Background: A finite projective plane has point-line duality, and affine planes lack it. You can get a projective plane by adding a "line at infinity," and this is reversible.

Years ago I stumbled across this in constructing a counterexample to a graph conjecture, and I never found out if it was something people had realized or used before:

We can also recover point-line duality from an affine plane by just deleting one of the parallel classes of lines. <cont...>

<...> Today I checked that this structure has the axiomatization:

Given any point P and line L, P not on L, there are:

A1: a unique line L' containing P parallel to L, and

A1': a unique point P' on L "parallel to" (i.e. not colinear with) P.

A2: something about nontriviality, I think "There is a point P and a line L, P not on L" is enough.

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@bmreiniger

You'll like this puzzle that I made recently: https://www.thingiverse.com/thing:3856293

That axiomatization is an encapsulation of why the game Dobble (Spot-It in the USA) works.

@bmreiniger sorry, misread your post!

Ben Reiniger@bmreiniger@mathstodon.xyz@christianp

I did see your post (in fact that's what reminded me of this), but hadn't gotten around to commenting. But Dobble/Spot-It is a projective plane (less two lines, conjecturally because of printing convenience); in my structure, there would be cards that shared no images, and more images that don't appear together on a card.