MAA Online - Columns

Mathematical Treks: From Surreal Numbers to Magic Circles -

Bibliography

Ivars Peterson

1. Calculation and the Chess Master

Chelminski, Rudy. 2001. This time it's personal. Wired 9 (October): 96-113. Available at http://www.wired.com/wired/archive/9.10/chess.html.
Devlin, Keith. 1997. Clash of the chess titans. MAA Online (May). Available at http://www.maa.org/devlin/devlin_5_97.html.
______. 1996. Reflections on Deep Blue. MAA Online (March). Available at http://www.maa.org/devlin/deepblue.html.
Ginsberg, Mathew L. 1998. Computers, games and the real world. Scientific American (November). Available at http://www.sciam.com/1998/1198intelligence/1198ginsberg.html.
Levy, David, and Monty Newborn. 1991. How Computers Play Chess. New York: W. H. Freeman.
Levy, David. 1983. Computer Gamesmanship: Elements of Intelligent Game Design. New York: Simon & Schuster.
Newborn, Monty. 1997. Kasparov versus Deep Blue: Computer Chess Comes of Age. New York: Springer-Verlag.
Peterson, Ivars. 1997. Computer triumphs over human champion. Science News 151 (May 17): 300.
______. 1996. The soul of a chess machine. Science News 149 (March 30): 200-201. Available at http://www.sciencenews.org/sn_edpik/mc_4.htm.
______. 1996. Chess champion sinks Deep Blue's figuring. Science News 149 (Feb. 24): 119.
Shannon, Claude. 2000. A chess-playing machine. In The World of Mathematics, vol. 4, James R. Newman, ed. New York: Dover.
Shaw, J. B. 1912. What is mathematics? Bulletin of the American Mathematical Society 18: 386-387.

A Web site devoted to the matches between Garry Kasparov and Deep Blue can be found at http://www.research.ibm.com/deepblue/home/html/b.html.

2. The Cow in the Classroom

Eberhart, J. G. 2001. Humor and music in the mathematics classroom. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings, Reza Sarhangi and Slavik Jablan, eds. See http://www.sckans.edu/~bridges/.
Enzensberger, Hans Magnus. 1997. The Number Devil: A Mathematical Adventure. New York: Metropolitan Books.
Fadiman, Clifton, ed. 1997. Fantasia Mathematica. New York: Copernicus.
______. 1997. The Mathematical Magpie. New York: Copernicus.
Frucht, William, ed. 1999. Imaginary Numbers: An Anthology of Marvelous Mathematical Stories, Diversions, Poems, and Musings. New York: Wiley.
Juster, Norton. 1971. The Phantom Tollbooth. New York: Alfred A. Knopf.
Leacock, Stephen. 2000. Mathematics for golfers. In The World of Mathematics, vol. 4, James R. Newman, ed. New York: Dover.
______. 2000. Common sense and the universe. In The World of Mathematics, vol. 4, James R. Newman, ed. New York: Dover.
______. 1997. A, B, and C—The human element in mathematics. In The Mathematical Magpie, Clifton Fadiman, ed. New York: Copernicus.
Sachar, Louis. 1995. Wayside School Gets a Little Stranger. New York: Morrow.
______. 1994. More Sideways Arithmetic from Wayside School. New York: Scholastic.
______. 1989. Sideways Arithmetic from Wayside School. New York: Scholastic.
Scieszka, Jon, and Lane Smith. 1995. Math Curse. New York: Viking.
Stueben, Michael (with Diane Sandford). 1998. Twenty Years Before the Blackboard: The Lessons and Humor of a Mathematics Teacher. Washington, DC: Mathematical Association of America.
Twain, Mark. 1985. Life on the Mississippi. New York: Penguin. Full text available at http://docsouth.unc.edu/twainlife/menu.html.
Vinik, Aggie, Linda Silvey, and Barnabas Hughes, eds. 1978. Mathematics and Humor. Reston, VA: National Council of Teachers of Mathematics.

Math humor can be found on the Web at http://www.mathacademy.com/pr/humor/index.asp and http://www.escape.ca/~dcc/phys/humor_ma.html.
Bibographical and other information about Stephen Leacock is available at http://www.nlc-bnc.ca/3/5/index-e.html.

3. A Passion for Pi

Bailey, David H., and Simon Plouffe. 1997. Recognizing numerical constants. In Organic Mathematics: CMS Conference Proceedings. Providence, RI: American Mathematical Society. See http://www.cecm.sfu.ca/organics/papers/bailey/.
Beckmann, Petr. 1971. A History of Pi. Boulder, CO: Golem Press.
Benjamin, Arthur. 2000. A better way to memorize pi: The phonetic code. Math Horizons 7 (February): 17.
Berggren, L., J. Borwein, and P. Borwein. 1997. Pi: A Source Book. New York: Springer-Verlag.
Blatner, David. 1997. The Joy of Π. New York: Walker. See http://www.joyofpi.com/.
Borwein, J. M., and P. B. Borwein. 1990. A Dictionary of Real Numbers. Pacific Grove, CA: Wadsworth & Brooks/Cole.
Castellanos, Dario. 1988. The ubiquitous pi. Mathematics Magazine 61 (April): 67-96 and 61 (June): 148-164.
Conway, John H., and Richard K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Cukier, Mimi. 1999. Pi mnemonics. Math Horizons 6 (April): 35.
Davis, Philip J., and William G. Chinn. 1985. 3.1416 and All That, 2nd ed. Boston: Birkhäuser.
Gardner, Martin. 1999. Slicing pi into millions. In Gardner's Whys & Wherefores. Amherst, NY: Prometheus Books.
Hayes, Brian. 1996. A question of numbers. American Scientist 84 (January-February): 10-14. Available at http://www.sigmaxi.org/amsci/issues/comsci96/compsci96-01.html.
Palais, Bob. 2001. Π is wrong! Mathematical Intelligencer 23(No. 3): 7-8.
Peterson, Ivars. 2001. Normal pi. Science News 160 (Sept. 1): 136-137. Available at http://www.sciencenews.org/20010901/bob9.asp.
______. 1999. Pi by the billions. Science News 156 (Oct. 16): 255.
______. 1995. A new formula for picking off pieces of pi. Science News 148 (Oct. 28): 279.
______. 1995. Next number, please. Science News 147 (May 20): 319.
______. 1995. Spying pi in the sky. Science News 147 (May 20): 319. See http://ourworld.compuserve.com/homepages/rajm/pinature.htm.
Sloane, N. J. A., and Simon Plouffe. 1995. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press. See http://www.research.att.com/~njas/sequences/index.html and http://www.research.att.com/~njas/sequences/book.html.
Watson, Bruce. 2000. Squaring the circle is no piece of Π. Smithsonian 31 (May): 71-82. See http://www.smithsonianmag.si.edu/smithsonian/issues00/may00/pi.html.

