"There is order in chaos"

Can we create a drawing machine that takes in any ordered set of $3D$ points and draws it using epicycles (a sequence of chaotic, rotating circles)?

To solve this, we must do the following:

Step 1. Take our $3D$ points and create $3$ functions from its $x$, $y$, and $z$ components.

drawing $= {(0, 0, 1), (0, 1, 2), (1, 1, 2), (1, 0, 1), ...}$

$x = {0, 0, 1, 1...}$

$y = {0, 1, 1, 0...}$

$z = {1, 2, 2, 1...}$

Step 2. Apply the Fourier Transform to each function. This will allow us to represent each function as a sum of revolving circles and vectors. We will use the discrete version of the transform because computers cannot sum up an infinite number of values.

Step 3. Save the output of the Fourier Transform, which is a set of frequencies, amplitudes, and phase shifts. Visually, this is just a circle.

Step 4. Create $3$ epicycle systems from the output above. Each system controls the $x$, $y$, or $z$ coordinate of the points being redrawn.

Step 5. Start an animation that goes from $t = 0$ to $t = 2*pi$. For the $3$ epicycle systems, we sum up $A * cis(f * t + phi)$, where $A$ = amplitude, $f$ = frequency, $t$ = time, $phi$ = phase shift, and $cis(theta) = cos (theta) + i sin(theta)$. We then use the real or imaginary component of the $3$ sums to create a $3D$ point, which is plotted.

Step 6. As the animation progresses, we see the $3D$ points connect to form our $3D$ shape.

My version extends Fourier Epicycles (which you may have seen in YouTube videos, social media posts, and STEM forums) to 3D space. The sources below do a much better job explaining the math:

myFourierEpicycles - Amazing visualization and math explanation

The Coding Train - Much of the code I used

3Blue1Brown - Great video on the Fourier Series and Transform

Please feel free to check out my code and email me if you have a new 3D shape that I can add 😄