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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, $T_1:$

enter image description here

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  • $\begingroup$ 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. $\endgroup$ – Jean Marie Mar 9 at 21:42
  • $\begingroup$ Okay sounds good $\endgroup$ – Ultradark Mar 9 at 21:45
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    $\begingroup$ There is a "bible" on Moiré patterns "The Theory of the Moiré Phenomenon" in 2 volumes by Isaac Amidror, a researcher in EPFL (Switzerland). $\endgroup$ – Jean Marie Mar 9 at 21:49
  • $\begingroup$ Thank you for the reference. $\endgroup$ – Ultradark Mar 9 at 21:52

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