When Bricks of Pi are being
manufactured, each block receives a four
digit
number stamped onto its outside face,
prior to firing.

These numerals are extracted out of the
infinite series that results from the
evaluation of Pi. You can lay them in any
order, or request a precise sequence
corresponding to the number of bricks
required to construct your entire
building.

pi to 50,000 decimal placeshttp://www.geocitie.../1216/numtab/pi.htm or is it an outside wall of that new science tower? [xenzag, Jul 20 2008]

Normal Numbershttp://en.wikipedia.../wiki/Normal_number In mathematics, a normal number is, roughly speaking, a real number whose digits (in every base) show a uniform distribution... [futurebird, Jul 21 2008]

Search pi for stringshttp://www.angio.net/pi/bigpi.cgi My SS number doesn't occur in the first 200 million digits, but a string of the first 8 digits does, and also the strings 00000000, 11111111, 22222222, 121212121, and 12345678. [ldischler, Jul 21 2008]

Pi in the skyhttp://www.angio.net/pi/piquery Dave Andersen's program allows you to enter up to a 120-digit pattern for which it will search through the first (now updated to) 200,000,000 decimal places of Pi. It also shows you many of the digits which surround the location of a pattern match (very helpful!); however, in order to find multiple occurrences of the same pattern it's necessary to click on a "Find Next" link for each one. As an example, here's the output you'd get searching for the pattern 99999999 (eight nines): [xenzag, May 12 2017]

(I realize that this is widely believed, but I find it difficult to prove - it's a different statement from "there are infinitely many digits in the decimal expansion of pi" or from "pi is non-repeating".)

I just went to that "pi to 50,000 decimal places" link, opened my find-in-this-page editor, and typed in random sequences of numbers. I got most 5-digit sequences, and a few 8-digit sequences.

"Any arbitrary sequence of numbers" would mean we could somewhere find the digital equivalent of _Hamlet_--that's too much to expect. You'll need to get your house in order.

It seems that given the infinite nature of
the pi sequence, then [snictown] could
be right - this is a quote from the link.
"Now, if this be true, that p is
completely random, then every
conceivable sequence of numbers must
somewhere occur in p . That is, if one
considers any number, 12345678 for
instance, that number will sooner or
later be found in p if the search is
pressed far enough. To assume
otherwise would be to prejudice the
randomness of p."

First, I don't know of any proof that pi is a random sequence of digits. I'm aware that lengthy sequences have been tested and found to be random, but that's different from being able to assert that the entire, infinite, sequence of digits in pi follows a completely random distribution (or are there, for example a fraction of a percentage more sevens in the sequence than you'd expect?).

Second, it seems common sense to say that "if the digits in pi follow a random distribution then every conceivable sequence of numbers must somewhere occur in pi" but I'm not sure that's actually true. To put it another way, if pi is random, then does that 'prove' that "1234567890" occurs somewhere in pi, or does it just make it very probable?

Either way, I'm not that concerned. If I
understand
Gödel's incompleteness theorem well
enough, it
leaves enough room for no proof or
perhaps of more
importance, no disproof. As Pi DOES
have a verifiable
sequence, then I see no problem.

//it has been proven that any arbitrary sequence of numbers appears somewhere in the sequence of pi.//

[sninctown] It is conjectured, not proven.

//Second, it seems common sense to say that "if the digits in pi follow a random distribution then every conceivable sequence of numbers must somewhere occur in pi" but I'm not sure that's actually true.//

[hippo], if pi were to be proved a normal number, it would be a disjunctive sequence, and hence every finite string would be represented.

[baconbrain] If it were a normal number, it would show uniform distribution in all bases. So Shakespeare, as well as Beethoven, and everything else (converted to binary), would have a representation in the binary representation of pi. As would this anno.

However, we are a long way from finding pi to be a normal number.

[HegelStone] Not true, discussed before on the 'bakery, but it makes a great tagline!

//So Shakespeare, as well as Beethoven, and everything else (converted to binary), would have a representation in the binary representation of pi.//

Even messages could be found there, and will, with enough computing power. Hidden messages from God himself. And one day we’ll see a book titled, “The Pi Code: Finding God in Transcendental Numbers.”

If you make certain of the bricks removable, then when the mood strikes you you could switch them around.

