Sometimes we like to make short animations that loop. This is particularly useful for testing ideas. We make just a few frames yet we can view a continuous movie. If these loops include noise textures, we have a problem. How can we 'close the loop' and keep the changes continuous. One way is to create a noise space that is inherently circular and finite. But that means re-building the random number generators to loop over the particular dimensions of the space.
A better way is to create the looping behaviour by plotting the changes through a circular path in noise space. The example that follows is a 2D slice through a 4D space suitably coloured. This is like a picture frame in space showing what is in the frame from the solid texture. The frame is moving in a circle but not in either of its two dimensions. The frame is in <x, y> space and moving through a circle in <z, w>.
The movie presented here plays four times through the loop. Notice that the ends of the loop are truly seamless. Most people don't even realise it is a loop until they have been told.
Incidentally, the background on all these pages is an example of filtered noise that repeats seamlessly in space.