  The correspondence between perfect difference sets and transitive projective
planes is well-known. We observe that all known dense (i.e., close to
square-root size) Sidon subsets of abelian groups come from projective planes
through a similar construction. We classify the Sidon sets arising in this
manner from desarguesian planes and find essentially no new examples, but there
are many further examples arising from nondesarguesian planes. We conjecture
that all dense Sidon sets arise in this manner. We also give a brief bestiary
of somewhat smaller Sidon sets with a variety of algebraic origins, and for
some of them provide an overarching pattern.
