Moore neighborhood tiles for visualizing 2-state cellular automata

These five tiles fit together like jigsaw pieces and can be used to explore 2-state Moore neighborhood cellular automaton, the most famous example being Conway's Game of Life.

Note: the tiles are shown with two colors for clarity though the print models are one-color. More on that below.

Each tile overlaps a 2x2 window of the automaton universe, and the jigsaw boundary constrains them to agree on which cells are present or absent. In addition, they are marked so that the Moore neighborhood (the 3x3 window around the cell) is shown in reduced size at each cell position.

For example, these two patterns, the block and the fishhook eater are well known still life patterns (unchanging when the generation rule is applied) and this can be verified by examining their tile layouts.

Observe that within each white circle, there is a cluster of either 3 or 4 black dots, representing the cell and its 2 or 3 neighbors, consistent with the rule for survival. Moreover, every cluster of white dots has cardinality less than or equal to 2 or greater than or equal to 4, but never 3. This is is consistent with the rule of birth on 3, which must not happen in a still life.

2x2 windows that are all 0 are understood as empty, though it would be possible to print a tile for them (I may add one later for completeness).

While it is possible to verify these rules just by looking at the pattern, collecting the neighborhood in this way makes verification less error-prone.

Here's an example of the simplest Life oscillator, the blinker.

In this case, one can see that the middle cell has 2 neighbors and survives, while the end cells each have only 1 neighbor each, and will die. Moreover, the clusters of 3 white dots above and below the middle cell each indicate birth.

Similarly, the two distinct shapes of the glider can be visualized in terms of their next state.

v2 tiles

There is an alternative set of v2 tiles that work the same but do not split dots across tiles, resulting in a more attractive display of Moore neighborhoods. The two kinds of tiles are not compatible with each other. I will provide some pictures of the v2 tiles soon. They work by assigning each shared dot to one tile in a consistent fashion so that if two adjacent tiles share a dot, it is assigned to the one going clockwise around the corner. The half-dot tiles are still somewhat easier to understand  so it may be a matter of personal preference. The v2 tiles look like this when printed with the top layers in a second color.

Using two colors

The models for each tile print in one color, and the tiles can be used this way, observing the raised part. I have found that using a different color makes the visualization much clearer and was able to do this by switching filaments during printing on a Flashforge Adventurer 3 Pro 2. I used the slicing software to estimate the length of filament for printing the base below the raised part and cut the filament there before loading, substituting the new color when it ran out. By recording the minute when this happens, it is possible to avoid doing any measurement on subsequent runs.

A more straightforward approach is simply to paint the raised part as shown below:

I used inexpensive white acrylic paint on a piece of flat cardboard as a stamp pad in reverse. By pressing the tile against the cardboard, only the raise part will be painted. I recommend this approach for anyone nervous about switching filaments on the fly. It is more than sufficient for producing usable tiles.