago interest into an intriguing and entertaining mathematical pursuit. 't Hooft shared the 1999 Nobel Prize in Physics for his part in elucidating the theory of the electroweak interaction in particle physics.

As described in a paper available at his Web page, 't Hooft focuses only on the metal strips, each one having a certain number of holes a fixed, integral distance apart. You can fasten these strips together in various ways using nuts and bolts.


The Meccano strips shown above have 7 holes (blue, length 6) and 4 holes (yellow, length 3). They can be fastened together at any of the holes, using bolts. All images courtesy of Gerard 't Hooft.

What sorts of planar geometric figures can you make from these strips?

"One can basically construct all of the fundamental figures that can be generated using the axioms of Euclidean geometry," 't Hooft contends.

"Usually, Euclidean geometry refers to compass and straightedge as the only legitimate tools," he continues. "Here, I replace compass and straightedge by Meccano strips."

In doing so, 't Hooft says that he finds, perhaps surprisingly, that "Meccano strips are more versatile that compass and straightedge."

In his constructions, 't Hooft assumes that the strips are strictly two-dimensional and that as many of them as desired may be completely or partly overlapping. The bolts take up no space.

't Hooft shows how it's possible to combine strips to construct new units that behave as strips in which the holes can be chosen at any rational position (p/q) on the strip. It's also possible to construct strips of length a + √(b), where a and b are any integers.


This construction produces a generalized strip that effectively has holes at the positions A, A + 1, . . . B, and C, where distance BC is the rational number qr/d. The resulting construction is equivalent to the strip shown to the right, which isn't available as a standard part.

His constructions also include examples in which strips can be combined so that one of the end points can move only along a perfect straight line segment. This feature can then be deployed to not only bisect but also trisect an angle.

't Hooft goes on to describe ways to construct a rigid regular pentagon. "We start with five strips of the same length, but to fix them rigidly . . . more strips are needed," he notes. His most straightforward construction of such a pentagon requires 11 strips (below).


One way to construct a rigid regular pentagon (brown).

Other polygons are also accessible. A bit of serendipity, 't Hooft says, led to a regular heptagon of only 15 pieces.

There's a lot more to explore in this Euclidean realm of integral Meccano strips. 't Hooft has a number of suggestions for further investigations.

And, luckily, Meccano sets are still available, though the brand name and assets, originally British, have been transferred to a Japanese company (Nikko Toys). Kids in the United States may be more familiar with the Erector Set brand, a similar toy construction kit. Erector sets are now also made by Meccano factories in France and China.

References:

Peterson, I. 2006. Constructing difference engines. MAA Online (May 1).

't Hooft, G. Preprint. Meccano math.

Flatland: The Movies

Like shadows, the denizens of Flatland flit about freely on the surface of their two-dimensional world. All Flatland's inhabitants—straight lines, triangles, squares, pentagons, and other figures—are trapped in their planar geometry. They lack the power (and the imagination) to rise above or sink below the flat surface of their strictly limited realm.

That's the idea behind a remarkable book called Flatland: A Romance of Many Dimensions, written in 1884 by Edwin A. Abbott (1838–1926), the head of a school for boys in London.

Conceived as a satire, this slim volume has long served as a gateway to the fourth dimension and beyond for many explorers of geometry. Still available in a variety of editions, Flatland has attracted all sorts of attention over the years. 

Mathematician Thomas F. Banchoff of Brown University has looked extensively into Abbott's life and background, pursuing the question of how Abbott came to write this book. Banchoff himself provided some insights in the introduction to a Princeton University Press edition of Flatland.

Mathematician Ian Stewart of the University of Warwick produced not only an annotated edition of Flatland but also an entertaining sequel that he titled Flatterland: Like Flatland, Only More So.

In 2007, two teams of filmmakers have created colorful animated versions of Abbott's classic. One production is the work of filmmakers Jeffrey Travis (director) and Seth Caplan (producer). About 30 minutes long, it features the voices of Martin Sheen (Arthur Square) and Kristen Bell (Square's granddaughter, Hex).

The movie is part of an educational DVD that also includes the original text of the book, teacher notes, games, and quizzes. The movie made its Mathematical Association of America debut in August 2007 at MathFest 2007 in San Jose, California. After the showing, Banchoff led a discussion about Flatland and its use in the classroom. Banchoff was on the film's advisory board, along with Jon Farley, Sarah Greenwald, and John Benson.

The second production comes from independent filmmaker Ladd Ehlinger Jr., of Huntsville, Ala. Working with Tom Whalen (writer) and Mark Slater (composer), Ehlinger created an 83-minute, wide-screen epic that fleshes out Abbott's tale into a dramatic political fable that hinges on the invasion of Flatland by three-dimensional beings.

