National Council of Teachers of Mathematics -  2005 Annual Meeting
Anaheim, California
Session 652, Friday, April 8, 1:00 - 2:30 pm
Hilton Anaheim, Room Avila-B, on the 4th floor.

Lawrence H. Shirley, Professor of Mathematics and Associate Dean
College of Graduate Studies and Research
Towson University, Towson, Maryland 21252 ; phone 410-704-3500; personal webpage:

Abstract Using examples from the history of mathematics, we will observe the surprises, paradoxes, confusions, abstractions, and insights that result when mathematicians believe in "impossible" things. Information and activities for classroom enrichment will be included.


I. Believing in the impossible: good or bad?

II. Ancient mathematical power

III. Greek impossibilities
    a. Zeno's paradoxes                        Can Achilles win?
    b. Irrational numbers:    
              Keep it a secret!
    c. Construction problems                 square circles? trisect?

IV. Challenges and new ideas
    a. negative and imaginary numbers   numbers in new directions?
    b. Ptolemy and Copernicus              Where is Mars going?  
    c. more problems with infinitesimals  Angels, yes!  infinitesimals, no!

V. Revolution
    a. non-Euclidean geometry               Was Euclid wrong?
    b. infinite sets                                  beyond counting
    c. math is incomplete                       "This statement cannot be proven"

VI. "Pathological" results
    a. topology (see below)                     weird properties!
    b. four (and more) dimensions           Flatland and much more!
    c. fractals                                        infinite repetition!
    d. giant proofs                                 who can check them?

VII. ..but in the end: it's good!
    a. solutions                                     we can win some!
    b. challenges                                   ...but not all (at least, not yet)
    c. creativity                                      ...but we keep trying!

Instructions for topological objects:

Mbius Ring: Take a strip of paper and bend it around into a ring, but give one end a 180-degree twist before fastening the two ends.  This has only one face (side) and one edge.  Cutting along the ring--either down the middle or about one-fourth of the way in from the edge--yields interesting results.

Klein Bottle: Take a bag that home-delivery newspapers come in (or any long thin plastic bag) and cut off the closed end to yield a cylinder open on both ends.  Also cut a small hole about one-third of the way along from one end.  Take the other end, squeeze it into a neck, stuff it through the small hole, and flare it out to join the other end from the inside.  The resulting bottle has no inside or outside.  It is really a four-dimensional creature and in four dimensions the neck does not really pass through the side (there is no small hole).  If you could perceive four dimensions, you could also make a Klein Bottle by fastening two Mbius rings together completely along their edges--you can try this task in three dimensions, but you will get caught up in the twists! (see the link below for more on Klein Bottles)


-- Alice Through the Looking Glass (Chapter 5 "Wool and Water") and Lewis Carroll (or Charles Dodgson, or with more math emphasis)
--Biographies of historical mathematicians (
nearly 1700 names!) and other topics
----including Pythagoras, Zeno, Ptolemy, Al-Khwarizmi, Cardano, Copernicus, Descartes, Newton, Leibniz, Berkeley, Euler, Bolyai, Lobachevsky, Hamilton, Cantor, Russell, Godel, Mandelbrot, Wiles, and hundreds more!
--a great reference source on many aspects of higher mathematics: MathWorld ("Eric Weisstein's World of Mathematics")
--geometric construction problems (including details of circling squaring, cube duplication, and angle trisection)
--non-Euclidean geometry
--topology: four-color theoremKlein bottle details and pictures and Klein bottles for sale
--four-dimensional hypercube (tesseract) in 3-D (note: If you have red-blue 3-D glasses, you can use them. Otherwise, press the"stereo" button twice to get a double image. Then cross your eyes to produce a third image between the two. Watch that one. The page also has some instructions and other "cool" images below the tesseract. Enjoy!)
--Banach-Tarski Paradox--reassembling pieces of a ball to make a larger one (it's OK not to believe this one!)
--fractals, and fractint, which is software for playing with fractal images.
--the "enormous" theorem of classification of finite simple groups
--the Riemann zeta function (Has it now been provenNot confirmed)
Here is a page of Hilbert's 23 problems, presented in 1900 as a "homework assignment" for pure mathematicians for the 20th century. Click on its "table of contents to see the actual problems.
--In 2000, the Clay Mathematics Institute offered $1 million prizes for the solution of each of the currently unsolved "Millennium Problems"
--For hot news from mathematics, MathWorld Headline News (also, from this link you can also jump to the rich MathWorld files of mathematics content)
--Believe in Unicorns