A history of pi can be found at http://www-history.mcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html. Accompanying chronology: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Pi_chronology.html.
The "Pi Pages" are available at http://www.cecm.sfu.ca/pi/pi.html.
A Web page devoted to the "uselessness of pi and its irrational friends" is available at http://www.go2net.com/useless/useless/pi.html.
Mike Keith's "World of Words & Numbers" Web pages feature a poem ("Near a Raven") encoding the first 740 digits of pi and a short story ("Cadaeic Cadenza") that goes to even greater lengths to immortalize pi: http://users.aol.com/s6sj7gt/mikehome.htm.
Advice on memorizing digits of pi can be found at http://plaza.v-wave.com/vseward/pi_main.html.
Plouffe's Inverter can be found at http://www.lacim.uqam.ca/pi/. Simon Plouffe has a home page at http://www.lacim.uqam.ca/plouffe/.
MathSoft has an introduction to "The Miraculous Bailey-Borwein-Plouufe Pi Algorithm" at http://www.mathsoft.com/asolve/plouffe/plouffe.html.
A history of the computation of pi can be found at http://www.lacim.uqam.ca/pi/Pihistory.html.
The movie "Pi" has a Web site at http://www.pithemovie.com/.
The Exploratorium's "Pi Day" celebration page is at http://www.exploratorium.edu/pi/pi99.html.

4. Computing in a Surreal Realm

Beasley, John D. 1990. The Mathematics of Games. Oxford, England: Oxford University Press.
Berlekamp, Elwyn R., John H. Conway, and Richard K. Guy. Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed. Natick, MA: A K Peters.
Beyers, Dan. 1996. Complex calculations add up to no. 1. Washington Post (March 11).
Bogomolny, Alex. 2001. Taking games seriously. MAA Online (April). Available at http://www.maa.org/editorial/knot/April2001.html.
Conway, John H. 2001. On Numbers and Games, 2nd ed. Natick, MA: A K Peters.
Conway, John H., and Allyn Jackson. 1996. Budding mathematician wins Westinghouse competition. Notices of the American Mathematical Society 43 (July): 776–779. Available at http://www.ams.org/notices/199607/comm-conway.pdf.
Conway, John H., and Richard K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Gardner, Martin. 2001. Surreal numbers. In The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton.
______. 2001. Nothing. In The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton.
Healy, Michelle. 1996. Surreal numbers place first in science search. USA Today (March 11).
Knuth, Donald E. 1974. Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. Boston, MA: Addison-Wesley. See http://www-cs-faculty.stanford.edu/~knuth/sn.html.
Lipkin, R. 1996. Top projects capture Westinghouse awards. Science News 149 (March 16): 167.
Lurie, Jacob. 1998. The effective content of surreal algebra. Journal of Symbolic Logic 63 (June): 337–371.
Matthews, Robert. 1995. The man who played God with infinity. New Scientist 147 (Sept. 2): 36-40.
Shulman, Polly. 1995. Infinity plus one, and other surreal numbers. Discover 16 (December): 96-105.
Steen, Lynn Arthur. 1978. What's in a game? Science News 113 (April 1): 204–206.

A brief definition of surreal numbers (along with references) can be found at http://mathworld.wolfram.com/SurrealNumber.html.

5. Pythagoras Plays Ball

Bradley, Michael J. 1996. Building home plate: Field of dreams of reality. Mathematics Magazine 69 (February): 44-45.
Kreutzer, Peter, and Ted Kerley. 1990. Little League's Official How-to-Play Baseball Book. New York: Doubleday.
Thorp, John, and Peter Palmer, eds. 1995. Total Baseball, 4th ed. New York: Viking.

The shape of home plate is included in Eric Weisstein's "World of Mathematics" compilation: http://mathworld.wolfram.com/HomePlate.html.

6. Recycling Topology

Gardner, Martin. 1989. Möbius bands. In Mathematical Magic Show. Washington, DC: Mathematical Association of America.
Jones, Penny, and Jerry Powell. 1999. Gary Anderson has been found! Resource Recycling (May). Available at http://www.mcmua.com/solidwaste/CreatingtheRecyclingSymbol.htm or http://www.afandpa.org/recycling/anders.pdf.
Long, Cliff. 1998. Bug bands and monkey saddles. Math Horizons 5 (April): 24-28. Available at http://www.wcnet.org/~clong/carving/carving.html.
______. 1996. Möbius or almost Möbius. College Mathematics Journal 27 (September): 277.
Peterson, Ivars. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. New York: Wiley. See http://www.isama.org/.
______. 1999. Chasing arrows. Muse 3 (January): 27-28. Available at http://home.att.net/~mathtrek/muse0199.htm.
Peterson, Ivars, and Nancy Henderson. 2000. Math Trek: Adventures in the MathZone. New York: Wiley.

Cliff Long has a Web page at http://www.wcnet.org/~clong/.

NOTE: Topologists generally apply the term "Möbius band" to not only the standard form (one half-twist) but also the symmetric version (three half-twists) and anything else "homeomorphic" to the standard form. For historical and cultural reasons, I apply the term only to the "standard" embedding of the Möbius band in three-dimensional space to distinguish this particular form from other embeddings.

