One of my more obscure hobbies is messing around with mathematics via Python programming, and I’ve recently completed a strange project connected to the Champernowne Constant Cn. It started with simply writing a Python prog to generate arbitrary chunks of C₃₆ – that constant in base-36 number system, which enables a simple encoding of English text since all the alphabet letters become valid digits. Having written this I calibrated its execution time in Pydroid 3 on my Intel 12th-gen Core i3 CPU, and learned that my own name ‘dick_pountain’ would take around 20,000 years to arrive if all printed to the screen, while Hamlet’s Soliloquy would take 2 million years. (Obviously I didn’t find these ridiculous times by actual execution, but rather calculated their positions in C₁₀ using the formula P(n)=1+k=1∑d−19k⋅10k−1+d(n−10d−1) and then extrapolated from shorter test samples.)
This set me to thinking that the position of a text fragment in C₃₆, or C₁₀ for that matter, says something about the uniqueness of that fragment. Now another, earlier, project of mine was a program to ‘cheat’ at Wordle (to which I’m addicted) by ranking the uniqueness of 5-letter words using the frequency distribution of alphabet letters in English. Having those frequency tables from that project at hand. I decided to combine the two projects: for any text T, find its position in C₁₀ (using a simpler ASCII encoding for convenience), then calculate the sum of the uniquenesses of all the separate words it contains and take the ratio of those two numbers. Since they can get very large indeed a lot of ‘log10ing’ is required even in Python with its huge floats. The ratio, which I’ve dubbed the text’s ‘ntropy’, might or might not mean something, but I’m damned if I can figure out what…
