Chaotic Itinerary but Regular Pattern

Mark three points on a sheet of paper to form a triangle. Use a die to let you pick one of these vertices randomly. Start the path by picking an arbitrary point—the current point—on the sheet and roll the die. Consider the segment between the current point and the random vertex obtained with the die and mark a dot exactly in the middle of this segment. This dot is the new current point. Roll the die again and repeat the operation. You will obtain a chaotic path, but the points drawn form a regular pattern.