Perfect and Pathological Mathematics

I. Course description

Quick! When is a coffee cup equivalent to a donut? Is it possible for a shape to have infinite surface area and finite area? How do you know? In this class, we will meet the fringe elements of the world of mathematics: we’ll encounter well-behaved and mathematically beautiful ideas and theorems, and we’ll spend a lot of time with the misbehaving miscreants that have stymied long-held mathematical assumptions. We’ll not only study the functions, curves and ideas that have reassured and rocked the world of math; we’ll also study the means by which a theorem, proposition or lemma becomes mathematically valid. In addition, we’ll explore the lives of the movers and shakers of the history of math and develop some ideas about the evolving nature of mathematics. Was it invented or discovered? What are the most pressing mathematical questions of our time?

II. Instructor’s educational preparation and current employment
  • B.A. in English, B.S. in physics, Southern Methodist University, Dallas, Texas
  • Currently completing work on a master’s degree in Applied Mathematics at the University of Missouri-Columbia
  • email:
III. Rationale for inclusion in a program for gifted students

The structure of normal math classes does not generally allow students enough space for deeper explorations of the topics they are expected to master. This class offers gifted students a chance to question that which they have already been taught and develop a deeper appreciation for the elegance and beauty of mathematics. With a class of gifted students, discipline issues are non-existent and students find other students with the same background and ability level. Given enough time to explore and contemplate the implications of difficult but not inaccessible topics, students can create and develop original problem-solving strategies that they could not get to through number-crunching or worksheets. These students can also comprehend the methods used to validate mathematical theorems and propositions, and given enough time and guidance they can go far beyond the expectations of most high school math curricula.

IV. Major topics covered

Week One:

  • David Hilbert’s 23 problems
  • The Nature and Magnitude of Negative Numbers and what Euler had to say about them
  • Different ‘types’ of numbers
  • Generalization vs. proof
  • Proofs of tests for divisibility
  • Proof by induction: introduction (using the sum of the first n odd natural numbers)
  • Reductio ad absurdum proofs: use Euclid’s proof of the infinitude of primes
  • Finite Sequences and Series, arithmetic and geometric sequences and series
  • Infinite Sequences and Series, arithmetic and geometric sequences and series
  • Convergence of infinite geometric sequences
  • Diophantine equations
  • Continued fractions, equipalindromic numbers
  • Use Fibonacci and Loucas sequences to develop and understand complicated notation
  • Zeno’s Paradox of Achilles and the Tortoise
  • Properties of primes (including Mersenne primes, Sophie Germainn Primes, Fermat primes), the sieve of Erastothenes, superprimes, superprime leaders
  • Relationship between prime and perfect numbers
  • Properties of palindrome numbers
  • Math in theatre: excerpts from Proof, Rosencrantz and Guildenstern are Dead, Arcadia
  • The “math montage” in film and how math gets treated in film: excerpts from Pi, Rushmore, Good Will Hunting, Stand and Deliver
  • Biography: Louis de Fermat, Sophie Germainn, David Hilbert, Leonard Euler, Blaise Pascal

Week Two:

  • Hilbert’s Hotel
  • Magnitudes of infinity, Cantor’s proof of the cardinality of integers, of rational numbers
  • Determine the size of the set of real numbers: Diagonalization (Cantor)
  • Euler Networks: properties of odd vertices
  • Stereographic projection
  • Projective geometry and Euclid’s Fifth Postulate
  • Fundamental ideas in topology: “a coffee cup is equivalent to a donut”
  • The interior, exterior and boundary of a set; closed sets, open sets
  • Historical ideas of ‘curve’: illustrate how a mathematical concept can evolve over time
  • Infinite curves: Von Koch snowflake, Sierpinski’s sieve
  • Proof that a given closed curve is infinite: apply series topics from week one to Von Koch Snowflake
  • Proof that a given infinite curve can bound finite area: apply series topics from week one to Von Koch snowflake
  • General properties of logarithms
  • Biography: Georg Cantor, Helge von Koch, Waclaw Sierpinski, David Hilbert

