another self-dual space


not a projective space though
(there are “parallel lines”, for example).

ten “lines” through ten “points”.
you can get this by using
2-element subsets of {0,1,2,3,4} as points.
the lines are then triples
(like {0,1}–{1,2}–{0,2})
such that each point of the triple
is the symmetric difference
(A \Delta B = (A \cap B') \cup (A' \cap B))
of the other two.

the point associated to {0,1}–{1,2}–{0,2}
in the points-to-lines correspondence
i’ve illustrated here is then {3,4}…
the complement of the union of
the three points of the line.

so you can re-create this at will.
you just have to fiddle out a
nice symmetric version of the picture.

anyhow. pick a big circle.
look at the *three* big circles
matching the dark dots inside.
the three “lines” meet at the
original point.

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  1. it should not go unremarked that
    the points-to-lines duality for this space…
    *unlike* those of the projective spaces…
    is *independent* of choice-of-coördinates.




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