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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?

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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.

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    @Thomas, the projective plane of order 4 has 21 points on 21 lines, 5 points on each line, according to https://en.wikipedia.org/wiki/Projective_plane#Finite_projective_planes – the affine plane of order 4 has 16 points, 4 on each line, according to https://en.wikipedia.org/wiki/Affine_plane_(incidence_geometry)#Finite_affine_planes – so removing a line from an order 4 projective plane leaves an order 4 affine plane.2015-11-14