How to model rules for generating geometric patterns?

For my problem it would be best to find a numeric representation of kazakh national ornaments for generating new ones. But other approaches are also fine.

The ornaments essentially consist of combinations of relatively basic ornaments. Usually the ornaments are symmetrical.

Here are few examples of basic elements:

(The images are a bit distorted)

And this is an example of a more complex ornament:

How could I encode an ornament's representation in as few numbers as possible? So that I could write a program that would generate an ornament, given some sequence of numbers

Any ideas are appreciated.

As I write this, I have thought that generating images of snowflakes may be somewhat relevant, although it's possibly just a fractal.

You have to realize that your question is actually not how to represent them but how to generate them.

Still you might get some ideas. But don't hold your breath, because it can get complicated

EDIT: In researching problems such as this you could start with L-systems, this paper seems to convey the idea.

Actually here's an attempt at an answer: Represent it as a set of grammar rules.