Math Trek: Adventures in the MathZone
Ivars Peterson and Nancy Henderson
Barry Cipra. What's Happening in the Mathematical Sciences, 4 volumes. (Providence, R.I.: American Mathematical Society, 1993, 1994, 1996, 1998).
Keith Devlin. Life by the Numbers. (New York: Wiley, 1998).
Harry Henderson. Modern Mathematicians. (New York: Facts on File, 1996).
Ivars Peterson. Islands of Truth: A Mathematical Mystery Cruise. (New York: W. H. Freeman, 1990).
Ivars Peterson. The Jungles of Randomness: A Mathematical Safari. (New York: Wiley, 1998).
Ivars Peterson. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. (New York: W. H. Freeman, 1998).
Charles Snape and Heather Scott. Puzzles, Mazes and Numbers. (Cambridge, England: Cambridge University Press, 1995).
Check out the Math Forum's Middle School Problem of the Week at http://mathforum.org/midpow/ .
Trek 1: Do Knot Enter
Geoffrey Budworth. The Knot Book. (New York: Sterling Publishing, 1985).
Kenneth S. Burton, Jr. Knots: A Step-by-Step Guide to Tying Loops, Hitches, Bends, and Dozens of Other Useful Knots. (Philadelphia: Running Press, 1998).
Martin Gardner. "The topology of knots." In The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. (New York: Copernicus, 1997).
Lee Neuwirth. "The theory of knots." Scientific American 240 (June, 1979): 110-124.
A remarkably complete guide to "Knots on the Web" can be found at http://www.earlham.edu/~peters/knotlink.htm .
"Mathematics and Knots" at http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm presents a grand visual tour of knots and their curious properties.
Untangle the mathematics of knots at http://www.c3.lanl.gov/mega-math/workbk/knot/knot.html .
One place to start to learn about cat's cradle and string figures is the Web site of the International String Figure Association at http://www.isfa.org/ . World-Wide Webs at http://members.xoom.com/_XOOM/darsie/string/index.html shows string figures from around the world.
You can learn about string figure mathematics at http://website.lineone.net/~m.p/sf/menu.html .
Trek 2: The MapZone
Kenneth Appel and Wolfgang Haken. "The solution of the four-color-map problem." Scientific American 237 (October, 1977): 108-121.
Lewis Carroll, with notes by Martin Gardner. More Annotated Alice: Alice's Adventures in Wonderland and Through the Looking-Glass. New York: Random House, 1990).
Lewis Carroll. Mathematical Recreations of Lewis Carroll: Pillow Problems and A Tangled Tale. (New York: Dover, 1958).
Martin Gardner. "The four-color map theorem." In Martin Gardner's New Mathematical Diversions from Scientific American. (New York: Simon & Schuster, 1966).
Ivars Peterson. "Maps of many colors." Science News Online. January 4, 1997. New approaches to proving the four-color map theorem and its variants.
Edward Wakeling, editor. Rediscovered Lewis Carroll Puzzles. (New York: Dover, 1995).
You can find a great introduction to the mathematics of map coloring at http://www.c3.lanl.gov/mega-math/workbk/map/map.html .
Learn about a new proof of the four-color theorem at http://www.math.gatech.edu/~thomas/FC/fourcolor.html .
Trek 3: The Crazy Roller Coaster
Stephen Barr. Experiments in Topology. New York: Dover, 1989.
John Fauvel, Raymond Flood, and Robin Wilson, editors. Möbius and his Band: Mathematics and Astronomy in Nineteenth-century Germany. (Oxford, England: Oxford University Press, 1993).
Claire Ferguson. Helaman Ferguson: Mathematics in Stone and Bronze. (Erie, Penn.: Meridian Creative Group, 1994).
Ivars Peterson. "Möbius in the playground." Science News Online. May 22, 1999. Building a Möbius-strip climber for a playground in Florida.
Ivars Peterson. "Minimal snow." Science News Online. March 6, 1999. Constructing the Costa surface (see Math Trek, page 27) out of packed snow!
Ivars Peterson. "Recycling topology." Science News Online. September 28, 1996. The Möbius strip in the recycling symbol.
Ivars Peterson. The song in the stone. Science News 149 (Feb. 12): 110-112. How Helaman Ferguson created his sculptures of the Costa surface.
Ivars Peterson. Three bites in a doughnut. Science News 127 (March 16): 168-169. The story of the discovery of the Costa surface.
Sculptor Helaman Ferguson has a Web site at http://www.helasculpt.com/ .
You can learn about how some sculptors have used the Möbius strip in their artworks at http://www.bangor.ac.uk/SculMath/image/mb.htm and http://www.bangor.ac.uk/SculMath/image/eternity.htm .
Möbius strips have played pivotal roles in a number of short
stories. The following stories appear in Fantasia Mathematica,
edited by Clifton Fadiman. (New York: Copernicus, 1997):
Trek 5: Mersenne's Fun House
Martin Gardner. "The remarkable lore of the prime numbers." Scientific American 210 (March, 1964): 120-128.
Ivars Peterson. "Mersenne megaprime." Science News Online. July 24, 1999. The latest GIMPS success.
Ivars Peterson. "Prime listening." Science News Online. June 20, 1998. Setting prime numbers to music.
Discover the largest known primes at http://www.utm.edu/research/primes/largest.html .
A biography of Marin Mersenne is available at http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Mersenne.html .
