I am working with a special cellular automata that uses hexagonal cells rather than square cells, a hexagonal grid, rather than a square grid, and the set of complex numbers, rather than a finite set, for the cellular state-space. In this case it seems more appropriate to refer to a cell's "value" rather than the cell's "state."
I need to have a way of calculating the density of values in relation to a particular cell, as well as a function that takes a set of cells and produces a density value for the set. The end goal is have a metric for measuring clusters of values for any arbitrary area, possible discontinuous or infinite, of the cellular automata's universe.
Here is my current model: $$\mathbf{C}:\mathrm{the\;set\;of\;all\;cells\;}$$ $$Value(x):\mathrm{calculates\;the\;value\;of\;the\;given\;cell,}\;x$$ $$Dist(a,b):\mathrm{calculates\;the\;Manhattan\;distance\;between\;two\;given\;cells,}\;a\;\mathrm{and}\;b$$ $$CellularDensity(x)=\sum_{c\in\mathbf{C}}\frac{Value(c)}{2^{Dist(x,c)}}$$ $$AreaDensity(\mathbf{A})=\sum_{a\in\mathbf{A}}CellularDensity(a)$$
Here is what is wrong with the current model:
The current model is does not work well when there is an infinite number on cells with non-zero values in the universe, such as when a pattern of values is defined.
I am looking to have this model replaced with something that can produce meaningful density values given a finite area, as defined by a set of cells, when there is an infinite number of cells with non-zero values present.