lexicons!
Perhaps I can shed some light on this matter. The
property of the decimal expansion of a number containing every possible
FINITE (my emphasis) sequence of digits is called (appropriately
enough) being a lexicon. Many numbers are lexicons.
All Borel normal are lexicons. This
includes all numbers which are truly random in a certain precise
mathematical sense. It is not known if PI is a lexicon.
However there are simply constructed lexicons for example 0.123456789
10 11 12 13 14 15 16 17 18 19 ... (I put the spaces for pedagogical
reasons). constructed by just counting! This number will contain the
library of congress infinitely often as ASCII code. 0, 1, 01, 11, 100,
etc will all occur infinitely often.
Now this number is not random but contains
every possible pattern of digits (finite patterns) infinitely often. It
doesn't contain the square root of 2, since this is infinite. But it
certainly contains Greg Bear's Blood Music infinitely often as well as
every episode of Monty Python's flying circus. Basically a random real
number (many equivalent definitions of random, see for example Greg
Chaitin) between 0 and 1 will have this remarkable property. But I am
not sure if PI is a lexicon. I believe that it is but I don't believe that it has been proven. -- Jim Cox
Martin responds: Another question about pi: Is pi normal ? It looks like e is not normal. See:
No computer program anywhere could ever generate the digits of the square root of two or of Pi or of any other irrational or transcendental number, Chuck. So no, the square root of two is not "encoded" in Pi.
Even the vast assortment of known mathematical functions we have to "generate" Pi are not really Pi generators, in reality they can never generate more than finite approximations of the actual infinite value of Pi. Any and all of those are so trivial in content compared to the true infinite entity as to be totally negligible as true absolute and complete representations of Pi itself. They all can only physically be used to generate exactly zero percent of the true infinite string of digits comprising the exact value of Pi, and all of the known calculated results therefore actually differ from the true infinite string of digits of Pi by an infinite amount of digits.
Nothing is "encoded" in Pi except Pi itself. Finally, the term "encoding" as you use it is the same as saying that the symbol "2" is an "encoding" of the abstract mathematical value 2, which is true bit is totally trivial.
Pete B
