2002: On Omega Numbers and the Omniscience of Pi

Correctly interpreted, you know, pi contains the entire history of the human race.

–Dr. Irving Joshua Matrix


Consider the quotation from Dr. Matrix, the arch-numerologist of Martin Gardner’s book “The Magic Numbers of Dr. Matrix“. You may laugh, but try to disprove his assertion. Amazingly, Dr. Matrix is probably correct!

Let us take the statement to its logical extreme: pi is omniscient. By this we mean that pi encodes in its digits the totality of human knowledge, past, present, and future–and probably more…. An apparently absurd claim–and one which is almost certainly true!

First, let us try a little experiment. Can pi “count”? That is, does the pattern

“1, 2, 3, …., n”

appear in pi’s digit sequence 3, 1, 4, 1, 5, …. for all n? (We split the digits of numbers larger than 9 in the pattern, for example, if n = 11, then the pattern is “1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1”.)

Define the sequence a(n) by: a(n) = the first place in the digit sequence of pi where the pattern “1, 2, …, n” appears. Is a(n) well-defined for all n, that is, does “1, 2, …, n” appear in pi’s digit sequence for all n? The first five terms of a(n) are

2, 149, 1925, 13808, 49703, ….

Now, the pattern “1, 2, 3, 4, 5, 6” does not occur before the 100,000-th term in the digit sequence of pi, which is as far as we perform the search. But this doesn’t show that the pattern doesn’t occur, much less prove that pi can’t “count”. Indeed, how would such a disproof look like, short of being an infinite list of all the digits of pi?

Furthermore, statistical studies of pi’s digits indicate that they are randomly distributed. Let’s suppose that this really is the case. Now, the probability of finding a pattern in a random string of n digits approaches 1 as n goes to infinity. (Similarly, the probability of rolling, say, six consecutive 6’s with a die approaches 1 as the number of trials gets large.) Hence, we are “almost sure” to find any pattern in the digits of pi!

For those of you who’d like to keep searching for evidence that pi can count, here is Mathematica code you can modify:

p = ToString[N[Pi, 50000]/10];
t = {1, 12, 123, 1234, 12345};
g[n_] := StringPosition[p, ToString[n]][[1]][[1]] – 2;
Table[g[t[[i]]], {i, 1, 5}]

(Admittedly, a faster Mathematica program can be written.) Exercise: Can you find the first occurrence of your phone number in the digits of pi?

Now, let’s consider the grand proposal. The totality of all human knowledge, while potentially infinite, can certainly be enumerated; let

s1, s2, s3, s4, ….

be, say, English sentences enumerating this knowledge. With a suitable assignment of code numbers to English letters/symbols, we can attach to each of these sentences s a unique code number, its Gödel number g(s). For example, suppose the code numbers 19, 20, 15, 16 have been assigned to the letters s, t, o, p, respectively. If s = “stop”, then g(s) = 219 x 320 x 515 x 716, where the bases are the primes in ascending order, and the exponents are simply the code numbers. The main idea is that given a sentence, we can calculate its Gödel number, and conversely, given the Gödel number, we can recover by factoring the sequence it encodes. (This is not a computationally efficient coding scheme, but it’s one of the most famous.)

Define the sequence A(n) by: A(n) = the first place in the digit sequence of pi where the pattern g(sn) occurs. That is, A(n) points to the first occurrence of (the code of) sn in the digit sequence of pi. If A(n) is well-defined for all n, or equivalently, if each sn appears in the digit sequence of pi, then pi codifies all human knowledge. This may appear very unlikely, but can you conceive of a disproof other than an infinite list consisting of all the digits of pi? In fact, the randomness of pi’s digits virtually assures that any sn will occur! Are we condemned to live and die with the perpetually unresolved suspicion that within pi lies the answers to all our questions–indeed, the very secrets of the universe?

A Recreation in Estimation: Omega Numbers

The infinite number of digits of pi is perhaps too much–after all, our knowledge at any one time is finite.

Concatenate the digits of (the Gödel numbers of) a current compilation of human knowledge s1, s2, s3, s4, …., sn. The resulting sequence is the digit sequence of an integer which we call an omega number. (Clearly, different encodings of the character set will produce different omega numbers.) An omega number directly encodes human knowledge in its digits. How do we even begin to compute this kind of number?

