From the page: "Thus, beginning with total linguistic relativism, Chuang Tzu ends with a sort of metalinguistics. Spillover words do not ward and sector, They play. They contain more than they contain - therefore, like the famous cleaver which never needs sharpening because the Taoist butcher can pass it between all tendons and joints, the Spillover word "finds its proper channel." The sage does not become trapped in semantics, does not mistake map for territory, but rather "opens things up to the light of Heaven" by flowing with the words, by playing with the words. Once attuned to this flow, the sage need make no special effort to "illumine," for language does it by itself, spontaneously. Language spills over.
Now recall that Saussure was studying the Latin anagrams, and that he found the key words of the poems spilling over into other words. Syllables of characters' names for example are echoed in the words describing those characters. At first the founder of modern linguistics considered these anagrams as conscious literary devices. Little by little however it became apparent that such a "reading" would not hold. Saussure began to find anagrammatc spillovers everywhere he looked - not only in all Latin poetry, but even in prose. He reached the point where he couldn't tell if he was experiencing a linguistic hallucination or a divine revelation. Anagrams everywhere! Language itself a net of jewels in which every gem reflects all others! He wrote a letter to a respected academic Latinist who had composed Latin odes - poems in which Saussure had detected anagrams. Tell me, he begged, are you the heir to secret tradition handed down from Classical antiquity - or are you doing it unconsciously? Needless to say, Saussure received no answer. He stopped his research abruptly with a sensation of vertigo, trembling on the abyss of pure nihilism, or pure magic, terrified by the implications of a language beyond language, beyond sign/content, langue/parole. He stopped, in short, precisely where Chuang Tzu begins."
From the page: "Evil numbers are easy to find, as one might expect with an arbitrarily defined property. Interestingly, π is doubly evil. Many numbers aren't evil. For example, e isn't evil. The important sums are 665 and 668 after 141 decimal digits -- e goes over 666 without hitting it. Chris Lomont notes that you can take every 15th digit of e, and show that e is somewhat evil. The farthest he had to reach for any constant was with Catalan's constant, which wasn't evil until he took every 28th digit.
It appears that most numbers are at least somewhat evil. With enough fiddling, almost anything is likely to be findable.
How common are evil numbers? (number of non-zero digits / sum of digits) gives a good estimate. For a base 10 number, the estimate gives 9/(1+2+3+4+5+6+7+8+9) = 20% = 1/5. Thus, any random number has a 1 in 5 chance of having this property.
A fifth is an estimate. Chris Lomont found a recursion that could give an exact value:
p[1] = 1/9;
p[k_ /; k <10] := p[k] = 1/9 + 1/9 Sum[p[j], {j, 1, k - 1}];
p[k_ /; k > 9] := p[k] = 1/9 Sum[p[k - d], {d, 1, 9}];
Timing[Table[p[100 n], {n, 1, 10}]][[1]]
The exact probability of 666 being hit is the following:
(large number edited out as it was breaking my blog)
Which is approximately .2000000000000000000000000000000000000000000000000000000000000002166222683713523944720537405934866672. That's very close to 1/5. The third term in the continued fraction expansion of p[666] is 184653222869167741981875869102352405779668736930185085305398884.
The probability of evilness converges quickly to 1/5, but it does so in a highly oscillatory way. Note that each term is being multiplied by 1.243n.
Figure 1. ListPlot[Table[(1/5 - p[n]) (1243/1000)^n, {n, 30, 900}]]
Another method Chris Lomont discovered for calculating evilness involved the series expansion of (1-t9)/(t10-10t+9). In this series expansion, coefficient n is identical to p[n].
It seemed like there should be a closed form for all of this, but it was beyond Chris's ability to calculate. Also, I wasn't able to calculate the evilness probability for the continued fraction of a random irrational number. This calls for wisdom. If anyone has insight, let them calculate the equations."