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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?

Xuan Huang
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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.

Gerry Myerson
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  • Wouldn't F need to have 2 elements for the plane to have order 4? (What I am asking is, what is the order? :) ) – Mariano Suárez-Álvarez Jul 18 '12 at 01:04
  • Could you draw a picture for the plane described? Some of lines are definitely incorrect. For instance I think 1,5,9,14 is definitely 1,5,9,13. – Xuan Huang Jul 18 '12 at 01:16
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    @Mariano, in this context, "order" means the number of points on a line. – Gerry Myerson Jul 18 '12 at 01:29
  • @Danielle, I don't vouch for the example at the link, and if you have found a mistake there, +1 for you. Have you thought about the suggestion to use that finite field? – Gerry Myerson Jul 18 '12 at 01:31
  • Yes, and I did figure that out. Thanks, Gerry. – Xuan Huang Jul 18 '12 at 06:15
  • Doesn't removing a line from an order four projective plane give us an order 3 affine plane? – Thomas Ahle Nov 14 '15 at 02:06
  • @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. – Gerry Myerson Nov 14 '15 at 03:24