I cannot seem to construct an affine plane of order 4. I have the construction for order 3- but cannot seem to come up with or find the construction for 4 anywhere. Could someone show me a picture of one, preferably with parallel lines indicated?
An example is given here, but without any motivation. You could probably try looking at $F\times F$, where $F$ is the field of 4 elements.
EDIT: Here's another way. Here is a drawing of an order 4 projective plane. Remove any one line, and the five points on that line, and what's left is an affine plane of order 4.