What do a doughnut and a mug have in common?

Yes, both a doughnut and a mug would be welcome at a morning tea. But more than that, they are the same object! Topologically, that is. I’ll explain what I mean.

Topology is a field of mathematics that studies the properties of objects that are preserved through continuous deformation. Bending, twisting, stretching, and compressing are allowed, but cutting, tearing, and joining are not. I often use the example of a doughnut and a mug as a starting point to try to describe topology to my non-mathematically minded (and mathematically minded) friends.

Imagine you have a doughnut made of malleable clay or something like Play-Doh. You can bend, twist, stretch, and compress the material to create a mug. This would be without making any cuts or tears or joining material to close any holes. Similarly we can deform the mug to get back the doughnut.

Here is an illustration of this process from Wikipedia:

When we can do this, we say the two objects are topologically equivalent. The important topological components of a doughnut and a mug is that they are three-dimensional, solid objects with one hole. The doughnut has the hole through the middle; the mug through the handle.

As a mathematical object, this is called a solid torus. The torus is a surface of revolution generated by rotating a circle about a coplanar axis. The solid torus is a torus with the inside filled; instead of a circle, we rotate a disk about such an axis. 

Generation of the torus (left) by rotating the red circle and the solid torus (right) by rotating the red disk . Both are rotated about the dotted axis and trace the blue circle.

By studying properties of objects that are preserved through these operations, we study intrinsic characteristics of the objects that are independent from their exact shape. Such characteristics are called topological invariants.

Consider this further and focus on the torus (akin to the surface of the doughnut or mug). We can subdivide the torus (or any surface) into “polygons” or “regions” consisting of a certain number of vertices, edges, and faces. Faces are bounded by edges and edges meet at vertices.

For example:

Example subdivisions of the torus with distinct vertices, edges, and faces marked with different colours. Can you create other subdivisions? What would it look like if you “unfolded” these subdivisions?

The Euler characteristic of a subdivision of a surface is 

for the number of vertices, edges, and faces respectively. (Note this is different to the used for the chromatic polynomial.) What is the Euler characteristic of the torus?

The Euler characteristic is well-defined, meaning it stays the same no matter the subdivision. Have a think about why this is true. What happens in the above subdivisions when we try to add an extra vertex, or an extra edge, or face?

The Euler characteristic is also a topological invariant, meaning it stays the same no matter the continuous deformations we apply! Consider the subdivisions you could have on the surface of a doughnut, mug, or any other continuous deformation of the torus. These will still provide a valid subdivision of the object no matter how we bend, twist, stretch, or compress, and thus the Euler characteristic will remain the same. Nifty, right?

All of this has been a very brief insight to some topology using the example of the torus (and hopefully I have convinced you to see the commonality between doughnuts and mugs too). There is so much more to topology and we have only scratched the surface of what kinds of things we can do!