Is a Mug the Same as a Donut?
According to topology, yes!
Mathematics is full of strange and unexpected results. One of which comes from the field of topology. This relatively recent area of study concerns itself with geometrical objects that can be deformed. While its roots date back to ancient Greece, the field was formalized in the 20th century. Since then, topology has become widely accepted as a key area of pure mathematics. Like many aspects of math, it remained mostly theoretical for most of its lifetime and has since become applied to a variety of other disciplines in unexpected ways.
Imagine a 3D surface that you can stretch, crumple, twist, and bend. This is a topology, and these objects form the fundamental building blocks of this theory. Mathematicians soon realize that a lot of geometrical theories stay the same even when you apply these actions to an object. In fancy terms, these types of deformations are called homeomorphisms. The animation below shows how a mug and a donut are fundamentally the same thing.
To get between the two, we only deform the object in ways that are acceptable in the field of topology. Really, you can think of these as any kind of stretching motion. I also really like this series of ceramic models showing the transformation between the two.