For those of you looking for an unusual mathematical recreation, I challenge you to estimate the number of digits of the following “easy” omega number. First, fix an encoding of English letters/symbols by numbers (e.g., A = 1, B = 2, etc.), then fix an enumeration of human knowledge, say for the sake of argument, all the sentences in the Encyclopedia Britannica. Compute the Gödel number of each sentence, string together the digits of all these numbers, and there you have it! 🙂

A Strange Recreation: The Pi-nary Representation of 1

Imagine a world where people have no fingers, and have only circular disks for hands. Suppose these beings don’t count in multiples of ten or two, but in multiples of pi. For them, pi is perfectly integral, and it’s a number such as 1 that’s transcendental! Can you help them find the “pi-nary” representation of 1? In particular, can you write 1 as the infinite sum of powers of the form (1/pi)^n (n a natural number)? Note that we, as ten-fingered creatures, use 10 instead of pi in the infinite series, e.g.,

1/2 = .4999…. = 4*(1/10) + 9*(1/10^2) + 9*(1/10)^3 + 9*(1/10)^4 + ….

Update, 9 April 2002:

As an example of what an omega number might begin to look like, consider this correspondence from Mark Ganson:

Hello,

I read the info. at your website about Omega numbers and Pi. The challenge was to generate my own Omega number in the form of nthPrime^asciiVal x nthPrime+1^asciiVal…

As you mentioned, this is not the most efficient encoding scheme one can come up with, but it does have the advantage of guaranteeing that every string will generate a unique number.

I decided to take it upon myself to create an Omega generator that would take arbitrary text and generate the Omega number for that text. It does not search Pi for the generated number, which would be most likely fruitless, anyway.

Here is the Omega number for the following string of text:

“Pi is the ratio of the circumference of a circle to the diameter of the same circle.”

506246415425669701312751127885079715076180420832761487006222712173805741968228177184817722250896602313723818928019855059163389792292160968375651284831096201235566970996292453859654863941406865830400208017246861427143630614908314227801537410105682847423167117327402389971344414808393459235047982364696594431030600885773990100711151265367004173737039821307028649538507096141596857892161948298938965275128495687602142990823208879144144689346963771799794073422734206137826982642993719933043716467115616358562871447067029524405734358286147131992059672233059802020803669224968397301987840883304501185119472467758667129990549015065595073950431706174843786253721651343753177281861090114311601240406555496859772124543523665163889363245243019634220412932138085062956946156573534694322425293963761675993980658949158428858406777131627957391648246314202300478847753305720967694599675655643471915622429054103878209 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If this string of digits exists in Pi, then we can say that Pi is self-referential. That is, contained within Pi, is the definition of Pi. This recursive strange loop would then up a proverbial Pandora’s Box of potential paradoxes, would it not?

Attached is the source code for the java applet that does the Omega number generation, along with the Omega.class file and Omega.html to get it going. It works fine in AppletViewer, which is part of the java development kit, but IE 5.5 shows a bit of a quirk in that the Omega number tends to disappear in the lower box at times. I haven’t tried Netscape 4.7 with it. Works flawlessly with Netscape 6.2. (Go figure.)

It could well be that the sum of all human (and nonhuman) knowledge is encoded within the digits of Pi in this manner. However, I assert to you that it is possible that this is not the case. While Pi is comprised of an infinite number of decimal digits, I have seen no proof that these digits do not repeat. It is possible, therefore, that Pi is a repeating decimal, like 1/3 = .33333…, 1/6 = 1.66666…, 1/7 = 0.142857142857142857142… It could be that Pi begins repeating after some really huge, though finite, number of digits. If that is the case, then Pi might still encapsulate the sum of all human knowledge, since the sum of all human knowledge is certainly finite. Then again, it might not. Or, it might currently contain all human knowledge until someone comes along and makes a new discovery or writes a new book that happens to not be encapsulated.

The next logical step would be to create an Omega number decoder. We could then plug in a given substring of the digits of Pi to see what might be encoded in there. The vast majority of such substrings would decode into utter nonsense, of course. But, among the garbage, would surely be magnificent works of art. What constitutes art and what constitutes garbage, being of such the subjective nature that it is, of course, we’d have to have a committee set up to determine what was art and what was not. A science fiction writer might come up with clever ideas to use such a decoding program in a short story set in a future where practically all of human knowledge had been lost and where a committee was set up to recover from the digits of Pi all of the magnificent works of art that had been lost. The committee uncovers a particular document about which the artistic value is hotly contested. The group offers diverse opinions to the public about the aforesaid document. There are many who take it to be a genuine factual work of art. Others take it to be a work of fiction. Some say it has no artistic value whatever. The document causes major cultural changes in this futuristic setting. Wars are fought. Empires crumble and others rise to take their places. In the final paragraph of the story, the contents of the controversial document are revealed. It begins: “In the beginning…”

Mark Ganson

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