Addition looks like a donut with a twist, in both finite and infinite, one-point compactified cases. (A reason we recognize them as “the same” operation.)
Imagine a clock laying on a table, and then placing clocks at each number which are rotated that number of steps. (Equivalently, so the number on the new clock matches the original clock at that spot.) If you trace where the numbers are on these clocks, you get bands that twist around the torus. This shape [0].
So we have:
1. Product of two circles.
2. A simple 2D shape with a hole.
3. Addition.
…all mixed up in one shape. So it shows up a lot of places.
Imagine a clock laying on a table, and then placing clocks at each number which are rotated that number of steps. (Equivalently, so the number on the new clock matches the original clock at that spot.) If you trace where the numbers are on these clocks, you get bands that twist around the torus. This shape [0].
So we have:
1. Product of two circles.
2. A simple 2D shape with a hole.
3. Addition.
…all mixed up in one shape. So it shows up a lot of places.
[0] - https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/05/Knot-S...