  Milnor and Thurston's famous paper proved monotonicity of the topological
entropy for the real quadratic family. Guckenheimer showed that it is H\"older
continuous. We obtain a precise formula for the H\"older exponent at almost
every quadratic parameter.
  Furthermore, the entropy of most parameters is proven to be in a set of
Hausdorff dimension smaller than one, while most values of the entropy arise
from a set of parameters of dimension smaller than one.
