Sunday 15 March 2015

Pi-ed to the galaxy

You may have heard that yesterday, March 14th, was 'Pi Day', or possibly 'American Pi Day', since it's in the usual American format that the date: 3/14/15 9.26:53 reads as the first ten digits of the number represented by the Greek letter pi. Usually the representation would be something like 3/14 1.59:26, but this year was of special interest to enthusiasts - and the precise time of the celebration was altered - because it is the only year of the century in which the two-digit year number can be incorporated, increasing the precision of the approximation.

But anyway. All this talk of pi reminds me of something that happened to a couple of friends one lazy Saturday afternoon...

Adam and Doug were sitting in the living room drinking tea when a package arrived at the door. Of course, it didn't arrive directly, but Adam watched through the window as the delivery man approached the door and posted a 'sorry we missed you' card, which is essentially the same thing. After a short sprint down the driveway and a forcibly-polite conversation in which Adam found himself conceding that the notion of 'being in' is fraught at the first hurdle with ill-defined notions of 'being', he returned with a small white box that Doug eyed gleefully as Adam slammed the door shut. It was funny, because he felt pretty 'in' right now.

 "It's here! Open it up!" said Doug. Adam did so, producing a small card with a disc in the jacket.

 "What is it?" he asked.

 "It's the Answer to the Great Question, of Life, the Universe and Everything!" exclaimed Doug. "Quick, let me see!"

"I thought we already settled that?" said Adam, handing over the card. "Isn't it-"

"Yes, yes, that's just how they market it. But this is so much more than that - look!"

Doug showed Adam the card. On the cover was the single Greek letter, pi. On the inside it read:

"CONGRATULATIONS on your purchase of the Answer to the Great Question, of Life, the Universe and Everything - and so much more! Contained in this innocuous little number is the answer to every question you could possibly imagine - and even some you can't! Simply load up the software, which contains an algorithm for calculating pi to whatever degree of accuracy you need. Enter your question and watch in amazement as the answer appears before your very eyes!*

*(Some computation time required).

HOW DOES IT WORK? Pi is perhaps the most famous of many numbers conjectured to be 'NORMAL', which means that its INFINITE decimal expansion contains every imaginable sequence of digits, however long, infinitely many times! THAT MEANS THE ANSWER TO YOUR BURNING QUESTION LIES HIDDEN DEEP WITHIN THE VERY FABRIC OF MATHEMATICS ITSELF! Simply assign a code, such as 'A' = '01', 'B' = '02', 'C' = '03' to all the letters and symbols in your language, then wonder as the SECRETS OF PI REVEAL THEMSELVES TO YOU. Try it now!"

Adam was bemused. "So it's saying we can read pi like some sort of message from the universe?" he asked.

"Exactly!" replied Doug. "And every conceivable answer is contained it in, because SOMEWHERE in there, the correct sequence of digits appears. We just need to pick a code. I suppose we may as well try the one they suggest. So my name, DOUG, would be coded using 04 for D, 15 for O, 21 for U and 07 for G. That's... 04152107. Let's just load up a few digits of pi here..."

Doug had inserted the disc into their supercomputer and pressed some buttons. The initial digits of pi flashed up on the screen:

"3.141592653589793238462643383279502884197169399"

Doug looked excited. "Let's have a read! So the first digits are 3 and 1, 31. Under our code that means... uhh. Hang on. Hm."

"I guess they haven't explained the full picture here," observed Adam. "There are too many two-digit numbers for the letters of the alphabet."

"No no, I'm sure it's fine" stammered Doug, trying to pick apart the card for a missing page. "Let's just-"

"It's ok - how about we just start again once we get to 27? So when we see 26 we get Z, then 27 will be A again, 28 is B and so on. We can keep wrapping round until we run out of numbers. A will be 01, 27, 53 and 79. Once we get to 99 it'll be U, so 00 can be V. I'll draw out a quick table...

A BCDEFGHIJKLMNOPQRSTUVWXYZ
0102030405060708091011121314151617181920212223242526
2728293031323334353637383940414243444546474849505152
5354555657585960616263646566676869707172737475767778
79808182838485868788899091929394959697989900

"Okay great," said Doug, nodding. "So we get 31, 41, 59, 26, 53, 58, 97, 93, 23, 84, 62... what's that? E, U, G, Z, A, F, S, O, W, F, J..."

"Eugzafsowfj?" interjected Adam, eyebrows raised. "Sounds like the 'very fabric of mathematics' has had one too many spins."

"Well, there's going to be some junk, obviously," replied Doug, a little defensively. "After all, EVERY combination of numbers is in pi somewhere!"

