Although there is no representation for four dimensions in ordinary space, some of the  simpler maths that we use to describe two and three dimensions still operates. For example we can calculate the distances between points using Pythagoras' theorem.  And we can smooth random numbers in an array of 4D points with a 4D convolution function.

There are many reasons to do this. One is to create textures that vary in a semi-random fashion over time. Another is to get a better quality of noise in two or three dimensions.

In the animation presented here, 4D filtered noise is used to vary a wave pattern as a function of 3D space. This creates the effect of underwater lighting. The noise pattern is, however, a function of 3D space and time. As time advances, the whole 3D pattern changes, smoothly and emulates the irregular ripple patterns.

Notice the way the lighting changes consistently on the wavy sandy bottom and on the red post. This is possible because the texture is represented at every time and every point in 3D.

More examples:    Stonelight      Waterfall     Loops