A

# Hausdorff, Felix (1868–1942)

Felix Hausdorff was a German mathematician who is considered to be one of the founders of modern topology and who also did significant work in set theory and functional analysis. Among several concepts named after him is the Hausdorff dimension, which gives a way of assigning a fractional dimension to a curve or shape. He also published philosophical and literary works under the pseudonym "Paul Monger."

Hausdorff studied at Leipzig and taught mathematics there until 1910, when he became professor of mathematics at Bonn. When the Nazis came to power, Hausdorff, a Jew, felt that as a respected university professor he would be safe from persecution. However, his abstract mathematics was denounced as useless and "un-German" and he lost his position in 1935. He sent his daughter to Britain but stayed with his wife in Germany. When in 1942 he could no longer avoid being sent to a concentration camp, he committed suicide together with his wife and sister-in-law.

## Hausdorff dimension

A Hausdorff dimension is a way to measureaccurately the dimension of complicated sets such as fractals. The Hausdorff dimension coincides with the more familiar notion of dimension in the case of well-behaved sets. For example a straight line or an ordinary curve, such as a circle, has a Hausdorff dimension of 1; any countable set has a Hausdorff dimension of 0; and an n-dimensional Euclidean space has a Hausdorff dimension of n. But a Hausdorff dimension is not always a natural number. Think about a line that twists in such a complicated way that it starts to fill up the plane. Its Hausdorff dimension increases beyond 1 and takes on values that get closer and closer to 2. The same idea of ascribing a fractional dimension applies to a plane that contorts more and more in the third dimension: its Hausdorff dimension gets closer and closer to 3. As a specific example, the fractal known as the Sierpinski carpet has a Hausdorff dimension of just over 1.89.