"Conjecturally, right?"

"Yes yes, conjecturally. You're right, we'll get all manner of complaints if we don't say that."

"From who?" said Adam, confused, but Doug didn't seem to hear. He continued:

"But you see, it found the word 'SOW' - that's a good start. We just need to search deeper. So what we do now is input our question, and the algorithm will calculate the digits of pi, one by one, until it locates the question in our code. What comes after that will be the answer!"

Doug thought for a second, then said:

"You know what, let's use a completely different code from that one from now on, and don't say it out loud, just in case anyone decides to check what we do for themselves."

He glanced over his shoulder, then back again. Adam was about to say something, but thought better of it.

"We'll start with something simple," continued Doug, beginning to type, "WHEREISMYTOWEL... and it says you have to select an answer length... ok". The supercomputer whirred into action. After a minute, it came back with the words:

"INYOURTRAVELBAG"

"Wait, what?" said a shocked Adam. "How can-"

"I knew it!" exclaimed Doug, rushing to the corner of the room and producing a very sandy towel from an equally sandy suitcase. "I remember now, I was sleeping under it during my last desert stargazing visit. Hm, it's a bit dirty. But you see, it works! The power of pi! Don't you see how big this is? All the knowledge of the universe is at our fingertips - cures for all the diseases of the world, the secret to world peace... lottery numbers!"

"I suppose for that we'd need to encode for some numbers as well as letters," said Adam, still hesitant.

"Try something else!" cried Doug. He was bouncing.

"Ok, let's try... WHATISTHECUREFORTHECOMMONCOLD" said Adam, pressing 'enter'. The computer sprang into action again, noticeably more pained this time. It began to make thumping noises from within, but Doug didn't notice because he was shaking out his towel, muttering something about a mini-raft. Eventually, the answer came back:

"RAISINSBOILEDUNDERAFULLMOONKDQZ"

"Raisins boiled under a full moon, really?" said Doug. "Well, that's useful to know."

"What's that KDQZ at the end there?" asked Adam.

"Oh, just unused answer space, I guess. The computer doesn't understand when to stop reading - that's why you have to tell it. What are you asking it now?"

"I'm typing HOWMANYLEGSDOESASPIDERHAVE."

"Why are you asking it that? We know the answer is-"

"FORTYTWO."

"What?"

"That's the answer it gave - forty two. How about WHATISTHECAPITALOFSPAIN?"

"Wait a minute..."

"Oh - HJKSDMAFT, of course!"

"What's going on?"

Adam sighed and turned to his friend.

"Don't you see, Doug? The digits of pi have all the answers, yes-"

"Conjecturally," Doug reminded him.

"Yes - conjecturally, all the answers are there, but there's also a LOT of complete rubbish! If every string of numbers exists, infinitely often, within the digits of pi, then not only is every answer there, but so is every conceivable WRONG answer, as well as a load of pure gobbledegook. It's the the ultimate internet troll. Frankly it's amazing the code and questions we picked returned answers that made any sense whatsoever the first few times. Someone must have left an infinite improbability drive running somewhere nearby."

Doug typed another query into the computer.

"Yikes," he gasped as the response popped up. "Well, that's just obscene." He turned back to Adam. "So it's completely useless?" he asked, starting to look somewhat dejected.

"Perhaps not. I mean, it's an algorithm for calculating the digits of pi, at the very least. I guess if you wanted to generate a string of random numbers, you might argue that as far as anyone can predict them, pi's digits are random. Of course, they're really completely fixed, but if you don't know what's coming next it's much the same thing in practice."

Doug stood in thought. "I suppose it goes to show," he said, "that having the universe's information at your fingertips is no good if you have no way of making head or tail of it. It's one thing to claim that pi contains all the great works of art under some sophisticated encoding, but that doesn't help us to produce that art ourselves, nor does it detract from the achievements of those who do create it. But you know, now I can't decide whether it's fascinating that pi contains all that information, both true and false..."

"Conjecturally"

"Yes - conjecturally... or whether it's a complete banality! I mean, it's just an abstract notion, at the end of the day, isn't it? We could have constructed a number with the same properties just by taking some source of randomness and converting it into a string of digits. Or are there deterministic methods that don't require randomness? How many numbers even are 'normal' the way pi might be?"

"Who knows," said Adam, and added jokingly: "Let's ask pi."

As he entered the question, the computer shuddered visibly and smoke began to billow from the casing. With what sounded like a sigh of relief, it completed the calculation and spat out the response:

"SEEYOUNEXTWEEK"

Adam and Doug glanced at one another nervously. Adam went to the window and quietly shut the curtains.
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