Videos

Möbius strip as a configuration space

A Möbius strip can be regarded as the configuration space of pairs of (indistinguishable!) points on a circle.

Percolation

A dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.

After drawing n dots, how many components should we expect to see? When you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color.

Reflecting triangles in the plane

Start with a triangle in the plane; reflect that triangle across its three sides; repeat, reflecting the resulting triangles through their sides, over and over again.

Observe how, for only four shapes of triangles, the resulting set of triangle vertices is discrete. You can watch the same thing with more triangles.

Tensor products

The chain rule

Linear functionals and dual vector spaces

How can we multiply quickly?

Visualizing $\epsilon$s and $\delta$s

Why is $\infty$ not a number?