Three Dimensions

In three dimensions, the random numbers are placed in a 3D grid and the filter is a spherical volume.

There are many applications of 3D noise, see :

SIGGRAPH 1991 Course #14 Notes: Fractal Modeling in 3D Computer Graphics and Imaging, July 1991, Las Vegas, 93-96.

In this example, the splash at the bottom of the waterfall is made of 100 textured spheres. The texture is basically a 3D filtered noise. Where the value of noise falls below a certain threshold, the sphere becomes transparent. To animate this, we use a 4D filtered noise.