Many thanks for taking the time to convert the video to an animated GIF. It does help in providing an instant view of the animated spiral compression, albeit of much lower quality than the full video.

I've already edited my original post, and provided the MATLAB source code, as you requested. I considered including it when I first made the post, but thought that no one would care to see it. I'm glad that someone does! You can port it to another language, but you'll need to write (or find libraries of) appropriate functions for the polar plot generation, vector/matrix manipulation, and video capture/recording. With MATLAB, all of this is provided. As for the spiral compression algorithm I used, it's rather straightforward, I'm just mapping the entire spiral's linear time to an angular vector, and varying (increasing) the days/cycle amount at each frame, thus compressing the spiral. All this is done in just a single line:

th = 2*pi*(i-1)/c+pi/2;

where th is the angular vector, i is the linear time vector, c is the days/cycle amount (the cycle period), and the π/2 term is just to rotate the end of the spiral from the 3 o'clock to the 12 o' clock position in the polar plot. The rest of the code is just for the visualization and video capture.