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A complete graph of size n has n vertices and all n(n-1)/2 possible edges.

Feeling edgy about your understanding of graph theory?
Several interactive tutorials are available, as well as some more advanced material.

Learn about the short life and fast times of
Evariste Galois!
Learn more about
Galois theory at
Abstract Algebra Online.

Venn diagrams are useful tools for visualizing probability spaces, and they are themselves interesting combinatorial structures. Check out Probability Central's learning section for some basic introductory material on probability theory, or read the more advanced discussions at MathPages.

A bounded version of Conway's ``Game of Life'', such as the applet to the right, is just a finite state machine. Click on the pause button to make the glider move. Here is an intriguing introduction to Conway's ``Game of Life'' that explains this applet's behavior. Also, here's a list of pointers and references from The Santa Fe Institute.

Self-reference and self-similarity underlie the vivid field of fractal geometry. Here's a gentle introduction to fractal geometry, and a list of cool self-reference sites.

You can check into Hotel Infinity any time you like, but the elevators are brutally slow. Feeling a little existential angst over infinitude? Try the University of Toronto's MathNet explanation. Or, take a brief tour of cardinality and countability.

If you take the values in Pascal's triangle modulo 2, and plot the resulting bits as a bitmap, you get an image of the famous fractal known as the Sierpinski Gasket! The Pascal's Triangle Interface at Simon Fraser University allows you to generate images of the triangle of any size and modulus, and the Interactive Pascal's Triangle at Swarthmore lets you view the numbers.

Counting is even easier than 1-2-3. In this class we'll learn to ``count without counting,'' to find the sizes of sets without explicit enumeration. You can count on the Dictionary of Combinatorics to define for you a handful of useful combinatorial concepts.

Group theory applies to a wide-range
of real-world pursuits, from
information theory and cryptography
to solving the Rubik's cube.
A treasure trove of definitions and theorems from abstract algebra in general and group theory in particular is available at Abstract Algebra Online.
Be warned that this presentation of the material is somewhat dense.

Flavius Josephus,
37 - 100 A.D.
Further discussion of the Josephus problem is available online. The algorithm presented is motivated by the recurrence we proved in recitation.

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