Modulo rugs of 3D functions


Consider a 3D function $latex z=f(x,y)$. Now evaluate it at each point of a $latex n times n$ integer lattice grid. Compute $latex z mod n$ 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 $latex z=x^2+y^2$.

matrixmod01_318Figure 1: $latex z=x^2+y^2, n=318$

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 ru. Several examples of such are shown below.

matrixmod06_318Figure 2: $latex z=x^4-x^2-y^2+y^4, n=318$

matrixmod08_315Figure 3: $latex z=x^4-x^2-y^2+y^4, n=315$

matrixmod13_309Figure 4: $latex z= x^6-x^4-y^4+y^6, n=309$

matrixmod12_318Figure 5: $latex z=x^6-x^2-y^2+y^6, n=318$

matrixmod07_312Figure 6: $latex z=x^4-x^2+y^2-y^4, n=310$

All the above $latex…

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It is absolutely clear that this world and its contents are purely mental and nothing solid in it. To come out of suffering entirely you should know your mind thoroughly.

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