|Ivars Peterson's MathTrek|
September 4, 2000
Fermilab's Wilson Hall. Courtesy of Fermilab.
Named for physicist and accelerator builder Robert Rathbun Wilson (1914-2000), the building celebrates Wilson's vision and skill, not only as a scientist but also as an artist. Made leader in 1967 of the effort to build a world-class proton synchrotron and founding director of Fermilab, Wilson played a central role in designing the entire laboratory. He was also widely recognized as an accomplished sculptor. Wilson died on Jan. 16 at the age of 85.
The Fermilab grounds feature several of Wilson's sculptures (see http://www.fnal.gov/projects/history/sculpture.html). One of the most striking examples is a tubular form 8 feet in diameter, which sits in the middle of a circular pool atop the Norman F. Ramsey Auditorium in front of Wilson Hall. Having spent part of his youth on a Wyoming cattle ranch, Wilson was no stranger to the blacksmith shop, and he welded together the sculpture from 3 by 5-inch pieces of stainless steel.
"Möbius Strip" by Robert R. Wilson. Courtesy of Fermilab.
Wilson called the sculpture "Möbius Strip," but it isn't the usual rectangular strip with its ends taped together after giving one end a 180-degree twist (see Möbius and His Band, July 10, 2000). It's a three-dimensional version of this "one-sided" surface. The tube's cross section is essentially an equilateral triangle, and this triangle is rotated through 120 degrees along the tube.
Wilson used the same Möbius form as the basis for a gleaming bronze sculpture now installed near an entrance to the Science Center at Harvard University in Cambridge, Mass.
"Topological III" by Robert R. Wilson.
It's possible to create a variety of three-dimensional variants of the Möbius band. Thickening a standard Möbius strip so that its edge becomes as wide as the strip's side produces a three-dimensional object with a square cross section. The resulting form has two edges and two faces (see Möbius in the Playground, May 24, 1999).
You can construct a model of this surface from a length of foam rubber with a square cross section. After rotating one end through 180 degrees, you simply fasten the ends together to form the twisted loop. If instead you rotate one end through 90 degrees before linking the ends, you wind up with a twisted loop with just one edge and one face!
You can do the same thing with a length of foam rubber having a triangular cross section to get the shape on which Wilson based his intriguing Möbius sculptures.
Wilson is just one of many artists who have sculpted Möbius strips. One of the first to do so was Max Bill, who lives in Zumikon, Switzerland. Starting in the 1930s, he created a variety of "endless ribbons." You can see two examples crafted out of metal at http://www.bangor.ac.uk/SculMath/image/mb.htm and a remarkable one carved from granite at http://www.bluffton.edu/~sullivanm/baltimore/baltimore1.html.
Bill has been a strong advocate of using mathematics as a framework for art. "I am convinced it is possible to evolve a new form of art in which the artist's work could be founded to quite a substantial degree on a mathematical line of approach to its content," he argued in a 1949 essay.
English sculptor John Robinson has also explored various mathematical themes in his series of "symbolic" sculptures. Robinson's "Immortality" bronze, for instance, is a Möbius band in the form of a trefoil knot (see http://www.bangor.ac.uk/SculMath/image/main.htm and http://www.bradshaw-foundation.com/jr/1immorta.htm). Another sculpture, named "Journey," is at Macquarie University in Sydney, Australia (see http://www.bangor.ac.uk/SculMath/image/journey.htm). This tall, angular form, made of polished stainless steel, is based on the so-called Brehm model of the Möbius strip.
In 1980, Robinson independently discovered and created several versions of the same three-dimensional Möbius form that had inspired Wilson.
"I prepared 100 equilateral triangles, each with a hole in the middle, and threaded them onto a ring," Robinson recounts. Following the faces around the loop, he saw that he had a surface with three edges and three bands. "By changing one of the edges to meet another, I found I was left with only one edge and one band," he says. "It was a magical moment when I realized that I had created "Eternity."
Polished bronze versions of "Eternity" can be seen in Canberra, Australia, and Aspen, Colorado (http://www.bangor.ac.uk/SculMath/image/eternity.htm).
Visitors to Washington, D.C., can find several Möbius forms in very public places--if they know where to look or understand what they're seeing.
A Möbius strip in the form of an 8-foot-high, stainless-steel sculpture rests atop a tall pedestal in front of the National Museum of American History. Designed by José de Rivera, who titled the piece "Infinity," this swooping form has been revolving majestically since 1967.
"Infinity" by José de Rivera.
