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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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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-Alvarez Jul 18 '12 at 1:04
    
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 1:16
    
@Mariano, in this context, "order" means the number of points on a line. –  Gerry Myerson Jul 18 '12 at 1:29
    
@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 1:31
    
Yes, and I did figure that out. Thanks, Gerry. –  Xuan Huang Jul 18 '12 at 6:15
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