more interesting mathematics links at:

Print Sources and Suggestions for Further Reading (with Amazon--or other--links, if available)

Abbott, Edwin; and Stewart, Ian (2001) The Annotated Flatland: A Romance of Many Dimensions, Perseus Publishing.
Bell, E.T. (1937) Men of Mathematics, Simon & Schuster.
Boyer, Carl (1989) A History of Mathematics, Princeton University Press.
Burton, David M. (1991) The History of Mathematics: An Introduction, Wm. C. Brown, Publishers.
Calinger, Ronald (editor) (1995) Classics of Mathematics, Prentice Hall.
Carroll, Lewis (1988) The Annotated Alice: Alice's Adventures in Wonderland and Through the Looking Glass
    (intro,annotation by Martin Gardner), Penguin.
Casti, John (1995) Five Golden Rules: Great Theories of 20th Century Mathematics and Why They Matter, Wiley.
Casti, John (2001) Mathematical Mountaintops: The Five Most Famous Theorems of All Time, Oxford.
Davis, Philip and Hersh, Reuben (1981) The Mathematical Experience, Houghton Mifflin.
Devlin, Keith (2002) The Millennium Problems, Basic Books.
Dunham, William (1990) Journey through Genius: The Great Theorems of Mathematics, Wiley.
Eves, Howard (1990) An Introduction to the History of Mathematics, Saunders College Publishing.
Grinstein, Louise S. and Campbell, Paul J. (1987) Women of Mathematics, Greenwood Press.
Kasner, Edward, and Newman, James (2001,originally 1940) Mathematics and the Imagination, Dover
Katz, Victor (1998) A History of Mathematics: An Introduction, Harper-Collins.
Kline, Morris (1953) Mathematics in Western Culture, Oxford University Press.
Kline, Morris (1962) Mathematics: A Cultural Approach, Addison-Wesley.
Mazur, Barry (2003) Imagining Numbers, Farrar, Straus and Giroux.
Nahin, Paul J.(1998) An Imaginary Tale: The Story of -1 Princeton University Press.
National Council of Teachers of Mathematics (1969,1989) Historical Topics for the Mathematics Classroom (31st Yearbook), NCTM.
Osen, Lynn M. (1974) Women in Mathematics, The MIT Press.
Seife, Charles (2000) Zero: The Biography of a Dangerous Idea, Penguin Books.
Shirley, Lawrence (2000) "Twentieth Century Mathematics: A Brief Review of the Century"
    in Mathematics Teaching in the Middle School, vol. 5, no. 5, pp. 278-285.
Smith, D.E. (1929,1959) A Source Book in Mathematics (2 volumes), McGraw-Hill; Dover.
Smith, Sanderson M. (1996) Agnesi to Zeno, Key Curriculum Press.
Stewart, Ian (1996) From Here to Infinity, Oxford University Press.
Struik, D.J. (1967,1995) A Concise History of Mathematics, Dover.
Wills, Herbert, III (1985) Leonardo's Dessert, No Pi, NCTM.

Similar but shorter presentations of this topic:
Maryland Council of Teachers of Mathematics Annual Conference--October 18, 2002
National Council of Teachers of Mathematics Eastern Regional Conference, Baltimore--October 14, 2004

This page was last updated on April 13, 2005; links checked on April 4, 2005