  The two most important notions of fractal dimension are {\it Hausdorff
dimension}, developed by Hausdorff (1919), and {\it packing dimension},
developed by Tricot (1982).
  Lutz (2000) has recently proven a simple characterization of Hausdorff
dimension in terms of {\it gales}, which are betting strategies that generalize
martingales. Imposing various computability and complexity constraints on these
gales produces a spectrum of effective versions of Hausdorff dimension.
  In this paper we show that packing dimension can also be characterized in
terms of gales. Moreover, even though the usual definition of packing dimension
is considerably more complex than that of Hausdorff dimension, our gale
characterization of packing dimension is an exact dual of -- and every bit as
simple as -- the gale characterization of Hausdorff dimension.
  Effectivizing our gale characterization of packing dimension produces a
variety of {\it effective strong dimensions}, which are exact duals of the
effective dimensions mentioned above.
  We develop the basic properties of effective strong dimensions and prove a
number of results relating them to fundamental aspects of randomness,
Kolmogorov complexity, prediction, Boolean circuit-size complexity,
polynomial-time degrees, and data compression.
