thumb|478x478px|Approach to the scaling limit as approaches from below.thumb|489x489px|At the point of chaos , as we repeat the period-doublings, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal. Looking at the images, one can notice that at the point of chaos , the curve of looks like a fractal. Furthermore, as we repeat the period-doublings, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.
thumb|478x478px|Approach to the scaling limit as approaches from below.thumb|489x489px|At the point of chaos , as we repeat the period-doublings, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees, converging to a fractal. Looking at the images, one can notice that at the point of chaos , the curve of looks like a fractal. Furthermore, as we repeat the period-doublings, the graphs seem to resemble each other, except that they are shrunken towards the middle, and rotated by 180 degrees.