There's a tendency for the evolution of mathematics to branch. If we assume the basic line of mathematics to be basic arithmetic, then one of the earliest branchings would have to be algebra. More recent developments might include calculus, combinations and permutations, trigonometry, set theory, logic and computer science, and so on. One thing I'm curious about is whether there is a mathematics of patterns. You know, repetitions and regularities. Is there such a thing?

perhaps something like cellular automata are what you are looking for? Stephen Wolfram's books A New Kind Of Science covers them in depth. While my opinion is that Wolfram goes a little overboard with his enthusiasm for the subject, there are some interesting things that happen with CA rules.

There's a tendency for the evolution of mathematics to branch. If we assume the basic line of mathematics to be basic arithmetic, then one of the earliest branchings would have to be algebra. More recent developments might include calculus, combinations and permutations, trigonometry, set theory, logic and computer science, and so on. One thing I'm curious about is whether there is a mathematics of patterns. You know, repetitions and regularities. Is there such a thing?

 

Would fractals qualify as the mathematics of patterns you are looking for?

 

http://en.wikipedia.org/wiki/Fractal

 

I am not sure math begins with arithmetic. People did not begin with Arabic numbers, or equal symbols. For the Egyptians the problem is figuring land boundaries after flooding, and building buildings and monuments, and art which is sacred and controlled by geometry. Greeks used stones arranged in patterns to discover fundamental concepts.

Mathematics is all about finding patterns, but usually very abstractly. I assume you mean geometric patterns.

 

Fractals and tilings spring to mind here. All these thing have some notion of repetition and symmetry.

 

Fractals have self-similarity (or something close like self-affine). This means you can find repeated "pieces" of the pattern at different scales within the pattern. There are plenty of interesting things here and generating fractals can be a lot of fun!

 

Tilings or tessellations are as fun and interesting; these are coverings of a plane with regular polygons. If the tiling has a translational symmetry, then it can be categorized by a wallpaper group. This is essentially "crystallography in the plane".

 

Aperiodic tilings are fascinating things and are related to quasicrystals, which are highly ordered arrangements that are not periodic.

 

Fractals and tilings are a subject of research today.

Topology??

Topology??

 

Topology can be an important idea in fractals and tilings.

Thanks, everyone, for your replies.