The formation of spatial patterns characterized by nontrivial fractal dimension often results from processes where the outermost parts of the pattern screen inner sites from growth. The threshold-screening model uses a simplified measure of screening to define active sites on a growing pattern. As the strength of screening is increased, the pattern geometry changes from compact to one dimensional, passing through an intermediate critical state of nontrivial fractal dimension. Precise knowledge about the active-growth sites makes possible the use of information theory to compute the rate of entropy production for the different growing patterns. We find that the specific heat computed from the information-theoretic part of the entropy is singular at the critical point. © 1989 The American Physical Society.