The mathematical model for a tippe top is a sphere with an uneven concentration of mass along the vertical axis, making the lower half heavier than the upper half; a physical model is more practical if you cut the top off the sphere and replace it with a stem. In each case, the key feature is that the centre of gravity is lower than the centre of the sphere. When the top spins, with the help of friction, it slowly tips over — raising the centre of gravity — until it is upside down.

For a history of the tippe top and an overview of how it works see this page. You can also watch an animation of the tippe top in action.

If you want to really understand why it inverts, Richard Cohen was the first to provide a rigorous explanation in The Tippe Top Revisited Am. J. Phys. 45, 12 (1977). Or, if it’s important that you also know why it falls back down as it runs out of spin, see this paper.

Peter Woit’s weblog is an interesting source for information about the intersection of math and physics. His latest is a post on the early history of using group theory in quantum mechanics. While group-theoretic methods in physics (and chemistry) are uncontroversial these days, the original emergence of the subject was painful, with pro- and anti-group theory partisans. (Wolfgang Pauli termed group theory the “Gruppenpest”.)

John Baez, of This Week’s Finds in Mathematical Physics fame, has a new article The Octonions.

The octonions are a mysterious example in mathematics that have been drawing attention in physics. Initially discovered by Graves in 1843, the octonions provided the first example of a number system with nonassociative multiplication: (ab)c is not equal to a(bc). Lots of examples of nonassociative multiplication are known, but the octonions remain the most interesting. For a more elementary introduction to the subject, there’s the article on Wikipedia.