In an important field of mathematics called topology, two objects are considered to be equivalent, or “homeomorphic,” if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.

Consider, for example, a torus — the dougnut-shape object shown in the intro slide. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can’t tell their doughnuts from their coffee cups.

On the other hand, Moebius bands — loops with a single twist in them — are not homeomorphic with twist-free loops (cylinders), because you can’t take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.

Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren’t allowed to poke a hole in the sphere and pull out the inside. But in fact, “sphere eversion,” as it’s called, is possible.

Incredibly, the topologist Bernard Morin, a key developer of the complex method of sphere eversion shown here, was blind.

The Riemannian Tonnetz (“tonal grid,” shown above) is a planar array of pitches along three simplicial axes, corresponding to the three consonant intervals. Major and minor triads are represented by triangles which tile the plane of the Tonnetz. Edge-adjacent triads share two common pitches, and so the principal transformations are expressed as minimal motion of the Tonnetz. Unlike the historical theorist for which it is named, neo-Riemannian theory typically assumes enharmonic equivalence (G# = Ab), which wraps the planar graph into a torus.