the naturals are discrete, distant. a set of beads on a string
the rationals are a sponge. everywhere but somehow not filling spacing. vaguely fractal in nature
the reals are Platonic Matter. they are smooth and consistent. an infinitely stretchable uniform taffy (although it is interesting to note that despite often having really pretty constructions the reals end up being these awful pathological nightmares)
the ordinals are an infinite sequence of limit breaks. you only really care about the most interesting shapes they can take on.
unfortuately im not mathematics enough to know anything about the surreals or other weird constructions. they seem wild though