You could then invite people to come to your house, show them around, and when they say, "This is the same place you've been for the last five years," you could provably respond, "No, it's just a similar house in the same geographical location. Now give me my housewarming gifts."

Repeat until you have a full china set and no friends.

The key to the answer to the question being asked is not so much the randomness of pi, but the infinite-ness of its sequence of digits. It is in that infinity that one could expect to find any FINITE arbitrary sequence. I conjecture it would be possible to find a place, somewhere in that infinity, that begins, 314159265358979323846... and proceeds to duplicate all the initial digits of pi, until this particular starting point is reached. After the duplication, effective randomness would follow, for a nice long stretch, and eventually a NEW starting point could be found, which duplicates everything up to THAT point. And so on.... If you don't think this is possible, then what it really means is that you don't properly understand what "infinity" allows.

Here's how to build a counterexample: Take PI, translate it into binary, then read those digits back as decimal digits. The resulting number goes something like this: "11.0010010000111111..." That gets you another series of decimal digits that has roughly the same properties of infinity and non-repeatingness as PI -- except that it doesn't contain the digits 2 through 9. (It is thus not a normal number.) That sequence is still infinite, but doesn't contain the substring "2".

I agree that it's likely that PI is a normal number, but it's interesting that it's not (yet?) proven.

[jutta], you appear to be mixing conditions. I was not talking about a binary form of pi. But if I was, I would simply say that you ought to be able to take some arbitrary length of the initial ones and zeros (instead of some arbitrary length of Base-10 digits), and eventually find a duplicate of them, somewhere in the infinite sequence of ones and zeros.

A piece of trivia for those who haven't seen it before: pi can be computed as the sum of the infinite series 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 + 4/13 - 4/15 ... (don't try this without a computer; it takes a LOT of terms for the result to gradually converge into even the first few digits of pi)

I'm not talking about the binary form of PI, I'm just using it to build a decimal number that is just as infinite as PI, yet doesn't contain some sequences of digits. That is a counterexample to your claim that infinity implies normalness.

// Repeat until you have a full china set and no friends.
[marked-for-tagline]

[jutta], OK, I know there are contrived infinite sequences with properties such that you describe. Pi has not been shown to be such a sequence, just as it has not been shown to be random enough for "normality". I think, though, that it is random enough for most modest/finite arbitrary sequences of digits that one might care to specify, those sequences can be found somewhere among the digits of pi. I did say in my first post here that the key was less a matter of its randomess than a matter of its infinite-ness, and in saying that I did not mean to imply that the degree of randomness which it exhibits is discount-able.

I am compelled to quote from Gödel (via
wikipedia) "Gödel's first
incompleteness theorem
shows that any formal system that
includes enough of
the theory of the natural numbers is
incomplete: it
contains statements that are neither
provably true nor
provably false. Or one might say, no
formal system
which aims to define the natural
numbers can actually
do so, as there will be true number-
theoretical
statements which that system cannot
prove."

This is all a bit moot. [ldischler] said "Of course you can"; [jutta] said "?"; [snictown] said "yes. It's proven."; [jutta] said "I don't think so".

Since then, the cut and thrust of the idea has been sidelined and there has been a debate about the randomness of Pi. Pi is not a random number, although it's irrational sequence appears to be so, and it is known to over a trillion decimal places.

Therefore, I suggest that it does not matter whether the 42,000 combinations of brick-numbers form a section of Pi. They will look like it - and no-one will be able to realistically mount a counter-proof.

<I'm reminded of kids in a playround: "Your Mum *smells*", "No she doesn't", "Yes she does", "Prove it then", "No. You prove she doesn't smell"...>

Assuming that the infinity/normallness thing does apply to pi, then it may also apply to other numbers of the same ilk (pi+1, e, e-9, e+pi, phi, among others) In which case any sequence of arbitrary numbers must 'belong' to any number of arbitrary transcendental numbers.

So you have an arbitrary sequence of bricks which *may* be found in any of an infinite and uncountable number of transcendental numbers - am I alone in thinking "Big Deal"?

Anyway, if you want a house that's piesque, shouldn't it be circular?

At least the bricks aren't representations of sqrt(-1) - Who would want to live in an imaginary house?

