There's this fun space made of p-adic numbers upon which you can define a simple distance, and then circles have mind bending properties like the diameter (max edge to edge distance) and radius (distance from edge to center) being equal to each other.

Quirky stuff happens to disc area and perimeter as well, and open discs are also closed. The equivalent of Pi there is nuts.

Sadly I can't recall the details (it was a 2000-ish exercise on my maths course).

https://en.wikipedia.org/wiki/P-adic_number#Topological_prop...