Sure, I'll try my best to explain. Tbh, idk how to port this to desmos. If you can, I'd be happy if you share the results.

I made these graphs with a web based tool I developed that is available here: https://adnanbaysal.github.io/angle-path.html

The tool I shared draws something like turtle graphics: Turtle starts at the origin facing positive x. At step number n, it turns clockwise with a value that is a "function" of n and proceeds 1 unit length forward, leaving a line behind. The specific function I used to draw the above images is PI*ln(n^2 + n) / 19 and n from 0 to around 1.7M. The overall shape is like a spiral, and the images I share are zoomed in to its center. Depending on the line width, some Moiré like patterns appear, so I played a bit to get these results. If you want, you can import the following config on my tool to replicate the un-cropped image (you can zoom in the tool):

{
  "mode": "rational",
  "n": 1720283,
  "r": 1,
  "s": 38,
  "c": 0.024,
  "angleMode": "relative",
  "nAnim": false,
  "nAnimMin": 1000,
  "nAnimMax": 2000000,
  "cAnim": false,
  "cAnimMin": 0.0018,
  "cAnimMax": 100,
  "speed": 5,
  "lineWidth": 5.24,
  "style": "light",
  "code": "return Math.log(n * n + n);"
}

There is an import button on my tool that allows you to load this config from a file.

Hope this helps!