Beeps, Blips, and Computery Bits

Before I discuss the simulation itself. I want to show this little graphic to explain how hexagonal grids are actually just a square grid of cubes in disguise.

The “why” for doing this is the (x,y) coordinate system, which makes storing and working with the data much simpler. With that out of the way, let’s move on.

One of the most common types of cellular automata produces the output for the next state of the simulation by iterating over every “cell” in a grid and determining what it’s new value will be based on the values of its neighbors.

There are many ways to determine how this works. For instance, in Conway’s Game of Life the rules are as follows: any “living” cell remains living if it has 2 or 3 living neighbors, and dies otherwise, any “dead” cell with exactly 3 living neighbors becomes alive and remains dead otherwise. This is generally done on a square grid where each cell has 8 neighbors in total, but for our example we’ll use a hexagonal grid (represented on a square grid by ignoring 2 of the neighbors for each cell).

The above animation uses the following string to define its rules: “F.F..T.F.5;6;7;15;16;17;18;19;20”. We’ll get into what this means later, but it roughly defines the rules we outlined earlier. However, this only covers half of the functionality of the simulation.

A different kind of automata is the block automata, which takes groups of cells and translates them all together in a “block”. After that is done is shifts the frame of reference so now different cells are grouped into blocks, and then repeats.

One of the easiest ways to do this translation is to have a big table where you can look up a specific pattern of cells and output the other pattern that is matched with it.

The above animation uses the following rule set: “T.T.1,2;2,18;3,66;4,34;5,82;6,50;7,70;8,54;9,86;10,114;11,118;14,10;15,26;16,74;17,42;18,90;19,58;20,78;21,62;22,94;23,122;24,126;27,4;28,5;29,6;30,20;32,7;33,21;34,70;35,85;36,71;37,23;38,87;40,12;41,13;42,14;43,28;44,15;45,29;46,78;47,93;48,79;49,31;50,95.T.T.”, and here is where we can begin to talk about what each part of this actually means.

Each section is separated by a “.”, the first section being whether the look up table for block transitions has symmetry (T for yes F for no). What we mean by symmetry in this case is whether a set of patterns that can be turned into one another by simply rotating or mirroring the image should use the same rule or different rules. If we treat every pattern as unique with its own rule, then there is a total of 2^7 (or 128) rules because there are 7 cells in each block and 2 options for every cell. If we group similar patterns together then there are only 26 rules (13 for the rotations with the middle cell off and another 13 for the rotations with the middle cell on).

The second part (another T/F option), determines whether we preserve orientation when using symmetrical rules. Let’s say we have a block that is just the pattern 1 rotated right by one cell.

The rule then might be to change all pattern 1’s to this:

And if we weren’t preserving orientation, then this would be enough. However, if we are preserving orientation, then we must remember that the input was rotated to the right by one cell, so we must do the same to the output before we are done.

So, with the first two settings, we’ve defined how the table will be used, but it only has default values in it which won’t actually change anything. The next part of the is a list of patterns and their outputs (but only the ones that differ from the default). The above example takes each symmetry pattern and outputs a value representing a slightly rotated example of that pattern.

The next setting (once again a T/F), is whether the cell transitions are symmetrical. It is essentially the same thing as the block symmetry setting.

After that is a setting to determine what the default values for cell transitions should be. When “F”, the default value will be Off, and when “T” the default value will be whatever the input value was (i.e. the middle cell in a pattern).

Finally, the last setting is a list of rules that are the opposite of their default value (On instead of Off and vice versa).