7. Soap Films and Grid Walks

Courant, Richard, and Herbert Robbins. 1941. What is Mathematics? Oxford, England: Oxford University Press.
Graham, Ronald L., and Marshall W. Bern. 1989. The shortest-network problem. Scientific American 260 (January): 84-89.
Hildebrandt, Stefan, and Anthony Tromba. 1996. The Parsimonious Universe: Shape and Form in the Natural World. New York: Copernicus.
______. 1985. Mathematics and Optimal Form. New York: Scientific American Library.
Morgan, Frank. 1992. Minimal surfaces, crystals, shortest networks, and undergraduate research. Mathematical Intelligencer 14 (No. 3): 37-44.
Peterson, Ivars. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W. H. Freeman.
Steinhaus, Hans. 1969. Mathematical Snapshots, 3rd ed. New York: Oxford University Press.

8. Mating Games and Lizards

2000. Game of life allows all mating strategies. Cornell University press release. Dec. 5. Available at http://www.news.cornell.edu/releases/Dec00/strategies.hrs.html.
Beasley, John D. 1990. The Mathematics of Games. Oxford, England: Oxford University Press.
Kohler, Reto. 2000. Lizards and labor unions. Science Notes 2000. Available at http://scicom.ucsc.edu/SciNotes/0001/lizards.htm.
Leutwyler, Kristin. 2000. Mating lizards play a game of rock-paper-scissors. Scientific American (Dec. 5). Available at http://www.sciam.com/news/120500/4.html.
Mirsky, Steve. 1996. The lizard kings. Scientific American 274 (June): 26. Available at http://www.sciam.com/0696issue/0696scicit07.html.
Peterson, Ivars. 1999. Lizard game. Muse 3 (April): 26-27. Available at http://home.att.net/~mathtrek/muse0499.htm.
Sinervo, B., and C.M. Lively. 1996. The rock-paper-scissors game and the evolution of alternative male strategies. Nature 380 (March 21): 240-243.
Smith, John Maynard. 1996. The games lizards play. Nature 380 (March 21): 198-199.
Zamudio, Kelly R., and Barry Sinervo. 2000. Polygyny, mate-guarding, and posthumous fertilization as alternative male mating strategies. Proceedings of the National Academy of Sciences 97 (Dec. 19): 14427-14432. Abstract available at http://www.pnas.org/cgi/content/abstract/97/26/14427.

You can visit Barry Sinervo's "LizardLand" at http://www.biology.ucsc.edu/~barrylab/.
See more of the side-blotched lizard at http://www.wildherps.com/species/U.stansburiana.html.
You can find the "official" rock-paper-scissors strategy guide at http://www.worldrps.com/index.html. You can play the game at http://www.2street.com/rock-paper-scissors/.

9. Random Bits

Gardner, Martin. 1989. Random numbers. In Mathematical Carnival. Washington, DC: Mathematical Association of America.
Knuth, Donald E. 1969. The Art of Computer Programming. Reading, MA: Addison-Wesley.
Marsaglia, George. 1968. Random numbers fall mainly in the planes. Proceedings of the National Academy of Science 61 (September): 25-28.
Marsaglia, George, and Arif Zaman. 1994. Some portable very-long-period random number generators. Computers in Physics 8 (January/February): 117-121.
______. 1991. A new class of random number generators. Annals of Applied Probability 1 (No. 3): 462-480.
Peterson, Ivars. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.
______. 1991. Numbers at random. Science News 140 (Nov. 9): 300-301.
______. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W. H. Freeman.
Pickover, Clifford A. 1995. Random number generators: Pretty good ones are easy to find. Visual Computer 11:369-377.
RAND Corporation. 1955. A Million Random Digits with 100,000 Normal Deviates. New York: Free Press. See http://www.rand.org/publications/classics/randomdigits/.

A Web page devoted to George Marsaglia's Diehard battery of randomness tests can be found at http://www.stat.fsu.edu/~geo/diehard.html.
George Marsaglia has a Web page at http://stat.fsu.edu/template/homepage/marsaglia.html.

10. Spreading Rumors

Fan, C. Kenneth, Bjorn Poonen, and George Poonen. 1997. How to spread rumors fast. Mathematics Magazine 70 (February): 40-42.

11. Toward a Fairer Expansion Draft

Berry, Scott M. 2001. Do you feel a draft here? Chance 14 (No. 2): 53-57.
Brams, Steven J., and Alan D. Taylor. 1999. The Win-Win Solution: Guaranteeing Fair Shares to Everybody. New York: W. W. Norton.
______. 1996. Fair Division: From Cake-Cutting to Dispute Resolution. Cambridge, England: Cambridge University Press.
______. 1995. An envy-free cake division protocol. American Mathematical Monthly 102 (January): 9-18.
Dawson, C. Bryan. 1997. A better draft: Fair division of the talent pool. College Mathematics Journal 28 (March): 82-88.
______. 1996. A better draft: Fair division of the talent pool. Abstracts of Papers Presented to the American Mathematical Society 17 (No. 1): 148.
Peterson, Ivars. 1996. Formulas for fairness. Science News 149 (May 4): 284-285. Available at http://www.sciencenews.org/sn_arch/5_4_96/bob1.htm.

Bryan Dawson has a Web page at http://www.uu.edu/personal/bdawson/.

12. Cracking the Ball-Control Myth

Blackwell, David A., and M. A. Girshick. 1980. Theory of Games and Statistical Decisions. New York: Dover.
Sackrowitz, Harold, and Daniel Sackrowitz. 1996. Time management in sports: Ball control and other myths. Chance 9 (No. 1): 41-49. Available at http://www.public.iastate.edu/~chance99/091.timemanag.pdf.
Whittle, Peter. 1982. Optimization Over Time: Dynamic Programming and Stochastic Control, vol. 1. New York: Wiley.

Harold B. Sackrowitz has a Web page at http://www.stat.rutgers.edu/people/faculty/sackrow.html.

