I propose an alien-geometry topic challenge.
Its often hard to tweak the mathematics of a world (for example, you can't really make 2+2=5 in a reasonable way), but one way you can is by tweaking the geometry. This can lead to many interesting implications, including chemical, biological, and even cultural implications.
Here are some examples of 3-manifolds for inspiration; (3-manifolds are similar to our world locally). (4-manifolds are also similar if you include time.)
Also, the use of quotient spaces helps too.
EDIT: It appears to be finished. If someone wants to propose another tag challenge, go ahead.
alien-[empirical science]
tags than using purely abstract disciplines. Or, perhaps, asking about the physics of worlds where the only possible triangles are Penrose triangles, and the only possible bottles are Klein bottles... $\endgroup$ – NofP Jan 25 '18 at 16:40