Sure. But specifications and prime numbers as well as rule 101 are not tangible objects.
Rule 101 for instance requires a whole pile of infrastructure to even understand it, a prime number requires you to understand what prime numbers are and chaos theory is something many people claim to understand (you get all kinds of talks about butterflies) but to really grok what 'sensitivity to initial conditions' really means you have to dive pretty deep.
Contrast that with being given a pile of tiles and finding that no matter what you do as long as you lay the tiles according to the (very simple) instructions you can't help but discover that try as you want you can't make a repeating pattern.
Anyway, I'm glad to see you don't find it overly surprising, I'm a bit naive when it comes to things like this and find endless enjoyment in such simplicity.
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.
Rule 101 for instance requires a whole pile of infrastructure to even understand it, a prime number requires you to understand what prime numbers are and chaos theory is something many people claim to understand (you get all kinds of talks about butterflies) but to really grok what 'sensitivity to initial conditions' really means you have to dive pretty deep.
Contrast that with being given a pile of tiles and finding that no matter what you do as long as you lay the tiles according to the (very simple) instructions you can't help but discover that try as you want you can't make a repeating pattern.
Anyway, I'm glad to see you don't find it overly surprising, I'm a bit naive when it comes to things like this and find endless enjoyment in such simplicity.