If I voted to close on questions, I would vote to close on this question, even after the edits. The primary reason is that I have to work at figuring out the question. A secondary reason is that there seems to be a lot of incidental information that does not seem to support the main specific question I think you want to ask. (This does not mean it is a bad question, or even poorly presented. It means for this forum the audience is unlikely to receive it well.) Let me suggest the following to make it more accessible and favorable for this forum.
Start with a brief summary: " I am looking at structures in the plane with some regularity. I am trying to define and study certain characteristics of these patterns, and my main questions are : what references cover material like what is presented below, and what search terms/synonyms can I use in doing more research into these structures and these particular characteristics? "
Now I am not sure that I understand the characteristics through your examples and presentation, so it might help to give examples that represent a scale for each parameter. Say 5 examples of increasing D value, with some words on why it is increasing. However, no one wants to read several paragraphs to understand what you might mean by D, so try to encapsulate it into two or three sentences, if you can.
Telling us what is not an answer (specifically your related but not an answer section) is good, but we need to know more of why they aren't answers.
I would like to see some motivation, perhaps near the beginning. If you are studying these patterns in the context of charge distribution, or growth of microbial colonies, or cellular automata, that would be good to know.
A version of this question that is more brief, more relevant, and more motivated might make a good MathOverflow question. Evem if the present version "follows the rules", that does not automatically make it a good or acceptable question.
Gerhard "Has More Advice On Meta" Paseman, 2016.02.23.