  Fano manifolds are basic building blocks in geometry - they are, in a precise
sense, atomic pieces of shapes. The classification of Fano manifolds is
therefore an important problem in geometry, which has been open since the
1930s. One can think of this as building a Periodic Table for shapes. A recent
breakthrough in Fano classification involves a technique from theoretical
physics called Mirror Symmetry. From this perspective, a Fano manifold is
encoded by a sequence of integers: the coefficients of a power series called
the regularized quantum period. Progress to date has been hindered by the fact
that quantum periods require specialist expertise to compute, and descriptions
of known Fano manifolds and their regularized quantum periods are incomplete
and scattered in the literature. We describe databases of regularized quantum
periods for Fano manifolds in dimensions up to four. The databases in
dimensions one, two, and three are complete; the database in dimension four
will be updated as new four-dimensional Fano manifolds are discovered and new
regularized quantum periods computed.