It echoes with commentary on racism, social rigidity, political manipulation, and challenges to accepted belief. For more details, including film clips and music samples, see http://www.flatlandthefilm.com/. To be able to afford to bring his feature to film festivals and to the attention of film distributors, Ehlinger is currently selling a special "collector's edition" DVD of the production. It'll be interesting to see what comes out of these ventures, but, whatever the result, the original Flatland is as accessible as ever—in print.

References: Flatland: The Movie. Flatland: The Film. Abbott, E.A. 2005. Flatland: A Romance of Many Dimensions, with an introduction by Thomas F. Banchoff. Princeton, N.J.: Princeton University Press. ______. 2002. The Annotated Flatland: A Romance of Many Dimensions, with introduction and notes by Ian Stewart. Cambridge, Mass.: Perseus. ______. 1992. Flatland: A Romance of Many Dimensions. Dover. Banchoff, T.F. 1990. From Flatland to hypergraphics. Interdisciplinary Science Reviews 15(No. 4):364-372. Peterson, I. 2000. Views from Flatland. Muse 4(October):26-27. ______. 2000. A stranger from Spaceland. MAA Online (Dec. 20). The full text of Flatland is available at http://www.geom.uiuc.edu/~banchoff/Flatland/.

Waves of a Sea Battle

In June 1943, a British submarine, the United, attacked an Italian supply ship in Mediterranean waters. This encounter was recorded in a dramatic aerial photo that reveals not only details of the ensuing battle but also a fascinating array of wave patterns.


Surface water waves from a World War II sea battle. Courtesy of A.D. Kirwan, University of Delaware.

In the photo, the United is at point A. Some of its torpedoes missed, passing the Italian ship. Their wakes are visible, leaving the photo's frame at D. Some torpedoes, however, struck the freighter (B). The ship apparently lost control of its rudder and began steaming erratically before finally sinking at C. In the meantime, an Italian airplane attempted to sink the submarine, dropping a bomb and setting off an explosion at E.

"This photograph is fascinating for scientific as well as historical reasons," Katherine Socha wrote in the March 2007 American Mathematical Monthly. "Its most striking features are the patterns of circular wavefronts visible at the edges of the frame and around point E. The motion of such water waves has been the subject of scientific and mathematical study for years."

Socha noted that the battle photo reveals a distinctive circular wave pattern surrounding the point of impact at E: The separation between wavefronts increases with distance from the explosion. This observation suggests that the outer waves, with longer wavelengths, traveled away from the impact point faster than did the inner waves, with shorter wavelengths. The dependence of wave speed on wavelength is known as dispersion.

In contrast, ripples created by raindrops falling on the surface of a shallow pool show the opposite effect. In this case, longer wavelengths travel more slowly than do shorter wavelengths, so the pattern is reversed.

"It is fascinating that the same physical impetus (an object striking the water surface) can generate qualitatively opposite behavior," Socha said.

A simple mathematical model of surface water waves reveals that water's surface tension is the dominant factor when the water is shallow (as in a pond or pool) and the wavelengths are relatively short. On certain other scales (as on the open sea), the effect of surface tension is negligible and gravity is the dominant factor.

In these two situations, dispersion relations giving wave speed as a function of wavelength lead to opposite results. For ocean waves, the speed of the waves is proportional to the square root of the wavelength. For pond waves, the speed is inversely proportional to the square root of the wavelength.

In the gravity wave case, the wave speed, c_g is given approximately by the following formula, where g is the gravitational constant and lambda (ƛ) is the wavelength.


"Even fairly crude models may capture essential features of a physical situation, at least qualitatively," Socha said. "It is also surprising that, despite the many simplifying approximations, the model describes this qualitative difference in enough detail to be measured convincingly."

In the battle scene, under the assumption that the freighter at point C is about 100 meters long, you can establish a length scale for the photo and estimate the wavelength of the leading (longest) wave and the distance from the impact point to the leading edge of the wavefront generated by the explosion. This distance measurement, together with the approximate wave speed, allows you to determine the lag in time between the explosion and when the photo was taken.


Socha first encountered this remarkable photo when she was taking a graduate course in fluid dynamics given by Thomas M. Dillon. The photo and a question about the lag between time of impact and time of photograph were a final exam puzzle.

Dillon had obtained a copy of the photo from oceanographer A.D. Kirwan. Kirwan noted that the photo has a long history of use in oceanography circles, though exactly who took it and under what circumstances isn't known.

References:

Socha, K. 2007. Circles in circles: Creating a mathematical model of surface water waves. American Mathematical Monthly 114(March):202-216.