13. Math and a Music Education

1998. Study of arts, music may enhance young pupils' math and readings skills. Brown University press release. Feb. 12. Available at http://www.brown.edu/Administration/News_Bureau/1997-98/97-080i.html.
Chabris, Christopher F. 1999. Prelude or requiem for the "Mozart effect"? Nature 400 (Aug. 26): 826-827.
Edwards, Roy. 1996. Children learn faster to sound of music. New Scientist 150 (May 18): 6.
Elias, Marilyn. 1996. Singing class helps math, reading skills. USA Today (May 23).
Gardiner, Martin F., Alan Fox, Faith Knowles, and Donna Jeffrey. 1996. Learning improved by arts training. Nature 381 (May 23): 284.
Holden, Constance. 1999. Music as brain builder. Science 283 (March 26): 2007.
James, Jamie. 1993. The Music of the Spheres: Music, Science, and the Natural Order. New York: Copernicus.
Kliewer, Gary. 1999. The Mozart effect. New Scientist 164 (Nov. 6): 34-37. Available at http://www.mikebyde.freeserve.co.uk/MozartEffect/newscientist.html.
Rauscher, Frances H., Gordon L. Shaw, and Katherine N. Ky. 1993. Music and spatial task performance. Nature 365 (Oct. 14): 611.
Rauscher, Frances H., et al. 1995. Listening to Mozart enhances spatial-temporal reasoning: Towards a neurophysiological basis. Neuroscience Letters 185: 44-47.
Rothstein, Edward. 1995. Emblems of Mind: The Inner Life of Music and Mathematics. New York: Times Books.
Shaw, Gordon L. 2000. Keeping Mozart in Mind. San Diego, Calif.: Academic Press.
Steele, Kenneth M., et al. 1999. Prelude or requiem for the "Mozart effect"? Nature 400 (Aug. 26): 827.
Viadero, Debra. 1998. Music on the mind. Education Week 17 (April 8): 25-27. Available at http://www.grps.k12.mi.us/~music.whymusic/MusicMind.html.
Weiss, Rick. 1996. Pedagogics: Arts program pays off in math. Washington Post (May 27).
Wertheim, Margaret. 1995. Pythagoras' Trousers: God, Physics, and the Gender Wars. New York: Times Books.

Martin F. Gardiner has a Web page at http://www.brown.edu/Departments/Human_Development_Center/who/gardiner.html.
The "Skeptic's Dictionary" offers its own perspective on the Mozart effect at http://skepdic.com/mozart.html.
You can find the Mozart Effect Resource Center at http://www.mozarteffect.com/.

14. Sprouts

Anthony, Piers. 1969. Macroscope. New York: Avon.
Applegate, David, Guy Jacobson, and Daniel Sleator. 1999. Computer analysis of sprouts. In The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner, E. Berlekamp and T. Rodgers, eds. Natick, MA: A K Peters.
Copper, Mark. 1993. Graph theory and the game of sprouts. American Mathematical Monthly 100 (May): 478-482.
Eddins, Susan K. 1998. Networks and the game of sprouts. NCTM Student Math Notes (May/June).
______. 1993. Sprouts: Analyzing a simple game. IMSA Math Journal 2(Fall). Available at http://www.imsa.edu/edu/math/journal/volume2/webver/sprouts.html or http://www.imsa.edu/edu/math/journal/volume2/articles/Sprouts.pdf.
Gardner, Martin. 1989. Sprouts and Brussels sprouts. In Mathematical Carnival. Washington, DC: Mathematical Association of America.
Lam, T. K. 1997. Connected sprouts. American Mathematical Monthly 104 (February): 116-119.

The Web site of the World Game of Sprouts Association can be found at http://www.geocities.com/chessdp/.
You can play the game at http://www.math.utah.edu/~alfeld/Sprouts/.

15. Groups, Graphs, and Paul Erdös

Aigner, Martin, and Günther M. Ziegler. 2001. Proofs from THE BOOK, 2nd ed. New York: Springer-Verlag.
Albers, Donald J., and G. L. Alexanderson, eds. 1985. Mathematical People: Profiles and Interviews. Boston: Birkhäuser.
Baker, A., B. Bollobas, and A. Hajnal, eds. 1991. A Tribute to Paul Erdös. Cambridge, England: Cambridge University Press.
Bellman, Richard. 1984. Eye of the Hurricane. Singapore: World Scientific.
Chung, Fan, and Ron Graham. 1999. Erdös on Graphs: His Legacy of Unsolved Problems. Natick, MA: A K Peters.
De Castro, Rodrigo, and Jerrold W. Grossman. 1999. Famous trails to Paul Erdös. Mathematical Intelligencer 21 (No. 3): 51-63.
Graham, R. L., and J. Nesetril. 1996. The Mathematics of Paul Erdös I. Berlin: Springer-Verlag.
Hoffman, Paul. 1998. The Man Who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth. New York: Hyperion.
______. 1987. The man who loves only numbers. Atlantic Monthly 260 (November): 60-74.
Honsberger, Ross. 1996. From Erdös to Kiev: Problems of Olympiad Caliber. Washington, DC: Mathematical Association of America.
______. 1985. Mathematical Gems III. Washington, DC: Mathematical Association of America.
______. 1978. Mathematical Morsels. Washington, DC: Mathematical Association of America.
MacKenzie, Dana. 1999. Find your mathematical relatives. Science Now (Sept. 15). Available at http://www.academicpress.com/inscight/09151999/grapha.htm.
Newman, M.E.J. 2001. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences 98(Jan. 16):404-409. Available at http://www.pnas.org/cgi/content/full/98/2/404.
Odda, Tom [=Ronald L. Graham]. 1979. On properties of a well-known graph, or what is your Ramsey number? Annals of the New York Academy of Science 328: 166-172.
Peterson, Ivars. 1998. The Jungles of Randomness: A Mathematical Safari. New York: Wiley.
Schechter, Bruce. 1998. My Brain Is Open: The Mathematical Journeys of Paul Erdös. New York: Simon & Schuster.
Tierney, John. 1984. Paul Erdös is in town. His brain is open. Science 84 (October): 40-47.

The Erdös Number Project Web page can be found at http://www.oakland.edu/~grossman/erdoshp.html.
A biography of Paul Erdös is available at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Erdos.html.