Week Three:

  • Definitions of dimension
  • Topological dimension versus fractal dimension
  • Calculating fractal dimension (using logarithms)
  • Fractal dimensions greater than 2, less than 1 (Sierpinski’s carpet, Menger sponge, Cantor dust)
  • Peano (plane-filling) curves, Hilbert’s Peano curve, dragon curves
  • Review of complex numbers
  • Recursive equations
  • Mandelbrot Set: determining if a given point in the complex plane is in the Mandelbrot Set
  • Determining the Julia set of a given point in the complex plane
  • Using series approximations to find the slope of a line tangent to a given point on a non-linear curve
  • Use series approximations to calculate the area bounded by a given, non-linear curve
  • Library research
  • Biography: Giuseppe Peano, Benoit Mandelbrot
V. Pre-requisite knowledge

Students who take this class should have completed and be comfortable with at least Algebra I, Geomtry and Algebra II. A scholar who has had exposure to the topics covered in Math Analysis, higher-level algebra and trigonometry will be able to extend the ideas presented in this class to those topics.

VI. Learning objectives

Students will:

  • Question and analyze the mathematical mechanism of ‘proof’
  • Study different types of proof
  • Translate mathematical notation into ‘real-world’ language
  • Develop problem-solving strategies
  • Identify and characterize different types of problems
  • Derive simple but rigorous proofs
  • Study the history behind mathematical topics and mathematicians
  • Tackle famous problems and paradoxes in the history of math
  • Be exposed to the portrayal of mathematics in theatre and film
  • Extend their classroom knowledge to wider areas of mathematics
  • Create and characterize finite and infinite curves that bound finite area
  • Research, develop and deliver individual presentations on selected topics not covered in the class
  • Present, as a group, ideas learned in class to wider groups of scholars
  • Work independently and in groups to solve problems
  • Question the underlying assumptions of mathematical validity
  • Develop individual ideas about the evolving nature of mathematics
VII. Primary source materials

Handouts prepared by the instructor from the works listed under “Supplementary source materials” which detail each topic and provide an historical background for each topic.

VIII. Supplementary source materials
  • Excursions in Number Theory
  • Topics for Mathematics Clubs
  • An Introduction to Topology
  • Paradoxes and Vicious Circles
  • Gardner, Martin. Penrose Tiles to Trapdoor Cyphers.
  • The Unexpected Hanging and Other Mathematical Puzzles
  • Stoppard, Tom. Arcadia.
  • Rosencrantz and Guildenstern are Dead.
  • Carroll, Lewis. Alice’s Adventures in Wonderland
  • An Introduction to the Philosophy of Mathematics
  • Rotman, Brian. The Semiotics of Zero.
  • Biographical information from
IX. Computing and the Internet (if applicable)

Students may use the Internet, with proper citation, on individual research projects. This class does not rely on calculators (though some students choose to use them) or on programming ability. Every topic we study can be approached by hand and brain.

X. Typical classroom strategies

Each new topic begins with either a ‘knight’ problem or a ‘knave’ problem. There is a certain type of logical problem that involves ‘knights,’ who always tell the truth, and ‘knaves,’ who always lie. The knights of our class are topics or problems which lead to mathematically elegant solutions, and the knaves are often the anomalies the lead to conclusions that contradict previously accepted theory. Students approach the problem individually or in small groups, and their work leads to higher-level applications of the implications of the initial problem. Students are expected to clearly articulate their methods, understand and comment on the approaches of other students and modify their own ideas accordingly. Classes held in other locations (2-3 days per week) allow students to relax their notions of the conventional classroom and enter into a dynamic, academic relationship with each other and with the instructor. The ‘classroom,’ then, becomes a moving, mutable exchange of ideas and methods, and students can tackle problems at their own pace.