If pi contained *every* possible sequence of numbers within its digits, then we would have to conclude that somewhere within pi is a never-ending string of 8's, just as long as pi itself.

//If pi contained *every* possible sequence of numbers within its digits, then we would have to conclude that somewhere within pi is a never-ending string of 8's, just as long as pi itself.//

[2 fries shy of a happy meal]:
Short answer: Yes
Long answer: <Morpheus> Unfortunately, no-one can be told how infinite Pi is - you have to prove it for yourself. </Morpheus>

Anyone else think of Heinlein's "--and he built a crooked house" when the imaginary house came up? I know a tessarect isn't imaginary, but it should be mentioned somewhere in a discussion of mathematical houses.

//Anyway, if you want a house that's piesque, shouldn't it be circular?// pi is a *ratio* that *crops* up everywhere, not just in circles. Even in your imaginary house. In fact, specifically, in your imaginary house.

In an undergraduate lecture, a tutor put the numbers 1,4,1,5,9 up on the board. He said, "I have a shiny new textbook for the first student who can tell me the next two numbers in this sequence..."

All the students hastily got their notepads out and started to calculate whatever. All the students except [Jinbish] - he'd recognised the sequence straight away. A friend of his had memorised Pi at high school. [Jb] had been sitting next to him as he recited out loud 3.1415926 over and over again. [Jb] had almost fallen out with his repetitive mate... but now fate had smiled on his near madness-inducing episode.

"2 and 6!" and as the tutor looked up and smiled, and a hundred astonished faces turned round, he knew that both the textbook and infinite amounts of derision from the rest of the student body were now his!!

The ideas explored in this discussion are the same as
those in the Kate Bush Conjecture (see link) -
essentially, she sung the digits of Pi but with a few
out of sequence and the conjecture is that this
'wrong' sequence actually does occur somewhere in the
expansion of Pi.

[irrelevant aside in response to earlier discussions]

There was a science fiction story (may have been Contact by Carl Sagan) in which a supercivilisation has engineered the universe and mathematics in such a way as to leave a message encoded in the digits of pi.

Because tau looks like pi with just one vertical, which if you
think about that as being the denominator, simply makes all
sorts of intrinsic inherent sense and beauty, or at least as
much as an irrational number can have.

If you draw a square that's 2 units (inches, cm, metres, demi-furlongs whatever) and then draw a circle exactly in the middle whose edges touch the sides; Go on to sprinkle exactly 4000 grains of rice from a height such that they fall randomly across the square; you could use a sieve or some other sprinkly-outy device to assist with this - the important thing is that it's random (if any grains that fall outside the square, just sprinkle them again)

(Alternately, you could do this with a box with a lid that's got a 2x2 base, and has a circle inscribed in the middle. It might make less of a mess.)

Once you're finished, count the number of grains of rice inside the circle. You should have a 4 digit number, so just plop a decimal place after the 1st digit, and that should be that. That's what pi is. (Subject to local humidity, wind and gravitational conditions)

If you want pi to more decimal places, try the same thing with 40,000, 400,000 or 4,000,000 grains of rice (after booking out a suitable period in your events calendar)

Alternately, the same routine ought to be possible with lentils, mung beans or similar, and a weight-based approach where you start with 4g, 4kg or some 4-weight of something;

Sprinkle, shake or otherwise enturbulate and extract only the stuff that lands on the circle. That stuff, weighed again should have a value of pi, give or take the odd bean.

When you measure the length of pi, are you allowed to
ignore the zero between the 1 and the decimal? I'd have
thought pi base pi has length 2.

Though really it's still infinitely long, but mostly zeroes.

// Because tau looks like pi with just one vertical, which
if you think about that as being the denominator, simply
makes all sorts of intrinsic inherent sense and beauty //

Oh, you see I'm an engineer, not a mathematician, so if I
see anything with inherent sense and beauty I'm
automatically suspicious. If it's slightly broken and
irregular then I'm much more confident that it's going to
work in the real world.

Maybe I'll go add bipi as a separate thing, for people like
me.

//going to work in the real world// What does that mean? In the "real world" we also have particles that can be in two places at once at the quantum level. I like the reality that is Pi being such a conundrum of numerical contradictions.