TL;DR

Some words on hexagonal grids, and a pointer to something serious: Hexagonal Grids on Red Blob Games.

In recent post The Definitive Conway’s Game of Life I made a quick and vague reference to using a hexagonal grid for playing Conway’s Game of Life. So I thought that maybe some additional hints might be useful as well.

In that specific challenge, a simple representation of the whole grid based on two coordinates was sufficient:

• take any hexagon in your grid and consider it your origin;
• that hexagon has three pairs of opposite sides;
• choose one pair and consider that as your $X$ dimension. Make it increase as you move right (or up, if the two sides are stacked vertically);
• choose a different pair and consider that as your $Y$ dimension. Again, make it increase with the same rules as above.

This is actually all you need. From the origin, you can directly reach four of the six adjacent cells by only changing one unit (positive or negative) in one of the dimensions. As an example, assume that the grid is such that hexagons have a flat side on the up/down direction (which is actually not what was happening in the challenge), we might choose $X$ and $Y$ as follows:

^ Y
|
|                        >--------<
|             >--------<  ( 0,  1)  >--------<
|         --<  (-1,  0)  >--------<            >--
|             >--------<  ( 0,  0)  >--------<
|         --<            >--------<  ( 1,  0)  >--
|             >--------<  ( 0, -1)  >--------<
|                        >--------<
          ----
              \--------- 
                        \----------
                                   \-------->
                                            X

Now we’re left with the two missing cells, one right-up and another one left-down.

Starting from the origin, there are two ways to reach the right-up cell:

• go right-down first (increase in $X$ only, landing on $(1, 0)$), then up one step (increase in $Y$ only, landing on $(1, 1)$);
• go up first (increase in $Y$ only, landing on $(0, 1)$, the right-down (increase in $X$ only, landing on $(1, 1)$).

Both way are consistent: the right-up cell is at $(1, 1)$, i.e. it is obtained by increasing one single step in both dimensions.

It’s easy to see that the same can be done for left-down, yielding $(-1, -1)$, i.e. a decrease in both dimensions.

So, we’re left with the following displacements for all cells adjacent to the origin:

^ Y
|
|                        >--------<
|             >--------<  ( 0,  1)  >--------<
|         --<  (-1,  0)  >--------<  ( 1,  1)  >--
|             >--------<  ( 0,  0)  >--------<
|         --<  (-1, -1)  >--------<  ( 1,  0)  >--
|             >--------<  ( 0, -1)  >--------<
|                        >--------<
          ----
              \--------- 
                        \----------
                                   \-------->
                                            X

Expanding this to the whole plane… we get a unique pair of $(x, y)$ coordinates for each cell. It’s also easy to see that this is equivalent to a square grid, where the neighbors of any cells are the four adjacent ones (in the up, down, left, and right directions), plus the two diagonal ones on the right-up and the left-down directions. The other two cells in the other diagonal… are distant, sorry! The following picture marks with asterisks the cells in this mapping that are considered adjacent to the origin:

               |            |
     (-1,  1)  | *( 0,  1)* | *( 1,  1)*
               |            |
    -----------+------------+-----------
               |            |
    *(-1,  0)* |  ( 0,  0)  | *( 1,  0)*
               |            |
    -----------+------------+-----------
               |            |
    *(-1, -1)* | *( 0, -1)* |  ( 1, -1)
               |            |

In that challenge, though, the hexagons are arranged with flat sides on the left-right direction instead. Again, it’s easy to choose two dimensions and extend what we explained above, e.g. yielding this arrangement for the displacement of cells adjacent to the origin:

               ( 0,  1)          ( 1,  1)

      (-1,  0)          ( 0,  0)          ( 1,  0)

               (-1, -1)          ( 0, -1)

Again… it’s the same mapping on the square grid: adjacents are all four in the up, down, left, and right, plus the two diagonals in the right-up and left-down directions.

So much for the challenge. If you really want to understand what’s going on with the hexagons… your next stop MUST be Hexagonal Grids from Red Blob Games!