16. DNA Adds Up

Adleman, Leonard M. 1998. Computing with DNA. Scientific American 279 (August): 54-61.
______. 1994. Molecular computation of solutions to combinatorial problems. Science 266 (Nov. 11): 1021-1024.
Ball, Philip. 2000. DNA computer helps travelling salesman. Nature Science Update (Jan. 13). Available at http://www.nature.com/nsu/000113/000113-10.html.
Bass, Thomas A. 1995. Gene genie. Wired 3 (August): 114. Available at http://www.wired.com/wired/archive/3.08/molecular.html.
Chen, Junghuei, and David Harlan Wood. 2000. Computation with biomolecules. Proceedings of the National Academy of Sciences 97 (Feb. 15): 1328-1330. Available at http://www.pnas.org/cgi/content/full/97/4/1328.
Devlin, Keith. 1995. Test tube computing with DNA. Math Horizons 2 (April): 14-21.
Fallis, Don. 1996. Mathematical proof and the reliability of DNA evidence. Mathematics Magazine 69 (June-July): 491-497.
Faulhammer, Dirk, Anthony R. Cukras, Richard J. Lipton, and Laura F. Landweber. 2000. Molecular computation: RNA solutions to chess problems. Proceedings of the National Academy of Sciences 97 (Feb. 15): 1385-1389. Abstract available at http://www.pnas.org/cgi/content/abstract/97/4/1385.
Gifford, David K. 1994. On the path to computation with DNA. Science 266 (Nov. 11): 993-994.
Guarnieri, Frank, Makiko Fliss, and Carter Bancroft. 1996. Making DNA add. Science 273 (July 12): 220-223.
Guarnieri, Frank, and Carter Bancroft. 1996. Use of a horizontal chain reaction for DNA-based addition. In Proceedings of the Second Annual Meeting on DNA Based Computers. Providence, RI: American Mathematical Society.
Kari, Lila. 1997. DNA computing: Arrival of biological mathematics. Mathematical Intelligencer 19 (No. 2): 9-22.
Kiernan, Vincent. 1997. DNA-based computers could race past supercomputers, researchers predict. Chronicle of Higher Education 44 (Nov. 28): A23-24.
Lander, Eric S., and Michael S. Waterman, eds. 1995. Calculating the Secrets of Life. Washington, DC: National Academy Press. See http://www.nap.edu/catalog/2121.html.
Leutwyler, Kristin. 1995. Calculating with DNA. Scientific American 273 (September): 20.
Lipton, Richard J. 1995. DNA solution of hard computational problems. Science 268 (April 28): 542-545.
Liu, Qinghua, et al. 2000. DNA computing on surfaces. Nature 403 (Jan. 13): 175-179.
Ouyang, Qi, Peter D. Kaplan, Shumao Liu, and Albert Libchaber. DNA solution of the maximal clique problem. Science 278 (Oct. 17): 446-449.
Peterson, Ivars. 1996. Computing with DNA. Science News 150 (July 13): 26-27.
______. 1994. Molecular computing in a DNA soup. Science News 146 (Nov. 12): 308.
Phillips, Tony. 2000. The ABC of DNA computing. American Mathematical Society (February). Available at http://www.ams.org/new-in-math/cover/dna-abc1.html.
Pool, Robert. 1995. A Boom in plans for DNA computing. Science 268 (April 28): 498-499.
Rubin, Harvey. 1996. Looking for the DNA killer app. Nature Structural Biology 3 (August): 656-658.
Seife, Charles. 2000. RNA works out knight moves. Science Now (Feb. 16). Available at http://www.academicpress.com/inscight/02162000/grapha.htm.

Information about DNA-based computing research is available at a number of Web sites, including http://www.neci.nj.nec.com/homepages/eric/eric.html, http://crypto.stanford.edu/~dabo/biocomp.html, and http://www.cs.toronto.edu/~roweis/.

17. Computing with the EDSAC

Campbell-Kelly, Martin. 1992. The Airy tape: An early chapter in the history of debugging. IEEE Annals of the History of Computing 14 (No. 4): 16-26.
______. 1980. Programming the EDSAC: Early programming activity at the University of Cambridge. IEEE Annals of the History of Computing 2 (No. 1): 7-36.
Hayes, Brian. 1993. The discovery of debugging. The Sciences 33 (July-August): 10-13.
Wheeler, David J. 1992. The EDSAC programming systems. IEEE Annals of the History of Computing 14 (No. 4): 34-40.
Wheeler, Joyce M. 1992. Applications of the EDSAC. IEEE Annals of the History of Computing 14 (No. 4): 27-33.
Wilkes, M. V. 1949. Electronic calculating-machine development in Cambridge. Nature 164 (Oct. 1): 557-558.
______. 1985. Memoirs of a Computer Pioneer. Cambridge, MA: MIT Press.

Martin Campbell-Kelly has a home page at http://www.dcs.warwick.ac.uk/people/academic/Martin.Campbell-Kelly/. His EDSAC simulator can be found at http://www.dcs.warwick.ac.uk/~edsac/.
The University of Cambridge Computer Laboratory archive of EDSAC photos is available at http://www.cl.cam.ac.uk/Relics/archive_photos.html.
The EDSAC 99 meeting celebrated the 50th anniversary of EDSAC 1: http://www.cl.cam.ac.uk/UoCCL/misc/EDSAC99/.

18. Waring Experiments

Bell, Eric T. 1987. Mathematics: Queen and Servant of Science. Washington, DC: Mathematical Association of America.
Jagy, William C., and Irving Kaplansky. 1995. Sums of squares, cubes, and higher powers. Experimental Mathematics 4 (No. 3): 169-173.
Mahoney, Michael Sean. 1994. The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd ed. Princeton, NJ: Princeton University Press.
Peterson, Ivars. 2001. Surprisingly square. Science News 159 (June 16): 382-383.
Rademacher, Hans, and Otto Toeplitz. 1990. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. New York: Dover.
Ribenboim, Paulo. 1996. The New Book of Prime Number Records. New York: Springer-Verlag.
Schroeder, M. R. 1984. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Biology, Digital Information, and Computing. New York: Springer-Verlag.
Stewart, Ian. 1986. The Waring experience. Nature 323 (Oct. 23): 674.

A biography of Edward Waring is available at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Waring.html.
A discussion of Waring's problem (with references) can be found at http://mathworld.wolfram.com/WaringsProblem.html.

