  A complete set of mutually unbiased bases for a Hilbert space of dimension N
is analogous in some respects to a certain finite geometric structure, namely,
an affine plane. Another kind of quantum measurement, known as a symmetric
informationally complete positive-operator-valued measure, is, remarkably, also
analogous to an affine plane, but with the roles of points and lines
interchanged. In this paper I present these analogies and ask whether they shed
any light on the existence or non-existence of such symmetric quantum
measurements for a general quantum system with a finite-dimensional state
space.
