A new way to view continued fractions

[EdS2]

This might not be new to the world, but it's new to me... see the attached.

   

...it's a thought prompted by Joe Horn's nearby post of some screensaver images showing the binary for various irrational numbers:

(06-10-2022 06:50 AM)Joe Horn Wrote:  Want your DM42's powered-off screen to display some famous irrational numbers? The most information-dense way to do this is to use every pixel to represent one binary digit of the irrational number. That way you'll see the most significant 96 thousand bits (roughly as accurate as 28,898 decimal digits).
... I had HOPED that they would look at least slightly different from each other, but to my eye they all look like utterly random noise.

Continued fractions are another way to represent numbers, a way which isn't fixed to some number base, and a way which sometimes shows up interesting differences between kinds of numbers. Instead of a list of digits, you get a list of numbers: often with runs of small numbers but with occasional large numbers.

What I did was to convert each term into a run of pixels, with the last pixel a different colour. So a list of all-ones is a solid colour, whereas a list of all-twos shows up as fine vertical stripes. Short runs of ones turn into short horizontal lines. When one of the terms of a continued fraction is very large, we get a lot of pixels of the same colour which shows up as a horizontal band.

I've posted a zip file into the screensaver thread. There's one image missing, which might take a few days to generate, but the other numbers were pretty fast, using Simon Tatham's spigot program.