Across the Mall, a more complicated Möbius form stands guard at the entrance to the National Air and Space Museum. Created by sculptor Charles O. Perry of Norwalk, Conn., it is called "Continuum."
Several miles away in a plaza in front of the U.S. Patent and Trademark Office in Arlington, Va., stands another Möbius strip--a giant, calligraphic loop of twisted, red-painted steel--also created by Perry. A companion piece, a short distance away, presents an entirely different perspective on this intriguing topological shape.
The amazingly different ways in which artists can depict a Möbius surface are nice reminders that a topological form retains its essential character--in this case, its one-sidedness--no matter how much the figure is deformed, just as long as it isn't punctured or torn. Such infinite flexibility offers a vast playground for creative reconfiguration of an intriguing shape.
Charles Perry (right) helping with the installation of "Helix Möbius Mace" in Crystal City, Arlington, Va.
Perry's monumental sculptures can be found in many cities around the world. An astonishingly large proportion of them involve the Möbius strip in some way. If there's a Perry sculpture in your neighborhood or you happen to encounter one when you're traveling, take a close look at it and try to decipher its geometric message.
"My pieces are ordered unto themselves, as if space had a set of rules similar to the counterpoint of music," Perry remarks, perhaps a little cryptically. "At times, the works will have a single generating order which will grow to completion and stop, while others will have overlays of mutating effects. Most, however, are given intertwining themes which are meant to be played upon by sunlight. Many have some form of transparency to them. All have a sequential movement."
You'll find Perry sculptures in San Francisco, El Segundo, Costa Mesa, and Beverly Hills, Calif.; Sydney, Perth, and Ringwood, Australia; Dallas and Midland, Texas; Singapore; Fairfield, Westport, Greenwich, and Stamford, Conn.; Falls Church and Virginia Beach, Va.; Minneapolis, Minn.; Knoxville, Tenn.; Tampa, Jacksonville, and Maitland, Fla.; Atlanta, Ga.; Hanover, N.H.; Charlotte, N.C.; Tokyo and Kokubo, Japan; Bloomington, Ind.; and elsewhere. See some examples at http://www.cs.berkeley.edu/~sequin/SCULPTS/PERRY/.
Who would have thought that looking for Möbius could turn into a remarkably engrossing, wide-ranging mathematical treasure hunt?
Copyright 2000 by Ivars Peterson
Bill, M. 1993. The mathematical way of thinking in the visual art of our time. In The Visual Mind: Art and Mathematics, M. Emmer, ed. Cambridge, Mass.: MIT Press.
Brown, R. 1999. John Robinson's symbolic sculptures, knots and mathematics. In ISAMA 99, N. Friedman and J. Barrallo, eds. San Sebastián, Spain: University of the Basque Country. (See http://www.sc.ehu.es/mathema1/ISAMA99.htm.)
______. 1994. Sculptures by John Robinson at the University of Wales, Bangor. Mathematical Intelligencer 16(No. 3):62.
Burger, E.B., and M. Starbird. 2000. The band that didn't stop playing. In The Heart of Mathematics: An Invitation to Effective Thinking. Emeryville, Calif.: Key College Publishing (http://www.keycollege.com/Pages/ProdBurger.html).
Perry, C.O. 1998. Continuum, broken symmetry, and more. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings, R. Sarhangi, ed. (See http://www.sckans.edu/~bridges/.)
______. 1993. On the edge of science: The role of the artist's intuition in science. In The Visual Mind: Art and Mathematics, M. Emmer, ed. Cambridge, Mass.: MIT Press.
Silverman, A. 2000. The magician: Robert Rathbun Wilson 1914-2000. CERN Courier 40(March):13. Available at http://www.cerncourier.com/main/article/40/2/13.
A biography of Robert R. Wilson can be found at http://www.fnal.gov/directorate/public_affairs/ferminews/ferminews00-01-28/, and his sculpture is featured at http://www.fnal.gov/directorate/public_affairs/ferminews/ferminews00-01-28/sculpture.html and http://www.fnal.gov/projects/history/sculpture.html.
Information about architecture and sculpture at Fermilab is available at http://www.fnal.gov/pub/architecture.html.
You can learn about Saint-Pierre Cathedral in Beauvais, France, which inspired the design of Fermilab's Wilson Hall, at http://www2.art.utah.edu/cathedral/beauvais.html.
Comments are welcome. Please send messages to Ivars Peterson at email@example.com.
Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He also writes for the children's magazine Muse (http://www.musemag.com) and is working on a book about math and art.