The Tree of All Seasons/Desmos Art Class

Note: This was originally posted on the Making Math Visual blog. Crossposting here and moving these Desmos art class posts to Math Art Animation 101 blog.

Suzanne von Oy shared a graph titled The Tree of All Seasons awhile back as one of her #graphjam submissions. It's mesmerizing at the very least, and left quite a few of us wishing for a Desmos art course so Suzanne can teach us her magic.

I look forward to this course, but as I dove into the graph, I found myself quickly overwhelmed. So many variables and functions. I wasn't sure where to start with understanding the various parts. This left me wondering what the different parts are of the Desmos art course that will allow me to build this (and any) awesome graph.

So where would the course start? There are quite a few things going on just with the flowers in The Tree of All Seasons graph.

1. The flowers are all different sizes. Suzanne uses exactly one image to render all of the flowers, so I'm curious as to how she made them all different sizes.
2. The flowers start growing at different times. They start falling at different times. They start shrinking at different times. I imagine these are all related somehow.
3. Once the flowers shrink down to size zero, they stay at that size throughout the animation.

I started exploring question 1, or how I can make all of the flowers different sizes using just one copy of the flower image. You can make multiple copies of an image by using a list to define the center of the image. In this case we could use (x_1,y_1) to define the center of the flowers since the centers are all listed in the table.

I made some efforts to apply my previous knowledge of animating images in the calculator and failed very quickly. My next move was to dig into the function Suzanne uses to control the dimensions of the flowers. Even that was tough, but I found some success by looking at the function one piece at a time. Here's the first piece:

This is one way we can define the length and width of the flower image. At its max, the value is something like 0.833, and at its min, the value is 0.5. I imagine Suzanne started with her minimum value in mind and played around with the numbers in the mod function to find a nice maximum value. (Suzanne, let us know if you thought of this in a different way).

Here's a graph with just five flowers. The dimensions are controlled using the function above, which depends on the y-variable. Turn on the movable points and move them around to see how it works.

Question: What are other uses of this function that people might explore before working on the The Tree of All Seasons problem?

Note: I am interested in exploring what a Desmos art/animation course would look like for personal reasons, but also I am interested in exploring what it would look like because I'd love to see a course like this offered at the high school or college level as a math elective.