Modulo rugs of 3D functions

Consider a 3D function . Now evaluate it at each point of a integer lattice grid. Compute corresponding to each point and plot it as a color defined by some palette that suits your aesthetic. The consequence is a what we term the “modulo rug”.
For example, below is a plot of .

Figure 1:

We get a pattern of circles around a central circular system reminiscent of ogdoadic arrangements in various Hindu maṇḍala-s. From the aesthetic viewpoint, the best modulo rugs are obtained with symmetric functions higher even powers — this translates into some pleasing symmetry in the rug. Several examples of such are shown below.

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figure 6:

All the above are composite numbers. Accordingly, there is some repetitiveness in the structure. However, if is a prime then we have the greatest complexity in the rug. One example of such is plotted below.

Figure 7:

See also: 1) Sine rugs; 2) Creating patterns through matrix expansion.