In both hex and square grid systems, we have a mapping of a
two-dimensional plane. Because of this, any space in either system can be
represented by coordinates that denote that space’s location relative to two non-parallel
axes on that plane. The difference is, in a square grid those axes are
typically perpendicular, while in a hex grid the axes are typically at a 60°
angle.

We’ll call the space where the axes cross one another the
Origin, and designate it (0,0). From here, it’s easy to see in a square grid
how the coordinates are assigned. The number of spaces to the left or right of
the origin is the x-coordinate, while the number of spaces above or below is
the Y-coordinate.

Let’s focus on the x-coordinate for a moment. It’s easy to
think of the x-coordinate as describing the current space’s distance from the
y-axis, but that’s really only true in a square coordinate system, or any
system with the two axes being perpendicular. Another way to look at it is that
the x-coordinate identifies columns of spaces which run perpendicular to the
x-axis.

Similarly, the y-coordinate denotes rows of spaces
perpendicular to y-axis.

The end result is that we have coordinates that identify the
row (y-coordinate) and column (x-coordinate) relative to the two axes
individually. If we apply this to the hex system, we end up with a similar
coordinate system, except the “rows” and “columns” are hexes rather than
squares, and the axes are at a non-right angle.

You may notice that the “columns” and “rows” are adjacent
hexes that are opposite one another. In this case, the x rows run upper-left to
lower-right, while the y rows run upper-right to lower-left. There’s a third
row that runs horizontally, which would denote a hex’s position relative to a
vertical (z) axis.

This axis is totally unnecessary to uniquely identify spaces,
but it can be convenient for computation and describing distances, relative
positioning, etc. I will likely include this axis in most of my illustrations
for completeness and for reference, but remember that only two axes are
necessary to describe the location of a hex in a grid on a two-dimensional
plane.

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