Making pixel pictures of interesting real numbers
Making pixel pictures of interesting real numbers.
If we express a real number as a continued fraction, convert the list of terms into lengths, and fill an image raster with lines of those lengths, we can see some interesting phenomena.
And this is a lossless representation - it's convertible back to the (truncated) continued fraction.
In particular, most numbers come out as fuzzy gray patches with white bands. The bands correspond to large terms. The overall effect might be somewhat like a fingerprint.
Numbers which are rational will give rise to mostly blank images.
Numbers which are simple combinations of square roots will look like simple patterns.
Numbers from logs or pi will each have some unique-looking image.
But e and related numbers seem to have a much simpler and very different appearance.
Hat tip to Joe Horn who posted about making DM42 off-images using Simon Tatham's spigot program.