# Animate this Moire pattern. What mathematical tools could be used to analyze this moving pattern?

For a mathematical art project, I want to animate the following pattern I made on desmos. It seems to be a Moire pattern. However I cannot make the pattern move smoothly and continuously because desmos lacks the computing power.

The pattern is a projection of several roating $$3D$$ structures mapped to $$2D.$$

Essentially I made the (moving) shadow of these rotating $$3$$-dimensional structures.

The pattern consists of:

$$4$$ boxes that differ by rotations.

I've defined the "waves" in the boxes on desmos with these equations:

$$e^{ST\log(x)}+e^{ST\log(y)}=1.$$

$$e^{ST\log(1-x)}+e^{ST\log(y)}=1$$

$$e^{ST\log(x)}+e^{ST\log(1-y)}=1$$

$$e^{ST\log(1-x)}+e^{ST\log(1-y)}=1,$$

Where $$T$$ is a continuous slider $$T\in[0,1]$$ representing the flow of mathematical time.

$$S$$ gauges the spacing of the curves. For pretty evenly spaced curves I used these $$49$$ values for $$S:$$

$$S=\{15.3039,10.3612,7.9153,6.3793,5.3019,4.4955,3.8656,3.3584,2.9405,2.5903,2.2926,2.0367,1.8146,1.6204,1.4496,1.2983,1.1638,1.04377,.93622,.83959,.75256,.67401,.603,.53871,.48045,.42762,.37969,.33619,.29673,.26094,.22852,.19917,.17265,.14874,.12722,.10791,.090664,.075316,.061733,.049793,.039383,.030399,.0227475,.0163414,.0111008,.0069525,.0038286,.0016664,.0004081\}.$$

So essentially there should be $$2N$$ fluctuating waves inside each box. $$N$$ waves are attached to the ends of the box $$(0,0)$$ and $$(1,1)$$ and the other $$N$$ are attached to $$(0,1)$$ and $$(1,0).$$ Each half of the waves are spaced evenly apart along the diagonals of the box.

In the images there are $$392$$ curves.

In the animation each of the waves should move at the same constant velocity as they propagate through the box.

The boxes could also rotate. That would be cool.

Requests:

$$1)$$ What mathematical tools could I use to analyze this moving pattern?

$$2)$$ Please animate this pattern.

A picture of the animation frozen at time, $$T_1:$$ • Nice pattern. I have taken the liberty to suppress the tag "harmonic functions" which is related to functions such that their laplacian is $0$ which a priori have not their place here. – Jean Marie Mar 9 at 21:42
• Okay sounds good – Ultradark Mar 9 at 21:45
• There is a "bible" on Moiré patterns "The Theory of the Moiré Phenomenon" in 2 volumes by Isaac Amidror, a researcher in EPFL (Switzerland). – Jean Marie Mar 9 at 21:49
• Thank you for the reference. – Ultradark Mar 9 at 21:52