You can learn more about the Great Internet Mersenne Prime Search (GIMPS) at http://www.mersenne.org .
The Goldbach conjecture is featured at http://www.pass.maths.org.uk/issue2/xfile/index.html .
Trek 7: The Fractal Pond Race
Martin Gardner. "Mandelbrot's fractals." In Penrose Tiles to Trapdoor Ciphers . . . and the Return of Dr. Matrix. (Washington, D.C.: Mathematical Association of America, 1997).
"Exploring Fractals," developed by Mary Ann Connors of the University of Massachusetts, introduces high school students and teachers to the basic ideas of fractal geometry at http://www.math.umass.edu/~mconnors/fractal/fractal.html .
David G. Green of Clark Stuart University in Australia presents ideas about fractals and scale at http://life.csu.edu.au/complex/tutorials/tutorial3.html .
The Center for Polymer Studies at Boston University presents "On Growth and Form: Learning Concepts of Probability and Fractals" at http://argento.bu.edu/~trunfio/cps-ogaf.html .
A gallery of fractal shapes can be found at http://www.geom.umn.edu/graphics/pix/General_Interest/Fractals/ .
Find out how to create fractal designs using pattern blocks at http://ng.netgate.net/~millar/pblock/index.htm .
Trek 11: Tilt-A-Whirl Madness
James Gleick. Chaos: The Making of a New Science. (New York: Viking, 1987).
Richard L. Kautz and Bret M. Huggard. "Chaos at the amusement park: Dynamics of the Tilt-A-Whirl." American Journal of Physics 62 (January, 1994): 59-66.
Edward N. Lorenz. The Essence of Chaos. Seattle: University of Washington Press, 1993.
Ivars Peterson. "Tilt-A-Whirl chaos (I)." Science News Online. April 22, 2000. Why the Tilt-A-Whirl is a chaotic aumsement park ride.
Ivars Peterson. "Tilt-A-Whirl chaos (II)." Science News Online. April 29, 2000. Taking advantage of the sensitive dependence on initial conditions to control chaos.
The Sellner Manufacturing Company, which makes the Tilt-A-Whirl, has a Web site at http://www.whirlin.com/ .
Trek 13: Luck on the Boredwalk
Joseph Gallian. "Weird dice." Math Horizons (February, 1995): 30-31.
Martin Gardner. "Sicherman dice, the Kruskal count and other curiosities." In Penrose Tiles to Trapdoor Ciphers . . . and the Return of Dr. Matrix. (Washington, D.C.: Mathematical Association of America, 1997).
Darrell Huff and Irving Geis. How to Take a Chance. (New York: W. W. Norton, 1959).
Ivars Peterson. "Unfair dice." Science News Online. October 24, 1998.
Ivars Peterson. "Tricky dice." Science News Online. October 4, 1997. (See Math Trek, page 70).
Ivars Peterson. "Monopoly dollars and sense." Science News Online. June 7, 1997.
Warren Weaver. Lady Luck: The Theory of Probability. Garden City, N. Y.: Doubleday, 1963).
Trek 17: The Code-Locked Door
Nancy Garden. The Kids' Code and Cipher Book. (Hamden, Conn.: Linnett Books, 1981).
Martin Gardner. "The binary system." In Martin Gardner's New Mathematical Diversions from Scientific American. (New York: Simon & Schuster, 1966).
Eugene Eric Kim and Betty Alexandra Toole. "Ada and the first computer." Scientific American (May 1999): 76-81.
Learn about Ada Byron, Lady Lovelace at http://www.agnesscott.edu/lriddle/women/love.htm .
Trek 19: The Wild Game Hall
Martin Gardner. The Universe in a Handkerchief: Lewis Carroll's Mathematical Recreations, Games, Puzzles, and Word Plays. (New York: Copernicus, 1996).
Martin Gardner. "The ellipse." In Martin Gardner's New Mathematical Diversions from Scientific American. (New York: Simon & Schuster, 1966).
Ivars Peterson. "Around the dartboard." Science News Online. May 17, 1997. Why the dartboard is numbered the way it is.
Ivars Peterson. "Billiards in the round." Science News Online. March 1, 1997. Billiards on tables of different shapes.
Trek 23: Way Out!
Mathematician John Sullivan of the University of Illinois has a variety of photos and images of Klein bottles at http://www.math.uiuc.edu/~jms/photos/MathArt/Klein/ .
When you lay down and flatten a Möbius strip into a triangle, you end up with a shape that has the same geometry as the standard recycling symbol, which looks like three bent arrows chasing each other around a triangular loop.
If you examine printed recycling symbols, you'll sometimes see a version that doesn't look like a standard Möbius strip. How could that happen? One possibility is that someone drew just one bent, twisted arrow, made two copies of it, and put the three arrows in a triangle pattern. That's probably what happened when the illustrator created the diagrams on pages 10 and 20 in Math Trek.
In this case, the chasing arrows form a one-sided band that has three half twists instead of just one. If you were to lay a string along its edge until the ends met and pulled the string tight, you would end up with a knot in the string. If you did this with a standard Möbius strip, you wouldn't get a knot.
The alternative version of the recycling symbol also appeared on the cover of the Oct. 9, 1999, issue of The Economist magazine. If you look closely, you can probably find other examples.
See also "Recycling Topology" at http://www.sciencenews.org/sn_arch/9_28_96/mathland.htm.