19. Old and New Arithmetic

Swetz, Frank J. 1987. Capitalism and Arithmetic: The New Math of the 15th Century. La Salle, Ill.: Open Court.
Phillips, Tony. 2001. The romance of double-entry bookkeeping. American Mathematical Society (October). Available at http://www.ams.org/new-in-math/cover/book1.html.

20. Matchsticks in the Summer

Gardner, Martin. 1969. Eight problems. In The Unexpected Hanging and Other Mathematical Diversions. New York: Simon and Schuster.
Grabarchuk, Serhiy, Bill Ritchie, and Steve Wagner. 2000. Matchstick puzzles from the 4th dimension. Available at http://www.g4g4.com/exchange9.htm.
Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History. Washington, DC: Mathematical Association of America.
Harborth, Heiko. 1994. Match sticks in the plane. In The Lighter Side of Mathematics: Proceedings of the Eugene Strèns Memorial Conference on Recreational Mathematics and Its History, Richard K. Guy and Robert E. Woodrow, eds. Washington, DC: Mathematical Association of America.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.
______. 1986. Games mathematicians play. Science News 130 (Sept. 20): 186-189.

Heiko Harborth has a Web page at http://bmi1.math.nat.tu-bs.de/dm/mitarb/harborth.html.
You can find a variety of matchstick puzzles at http://www.puzzles.com/PuzzlePlayground/Matches.htm.

21. Tricky Crossings

Ascher, Marcia. 1990. A river-crossing problem in cross-cultural perspective. Mathematics Magazine 63 (February): 26-29.
Ball, W. W. Rouse, and H. S. M. Coxeter. 1974. Mathematical Recreations and Essays. Toronto: University of Toronto Press.
Brandreth, Gyles. 1985. Classic Puzzles. New York: Harper & Row.
Gardner, Martin. 2001. The growth of recreational mathematics. In The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton.
Guy, Richard K., and Robert E. Woodrow, eds. 1994. The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History. Washington, DC: Mathematical Association of America.
Kasner, Edward, and James R. Newman. 1989. Mathematics and the Imagination. Redmond, WA: Microsoft Press.
Perelman, Y. I. 1984. Fun with Maths and Physics. Moscow: Mir Publishers.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York: W.H. Freeman.
______. 1986. Games mathematicians play. Science News 130 (Sept. 20): 186-189.
Singmaster, David. 1994. The utility of recreational mathematics. In The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History, Richard K. Guy and Robert E. Woodrow, eds. Washington, DC: Mathematical Association of America.
Snape, Charles, and Heather Scott. 1995. Puzzles, Mazes and Numbers. Cambridge, England: Cambridge University Press.

22. Beyond the Ellipse

Bell, Eric Temple. 1987. Mathematics: Queen and Servant of Science. Washington, DC: Mathematical Association of America.
Gardner, Martin. 1995. The ellipse. In New Mathematical Diversions. Washington, DC: Mathematical Association of America.
Sekino, Junpei. 1999. n-ellipses and the minimum distance sum problem. American Mathematical Monthly 106 (March): 193-202.

Information about the Canada/USA Mathcamp is available at http://www.mathcamp.org/.
Learn more about the ellipse at http://mathworld.wolfram.com/Ellipse.html.

23. Trouble with Wild-Card Poker

Beasley, John D. 1989. The Mathematics of Games. Oxford, England: Oxford University Press.
Emert, John, and Dale Umbach. 1996. Inconsistencies of "wild-card" poker. Chance 9 (No. 3): 17-22.
Gadbois, Steve. 1996. Poker with wild cards—A paradox? Mathematics Magazine 69 (October): 283-285.
Packel, Edward W. 1981. The Mathematics of Games and Gambling. Washington, DC: Mathematical Association of America.
Scarne, John. 1991. Scarne on Cards. New York: New American Library.

24. Prime Theorems

Apostol, Tom M. 1996. What is the most surprising result in mathematics? Math Horizons 4 (November): 8-31.
______. 1996. A centennial history of the prime number theorem. Engineering & Science 59 (No. 4): 18-28.
Conway, John H., and Richard K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Dudley, Underwood. 1978. Elementary Number Theory, 2nd. ed. New York: W. H. Freeman.
Guiasu, Silviu. 1995. Is there any regularity in the distribution of prime numbers at the beginning of the sequence of positive integers? Mathematics Magazine 68 (April): 110-121.
Guy, Richard K. 1994. Unsolved Problems in Number Theory, 2nd. ed. New York: Springer-Verlag.
Hawkins, David. 1958. Mathematical sieves. Scientific American 199 (December): 105-112.
Kramer, Edna E. 1970. The Nature and Growth of Modern Mathematics. New York: Hawthorn Books.
Mollin, R. A. 1997. Prime-producing quadratics. American Mathematical Monthly 104 (June-July): 529-544.
Peterson, Ivars. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W.H. Freeman.
Ribenboim, Paulo. 1995. Selling primes. Mathematics Magazine 68 (June): 175-182.
______. 1996. The New Book of Prime Number Records. New York: Springer-Verlag.
Zagier, Don. 1977. The first 50 million prime numbers. Mathematical Intelligencer 0: 7-19.

An introduction to the prime number theorem is available at http://mathworld.wolfram.com/PrimeNumberTheorem.html.
A history of the prime number theorem can be found at http://www-history.mcs.st-and.ac.uk/history/HistTopics/Prime_numbers.html.
For a list of some open problems and conjectures involving primes, see http://www.utm.edu/research/primes/notes/conjectures/.

