When I initially saw that 3-layer Seifert box, my mind jumped to an infinitely layered structure similar to that one. Does anything necessarily prevent such a structure from existing?
@Rithaniel I didn't find a three-layered version of the box
As far as I can tell, nothing prevents you from doing it infinitely many times, other than the fact that you'll have to go infinitely far in and infinitely far out
where you have a cake with two colors: the first layer is one color, the second layer is the other color, the third layer is the first color again, and they alternate like that
and the layers get half as thin each time so they fit in a finite space
and the question is, if you look at the top, what color do you see
Though, you could argue that the end result would be an artifact of the nature of the infinite process, in which case you'd need something like the Ramanujan sum to assign values to divergent series.
there would be no top side of the cube, is what I mean
it is as if you would need some access to a "time beyond infinity"
the situation is isomorphic to the question "if the lamp toggles between on and off repeatedly every second, what would be the state of the lamp after an infinite amount of time?"
or, "if I tell you that $f: (\omega+1) \to 2$ satisfies $f(0)=0$ and $f(n+1)=1-f(n)$, what is $f(\omega)$?"
you can treat the bottom side of the cube as $0$, and then go up, and add $1$ every time you change colour
