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
Venn diagrams are useful tools
for visualizing probability spaces,
and they are themselves interesting
learning section for some basic introductory material on probability theory, or read the
more advanced discussions at
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
to Conway's ``Game of Life'' that
explains this applet's behavior.
Also, here's a list of
pointers and references
The Santa Fe Institute.
You can check into
any time you like, but
the elevators are brutally slow.
Feeling a little existential angst
over infinitude? Try
the University of Toronto's
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
Pascal's Triangle Interface
at Simon Fraser University allows you to generate images of the
triangle of any size and modulus, and
Interactive Pascal's Triangle at Swarthmore lets you view
Counting is even easier
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.
37 - 100 A.D.
Further discussion of the
is available online.
The algorithm presented is motivated by the
recurrence we proved in recitation.