25. Champion Numbers

2001. Researchers discover largest multi-million-digit prime using Entropia distributed computing grid. Entropia and GIMPS press release. Dec. 6. Available at http://www.mersenne.org/13466917.htm.
Buske, Dale, and Sandra Keith. 2000. GIMPS finds another prime! Math Horizons 7 (April): 19-21.
Conway, J. H., and R. K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Francis, Richard L. 1996. Math bite: Recitation of large primes. Mathematics Magazine 69 (October): 260.
Gillmor, Dan. 1996. Move over, supercomputers. San Jose Mercury News (Nov. 23).
______. 1996. Researchers discover prime example of mathematicians' love. San Jose Mercury News (Sept. 3).
Koshy, Thomas. 1998. The ends of a Mersenne prime and an even perfect number. Journal of Recreational Mathematics 29 (No. 3): 196-202.
Morrison, Philip. 1998. Numbers: Prime or choice? Scientific American 279 (November): 123-124. Available at http://www.sciam.com/1998/1198issue/1198wonders.html.
Peterson, Ivars. 2001. Searchers capture a champion megaprime. Science News 160 (Dec. 15): 372.
______. 2000. Great computations. Science News 157(March 4):152-153. Available at http://www.sciencenews.org/20000304/bob1.asp.
______. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W. H. Freeman.
______. 1998. Calculating a record prime. Science News 153 (Feb. 21): 127.
______. 1997. Lucky choice turns up world-record prime. Science News 152 (Sept. 13): 164. Available at http://www.sciencenews.org/sn_arc97/9_13_97/fob1.htm.
______. 1992. Striking pay dirt in prime-number terrain. Science News 141 (April 4): 213.
Ribenboim, Paulo. 1995. The New Book of Prime Number Records. New York: Springer-Verlag.
Stewart, Ian. 1997. Big game hunting in primeland. Scientific American 276 (May): 108-111.

Chris Caldwell's "Prime Pages" can be found at http://www.utm.edu/research/primes/.
A biography of Marin Mersenne is available at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Mersenne.html.
Learn more about Mersenne numbers at http://mathworld.wolfram.com/MersenneNumber.html and Mersenne primes at http://mathworld.wolfram.com/MersennePrime.html.
The Great Internet Mersenne Prime Search (GIMPS) Web site is at http://www.mersenne.org/.
A poster showing all 4,053,946 decimal digits of the largest known prime, found in 2001, is available from Perfectly Scientific at http://www.perfsci.com/.

26. A Perfect Collaboration

Bell, E. T. 1965. Men of Mathematics. New York: Simon and Schuster.
Conway, J. H., and R. K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Dunham, William. 1999. Euler: The Master of Us All. Washington, DC: Mathematical Association of America.

You can learn more about perfect numbers at http://www-history.mcs.st-and.ac.uk/history/HistTopics/Perfect_numbers.html and at http://mathworld.wolfram.com/PerfectNumber.html.
A biography of Leonhard Euler is available at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Euler.html.

27. Fragments of the Past

Boyer, C. B. 1985. A History of Mathematics. Princeton, NJ: Princeton University Press.
Joseph, G. G. 1992. The Crest of the Peacock: Non-European Roots of Mathematics. New York: Viking Penguin.
Katz, Victor J. 1997. The transmission of mathematics from Islam to Europe. Abstracts of Papers Presented to the American Mathematical Society 18 (No. 1): 5.
Kline, Morris. 1972. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press.
Mahoney, M. S. 1994. The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd ed. Princeton, NJ: Princeton University Press.

A biography of Al-Sabi Thabit ibn Qurra al-Harrani can be found at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Thabit.html.

28. More than Magic Squares

Ball, W.W.R., and H.S.M. Coxeter. 1974. Mathematical Recreations & Essays. Toronto: University of Toronto Press.
Cipra, B. 2001. Number fun with Ben. Science Now (Apr. 30). Available at http://www.academicpress.com/inscight/04302001/grapha.htm.
Gardner, Martin. 1998. Some new discoveries about 3 x 3 magic squares. Math Horizons 5 (February): 11-13.
______. 1988. Magic squares and cubes. In Time Travel and Other Mathematical Bewilderments. New York: W. H. Freeman.
______. 1988. Magic with a matrix. In Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles & Games. Chicago: University of Chicago Press.
______. 1961. Magic squares. In The 2nd Scientific American Book of Mathematical Puzzles & Diversions. New York: Simon & Schuster.
Hofstadter, D. R. 1979. Gödel, Escher, Bach: An Eternal Golden Braid. New York: Basic Books.
Pasles, Paul C. 2001. The lost squares of Dr. Franklin: Ben Franklin's missing squares and the secret of the magic circle. American Mathematical Monthly 108 (June-July): 489-511.
______. 2000. Benjamin Franklin, magician? Franklin Gazette (September). Available at http://www.pasles.org/Franklin/Gazette/article.html.
Phillips, Tony. 1999. Math and the Musical Offering. American Mathematical Society (March). Available at http://www.ams.org/new-in-math/cover/canons.html.
Rothstein, Edward. 1995. Emblems of Mind: The Inner Life of Music and Mathematics. New York: Times Books.

Learn more about magic squares at http://mathworld.wolfram.com/MagicSquare.html.
Victor E. Hill has a Web page at http://www.williams.edu/Mathematics/vhill/.
Paul C. Pasles has a Web site at http://www.pasles.org/.
The full text of The Autobiography of Benjamin Franklin can be found at http://earlyamerica.com/lives/franklin/index.html.
Web pages devoted to magic squares and a classroom lesson plan concerning Franklin order-8 squares are available at http://mathforum.com/alejandre/magic.square.html.

29. Rolling with Reuleaux

Bogomolny, Alex. 2001. The theorem of Barbier. MAA Online (September). Available at http://www.maa.org/editorial/knot/Barbier.html.
Casey, James. 1998. Perfect and not-so-perfect rollers. Mathematics Teacher 91 (January): 12-20.
Gardner, Martin. 2001. Curves of constant width. In The Colossal Book of Mathematics: Classic Puzzles, Paradoxes, and Problems. New York: W. W. Norton.
Honsberger, Ross. 1970. The theorem of Barbier. In Ingenuity in Mathematics. Washington, DC: Mathematical Association of America.
Kuperberg, Greg. 1999. Circumscribing constant-width bodies with polytopes. New York Journal of Mathematics 5 (July 9): 91-100.
Moon, Francis C. 1999. Franz Reuleaux's small machines. Cornell Engineering Magazine 5 (Spring). Available at http://www.mae.cornell.edu/Reuleauxcoll/Sp.feat5.html.
Peterson, Ivars. 1999. Covering up. Muse 3 (July/August):36. Available at http://home.att.net/~mathtrek/muse0799.htm.
Rademacher, Hans, and Otto Toeplitz. 1990. The Enjoyment of Mathematics. New York: Dover.
Smith, Scott G. 1993. Drilling square holes. Mathematics Teacher 86 (October): 579-583.

