  We observe that for a and b relatively prime, the "abacus construction"
identifies the set of simultaneous (a,b)-core partitions with lattice points in
a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise
polynomial functions on this simplex. We apply these results to rational
Catalan combinatorics.
  Using Ehrhart theory, we reprove Anderson's theorem that there are
(a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we
prove Armstrong's conjecture that the average size of an (a,b)-core is
(a+b+1)(a-1)(b-1)/24. Our methods also give new derivations of analogous
formulas for the number and average size of self-conjugate (a,b)-cores.
  We conjecture a unimodality result for q rational Catalan numbers, and make
preliminary investigations in applying these methods to the (q,t)-symmetry and
specialization conjectures. We prove these conjectures for low degree terms and
when a=3, connecting them to the Catalan hyperplane arrangement and quadratic
permutation statistics.
