Patterns in the solidification of a pure material are dealt with in this article. Results, deduced from a simple model based on heat conduction in the two phases and the effect of surface tension on the equilibrium temperature at the moving front, present a guide for experimental work. By introducing far-field conditions imitating what can be achieved in an experiment, we explain how the depths of the phases and the width of the container influence the patterns that can be seen if one advances the control variable to the critical point and then just beyond. Our new result is the existence of a third critical point. It occurs at small wave numbers and it is independent of surface tension. It appears because we take the depths of the phases into account. These depths are input values that offer the possibility of controlling crest-to-trough conduction, stabilizing in the solid, destabilizing in the liquid. The new critical point, and the patterns attending its appearance, can be found in cells of easily attainable widths.