  I introduce the problem of finding maximal sets of equiangular lines, in both
its real and complex versions, attempting to write the treatment that I would
have wanted when I first encountered the subject. Equiangular lines intersect
in the overlap region of quantum information theory, the octonions and
Hilbert's twelfth problem. The question of how many equiangular lines can fit
into a space of a given dimension is easy to pose -- a high-school student can
grasp it -- yet it is hard to answer, being as yet unresolved. This contrast of
ease and difficulty gives the problem a classic charm.
