## Fractal Landscapes; Variations

These images show the effect of changing the scaling parameter. At each stage in the construction, a random displacement of the midpoints is made according to the formula avg + scale*rand where avg is the average of the adjacent corners, rand is a random number (for these examples, normally distributed about zero with a variance of 1), and scale is a scaling parameter which is proportional to 2d*level where level is k, k-1, k-2, ..., 1 for the first iteration, the second iteration, the third iteration,..., the kth iteration. So the amount of random displacement decreases by a factor of 2d when going to the next iteration (random displacements become smaller as the construction proceeds to smaller and smaller scales). So we expect those landscapes that are made with larger d to have larger extreme variations (due to the first iteration) but relatively smoother surfaces than landscapes made with smaller d (which will have smaller extreme variations but rougher surfaces). You can see that in the examples below. (Imagine walking on these surfaces; for large d you will be going up and down to great heights and depths, but the surfaces of the 'mountains' will be relatively smooth. While for the smaller d surfaces you will not go to great heights or depths, but you will have to walk over very rough surfaces). The fractal dimension of these landscapes is related to the exponent d; larger d correspond to smaller fractal dimensions.

Click on the image for a larger view. Note that the z-scale is not the same for all images.

d=0.1

d=0.3

d=0.5

d=0.7

d=0.9

d=.11

d=.13

d=.15

d=2.0