But isn't there about as much infrastructure in 2-dimensional geometry? There are 5 Euclidean axioms and you probably need some topological axioms too for the notion of neighborhoods and space-filling tilings. As for prime numbers you need 9 axioms. Rule 110 probably requires the least amount of infrastructure of these three examples.
We just happen to live in a space in which we obtain intuition about 2D geometry from very early on.
You need formal axioms to do precise and rigorous mathematics, but you don't need any axioms to see and manipulate geometric patterns in our heads. We seem to naturally posses this ability (probably because our brains use a lot of pattern-matching).
We already have the geometry manipulation program installed in our head, and maybe this program is even based on Euclid's (or Hilbert's) axioms. But our brains just run this program, they can use this abstraction without having been thought geometric axioms beforehand.
No. Geometric intuition is widely believed to be only a very rudimentary native skill. There appear to be structures in our brain that allow us to think spatially [1], but infants are generally unable to perform even very simple geometric reasoning. Our environments are full of geometric theorems however, that our brains become acustomed to within the first few years of our lifes.
We just happen to live in a space in which we obtain intuition about 2D geometry from very early on.