See the Reuleaux triangle at http://mathworld.wolfram.com/ReuleauxTriangle.html and the Reuleaux polygon at http://mathworld.wolfram.com/ReuleauxPolygon.html. See also http://mathworld.wolfram.com/Rotor.html.
Learn more about shapes of constant width at http://www.cut-the-knot.com/do_you_know/cwidth.html and http://mathworld.wolfram.com/CurveofConstantWidth.html.
Information about the Cornell Reuleaux collection is available at http://www.mae.cornell.edu/history.html. Francis Moon has a Web page at http://www.mae.cornell.edu/faculty/Moon.html.
Learn more about the Wankel rotary engine at http://www.monito.com/wankel/j-wankel.html.

30. Next in Line

Conway, John H., and Richard K. Guy. 1996. The Book of Numbers. New York: Copernicus.
Enzensberger, Hans Magnus. 1997. The Number Devil: A Mathematical Adventure. New York: Metropolitan Books.
Gardner, Martin. 1997. Strong laws of small primes. In The Last Recreations: Hydras, Eggs, and Other Mathematical Mystifications. New York: Copernicus.
Guy, Richard K. 1994. The strong law of small numbers. In The Lighter Side of Mathematics: Proceedings of the Eugène Strens Memorial Conference on Recreational Mathematics and Its History, R. K. Guy and R. E. Woodrow, eds. Washington, DC: Mathematical Association of America.
______. 1990. The second strong law of small numbers. Mathematics Magazine 63 (February): 3-20.
______. 1988. The strong law of small numbers. American Mathematical Monthly 95 (October): 697-712.
Peterson, Ivars. 1990. Islands of Truth: A Mathematical Mystery Cruise. New York, W. H. Freeman.
______. 1988. A shortage of small numbers. Science News 133 (Jan. 9): 31.
Sloane, N.J.A., and S. Plouffe. 1995. The Encyclopedia of Integer Sequences. San Diego, CA: Academic Press. See http://www.research.att.com/~njas/sequences/index.html and http://www.research.att.com/~njas/sequences/book.html.

31. Pennies in a Tray

Albers, Donald J. 1996. A nice genius. Math Horizons 4 (November): 18-23.
Boll, David W., Jerry Donovan, Ronald L. Graham, and Boris D. Lubachevsky. 2000. Improving dense packings of equal disks in a square. Electronic Journal of Combinatorics 7: R46. Available at http://www.combinatorics.org/Volume_7/Abstracts/v7i1r46.html.
Gardner, Martin. 1992. Tangent circles. In Fractal Music, Hypercards, and More. . . New York: W. H. Freeman.
Graham, Ronald L., and Boris D. Lubachevsky. 1996. Repeated patterns of dense packings of equal disks in a square. Electronic Journal of Combinatorics 3: R16. Available at http://www.combinatorics.org/Volume_3/Abstracts/v3i1r16.html.
______. 1995. Dense packings of equal disks in an equilateral triangle: From 22 to 34 and beyond. Electronic Journal of Combinatorics 2: A1. Available at http://www.combinatorics.org/Volume_2/volume2.html#A1.
Graham, R. L., B. D. Lubachevsky, K. J. Nurmela, and P. R. J. Östergård. 1998. Dense packings of congruent circles in a circle. Discrete Mathematics 181: 139-154.
Lubachevsky, Boris D., Ron L. Graham, and Frank H. Stillinger. 1998. Spontaneous patterns in disk packings. In Bridges: Mathematical Connections in Art, Music, and Science Conference Proceedings, R. Sarhangi, ed. (See http://www.sckans.edu/~bridges/.)
Nurmela, K. J., and P. R. J. Östergård. 1999. More optimal packings of equal circles in a square. Discrete & Computational Geometry 18: 111-120.
______. 1997. Packing up to 50 equal circles in a square. Discrete & Computational Geometry 22: 439-457.
Stewart, Ian. 1998. Tight tins for round sardines. Scientific American 278 (February): 94-96.

Dave Boll's pages on optimal circle packings can be found at http://www.frii.com/~dboll/packing.html.

32. Fair Play and Dreidel

Feinerman, Robert. 1976. An ancient unfair game. American Mathematical Monthly 83 (October): 623-625.
Trachtenberg, Felicia Moss. 1996. The game of dreidel made fair. College Mathematics Journal 27 (September): 278-281.

You can learn more about the dreidel at http://www.holidays.net/chanukah/dreidel.html or play the game at http://www2.priscilla.com/priscilla/hanukkah/dreidel1.html.

33. Euclid's Fourteenth Book

Cahill, Thomas. 1995. How the Irish Saved Civilization: The Untold Story of Ireland's Heroic Role from the Fall of Rome to the Rise of Medieval Europe. New York: Nan A. Talese/Doubleday.
Devlin, Keith. 1998. The Language of Mathematics: Making the Invisible Visible. New York: W. H. Freeman.
Girvan, Ray. 1999. The Mandelbrot monk. (April 1). Available at http://www.freezone.co.uk/rgirvan/udo.htm.
Mandelbrot, Benoit B. 1982. The Fractal Geometry of Nature. New York: W. H. Freeman.
Mlodinow, Leonard. 2001. Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace. New York: Free Press.
Peterson, Ivars. 1998. The Mathematical Tourist: New and Updated Snapshots of Modern Mathematics. New York: W. H. Freeman.

An online, interactive edition of Euclid's Elements can be found at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
Learn more about Euclid of Alexandria at http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Euclid.html.
Non-Euclidean geometry is introduced at http://www-history.mcs.st-and.ac.uk/history/HistTopics/Non-Euclidean_geometry.html.
Euclid's postulates are given at http://mathworld.wolfram.com/EuclidsPostulates.html.

Mathematical Treks: From Surreal Numbers to Magic Circles - Bibliography