Far far away in the distant future some explorers stumble across the remains of an alien civilization. You are the teams linguist, and it's your job to attempt to translate the hodge podge of symbols that appear to be their written language. Unfortunately this seems like it's way out of your grasp. They seem to be using a system that allows for thousands of symbols, and you can't seem to find much consistency in it at all!
After weeks of struggling, you have a break through! You had noticed that certain symbols that often were present at the beginning of texts had some things in common, and you don't think it's a coincidence. You're convinced that a bunch of what you thought were different symbols are actually the same just drawn different ways, now you just need to figure out how many separate symbols there really are.
Each symbol of the language is made up of a hollow circle with 12 smaller circles overlaid. These smaller circles are each either hollow or filled. Here are some pictures along with some equivalencies (and inequalities) that you're convinced you've figured out.
Assuming that these patterns extend to all symbols, the goal is to find the number of possible symbols in this language that are not equivalent. If this goes for a while without any correct answers, I'll post some more specific equivalency rules, but I feel like it's more fun to try to work them out yourself from examples.
BONUS: See if you can find an equation that will give the number of possible symbols for a ring with N dots instead of 12!
BONUS to the BONUS: See if you can find the equation for a ring with N dots that each have M possible states!