The Surprising Similarity Between a Coffee Cup and a Donut

I can’t tell the difference between a coffee cup☕ and a donut🍩.

Of course, they look different, but from a certain perspective, they are considered to be the same shape.

In the field of mathematics called topology, the two shapes are not distinguished. In fact, if you continuously deform a coffee cup, it can turn into a donut. If you expand the loop on the handle of a coffee cup, doesn’t it deform into a donut shape?

Two Sides of Topology

In mathematics, the term topology has two meanings: one refers to general topology, which extensively discusses topological spaces, and the other refers to geometric topology, which studies the properties of shapes. The “位” (i) in “位相幾何学” (topology) refers to “position,” while the “相” (sō) represents “form.”

A Vast Landscape of Fields

Both of these belong to the field of mathematics known as geometry. When we broadly classify mathematical fields, they fall into three main categories: algebra, geometry, and analysis. However, when broken down further, there are many diverse subfields within mathematics.

For example, here is what the AMS (American Mathematical Society) has established: https://mathscinet.ams.org/mathscinet/msc/msc2010.html
It’s safe to say that studying all fields is quite impossible 🙁