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I'm doing a seminar on finite geometry. In my presentation, I want to count the number of parallel lines in $\mathbb{F}_4^2$, where $\mathbb{F}_4$ is a field with $4$ elements. But I'm not sure how to count them. I read that the number of parallel lines can be counted only by knowing the cardinality of $\mathbb{F}_4^2$ which is 16 and the cardinality of $\mathbb{P}^1(\mathbb{F}_4)$, the projective line of $\mathbb{F}_4$, which is 5.

Does anyone know which argument is used here? Or if there is an alternative method to count the parallel lines in $\mathbb{F}_4^2$?

Hamilcar
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  • Some help comes from this question. – Dietrich Burde May 14 '18 at 18:44
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    Hint: Those lines can, indeed, have five distinct slopes (any element of the field and infinite slope for the vertical lines). Through each point there is a unique line of a given slope. Four points on any (affine) line, so the sixteen points of the (affine) plane are partitioned into, guess how many parallel lines (= lines sharing the same slope). – Jyrki Lahtonen May 14 '18 at 18:58
  • I'm a little bit confused, when there are five different lines (with different slopes) trough each point and four points on each line we have 20 points and not 16. Where is my misstake? In any case the number of parallel lines is equal the number of slopes or not since we have the square of our field – Hamilcar May 14 '18 at 20:39
  • See https://math.stackexchange.com/q/1925479/35416 for a picture. – MvG May 15 '18 at 20:26
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    When you write “the number of parallel lines”, what exactly is it you want to count? Given one line, the number of parallels to that? Or the number of pairs of parallel lines in all the plane? Or the number of bundles of parallels, i.e. the number of different “directions”? – MvG May 15 '18 at 20:29
  • Hamilcar: Fix a point $P$. There are five lines through $P$ (with different slopes). Each and every one of those lines contains four points, that is, $P$ and three others. Altogether $1+5\cdot3=16$ points. – Jyrki Lahtonen May 16 '18 at 12:11

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