The *diamond-square algorithm* is a popular method for generating images which resemble terrain, clouds or plasma. Its description on Wikipedia is as good as any and will not be repeated here.

In the code below, the image size is $N = 2^n + 1$. Since `plt.imshow`

interpolates quite well anyway, $n=6$ gives acceptable results for the image size it produces. For high-resolution images, increase `n`

accordingly.

*This code is also available on my github page.*

```
import numpy as np
import matplotlib.pyplot as plt
# The array must be square with edge length 2**n + 1
n = 6
N = 2**n + 1
# f scales the random numbers at each stage of the algorithm
f = 1.0
# Initialise the array with random numbers at its corners
arr = np.zeros((N, N))
arr[0::N-1,0::N-1] = np.random.uniform(-1, 1, (2,2))
side = N-1
nsquares = 1
while side > 1:
sideo2 = side // 2
# Diamond step
for ix in range(nsquares):
for iy in range(nsquares):
x0, x1, y0, y1 = ix*side, (ix+1)*side, iy*side, (iy+1)*side
xc, yc = x0 + sideo2, y0 + sideo2
# Set this pixel to the mean of its "diamond" neighbours plus
# a random offset.
arr[yc,xc] = (arr[y0,x0] + arr[y0,x1] + arr[y1,x0] + arr[y1,x1])/4
arr[yc,xc] += f * np.random.uniform(-1,1)
# Square step: NB don't do this step until the pixels from the preceding
# diamond step have been set.
for iy in range(2*nsquares+1):
yc = sideo2 * iy
for ix in range(nsquares+1):
xc = side * ix + sideo2 * (1 - iy % 2)
if not (0 <= xc < N and 0 <= yc < N):
continue
tot, ntot = 0., 0
# Set this pixel to the mean of its "square" neighbours plus
# a random offset. At the edges, it has only three neighbours
for (dx, dy) in ((-1,0), (1,0), (0,-1), (0,1)):
xs, ys = xc + dx*sideo2, yc + dy*sideo2
if not (0 <= xs < N and 0 <= ys < N):
continue
else:
tot += arr[ys, xs]
ntot += 1
arr[yc, xc] += tot / ntot + f * np.random.uniform(-1,1)
side = sideo2
nsquares *= 2
f /= 2
plt.imshow(arr, cmap=plt.cm.Blues)
plt.axis('off')
plt.show()
```

*Update (January 2020)*: The Matplotlib colormap `terrain`

can be used to create pleasing map-like images (here, with `n=9`

for a 1025 x 1025 pixel image):

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