Goodstein sequences are integer sequences with a very surprising property.

Start with any number. Rewrite the number as a sum of powers of 2. In turn, rewrite the exponents as powers of 2, then the exponents of exponents, etc. For example, we’d write 33 as:
.

To compute the next value in the sequence, replace every 2 with a 3, and subtract 1. The next value in the Goodstein sequence for 33 is:

which is equal to 22876792454961.

For the next step, we replace the 3s with 4s and subtract 1, etc. This sequence continues to increase very rapidly, right?

Wrong. Goodstein proved that for any starting value, the sequence eventually goes to zero. Even more surprisingly, the proof relies on properties of infinite ordinal arithmetic. The theorem cannot have an appreciably more elementary proof: the result is independent of the Peano axioms for arithmetic.

A minicourse on Goodstein sequences and some related examples can be found at this online course: Fast-Growing Functions and Unprovable Theorems

The Algebraic Surface Homepage has many interesting pictures of algebraic surfaces. A few examples of algebraic surfaces might be familiar from high school: spheres, ellipsoids, hyperboloids. Higher degree surfaces are much more exotic, though. For exmple, check out the septic with 99 singularities.

Another page with galleries is the Cubic Surface Homepage.

The Hausdorff dimension is used to define the dimension of fractals, for example, the dimension of the Sierpinski triangle is log(3)/log(2).

To find the d-dimensional Hausdorff measure of a set: cover the set with very small balls, sum the diameter to the power of d of each ball, and take the lim inf as the balls get smaller. For integer dimensions, the Hausdorff measure is equivalent to the Lebesgue measure. The Hausdorff dimension of a set is the point where the d-dimensional Hausdorff measure changes from infinity to zero, i.e. the dimension of a set is d^* if for d < d^* its measure is infinity and for d > d^* its measure is zero.

From the abstract for a paper by Dierk Schleicher:

… we construct a set E ⊂ â„‚ of positive planar measure and with dimension 2 such that each point in E can be joined to ∞ by one or several curves in â„‚ such that all curves are disjoint from each other and from E, and so that their union has Hausdorff dimension 1. We can even arrange things so that every point in â„‚ which is not on one of these curves is in E. These examples have been discovered very recently; they arise quite naturally in the context of complex dynamics, more precisely in the iteration theory of simple maps such as z → sin(z).

Sigfpe, our most prolific commenter, has the most disturbing photo ever on his weblog.

While going some boxes the other day, I found an index-card-sized piece of paper with a commutative diagram on it, but no other text. Where does it come from? I have no idea. For all I know if you leave any box alone long enough, it starts to sprout commutative diagrams.

Lieven Le Bruyn, an algebraist in Belgium, has a weblog. His particular interest is in noncommutative geometry. There are as many approaches to noncommutative geometry as there are noncommutative geometers, but he outlines his particular point of view in a three-part post, here, here, and here.

On ArXiv there is a new survey paper on finite-dimensional algebras and quivers. The paper is rather dense, so it would be tough going for someone not already familiar with the vocabulary of quivers, but it covers some of the surprising connections with Kac-Moody Lie algebras.