  Understanding the behavior of dense hadronic matter is a central goal in
nuclear physics as it governs the nature and dynamics of astrophysical objects
such as supernovae and neutron stars. Because of the non-perturbative nature of
quantum chromodynamics (QCD), little is known rigorously about hadronic matter
in these extreme conditions. Here, lattice QCD calculations are used to compute
thermodynamic quantities and the equation of state of QCD over a wide range of
isospin chemical potentials. Agreement is seen with chiral perturbation theory
predictions when the chemical potential is small. Comparison to perturbative
QCD calculations at large chemical potential allows for an estimate of the gap
in the superconducting phase, and this quantity is seen to agree with
perturbative determinations. Since the partition function for an isospin
chemical potential, $\mu_I$, bounds the partition function for a baryon
chemical potential $\mu_B=3\mu_I/2$, these calculations also provide rigorous
non-perturbative QCD bounds on the symmetric nuclear matter equation of state
over a wide range of baryon densities for the first time.
