Finite hyperbolic planes

Euclid’s fifth postulate says that given a line ℓ and a point P not on ℓ , there is a unique line through P parallel to ℓ. In the projective plane there are no parallel lines, and in the hyperbolic plane there are at least two lines through P parallel to ℓ. Recent posts have looked at finite projective and affine planes. This post will look at finite hyperbolic planes. Finite projective planes come up far more often in application than finite hyperbolic planes. I’ve seen several resources on finite geometries that don’t mention the hyperbolic case. Finite hyperbolic geometries have less structure than their projective or affine counterparts and so are less useful. The definition of a finite hyperbolic plane isn’t entirely standardized. We will use the following axioms for this post. Any two distinct points lie on exactly one line. Given a line ℓ and a point P not on ℓ, there exist at least two lines through P parallel to ℓ. There exist four points, no three collinear. Every line…

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