  Let A be an A_\infty ring spectrum. We use the description from [2] of the
cyclic bar and cobar construction to give a direct definition of topological
Hochschild homology and cohomology of A using the Stasheff associahedra and
another family of polyhedra called cyclohedra. This construction builds the
maps making up the A_\infty structure into THH(A), and allows us to study how
THH(A) varies over the moduli space of A_\infty structures on A.
  As an example, we study how topological Hochschild cohomology of Morava
K-theory varies over the moduli space of A_\infty structures and show that in
the generic case, when a certain matrix describing the noncommutativity of the
multiplication is invertible, topological Hochschild cohomology of 2-periodic
Morava K-theory is the corresponding Morava E-theory. If the A_\infty structure
is ``more commutative'', topological Hochschild cohomology of Morava K-theory
is some extension of Morava E-theory.
