There are however a few cellular automata in which class 3 behavior is obtained with random initial conditions, but in which no significant randomness is ever produced with simple initial conditions.
The pictures below show one example. And in this case it turns out that all patterns are in effect just simple superpositions of the basic nested pattern that is obtained by starting with a single black cell.
As a result, when the initial conditions involve only a limited region of black cells, the overall pattern produced always ultimately has a simple nested form. Indeed, at each of the steps where a new white triangle starts in the center, the whole pattern consists just of two copies of the region of black cells from the initial conditions.
The only way to get a random pattern therefore is to have an infinite number of randomly placed black cells in the initial conditions.