  We investigate a stationary process's crypticity---a measure of the
difference between its hidden state information and its observed
information---using the causal states of computational mechanics. Here, we
motivate crypticity and cryptic order as physically meaningful quantities that
monitor how hidden a hidden process is. This is done by recasting previous
results on the convergence of block entropy and block-state entropy in a
geometric setting, one that is more intuitive and that leads to a number of new
results. For example, we connect crypticity to how an observer synchronizes to
a process. We show that the block-causal-state entropy is a convex function of
block length. We give a complete analysis of spin chains. We present a
classification scheme that surveys stationary processes in terms of their
possible cryptic and Markov orders. We illustrate related entropy convergence
behaviors using a new form of foliated information diagram. Finally, along the
way, we provide a variety of interpretations of crypticity and cryptic order to
establish their naturalness and pervasiveness. Hopefully, these will inspire
new applications in spatially extended and network dynamical systems.
