A line has 1 dimension, double it and you double its measure (length, 2¹). A square has 2 dimensions, doubling the side will increase its measure (area) by 4 (2²). The same operation on a cube (3 dimensions) will increase its measure (volume) by 8 (2³). For fractals, doubling will cause an increase of its measure by 2^N, where N is not necessarily an integer.