11 March 2012

Most Astounding Fact

When reading through my blogroll, as is my wont, I found this video at Bad Astronomy, of Neil deGrasse Tyson telling us what he thinks the most astounding fact about our universe is:

The fact? That we are all made of stars.

It's a powerful fact; a beautiful, moving fact. It's also a fact that recently caught Miley Cyrus a fair amount of flak when she called Lawrence Krauss's way of putting it "beautiful".

But I don't think it's a very astounding fact - that is, not to me at least. Of course, I have a skewed perception: I've known it for as long as I can remember. It's one of the perks of being raised by a mother who read Carl Sagan books to her kids instead of Dick and Jane. Next to "billions and billions", his most famous line must be "we're made of star stuff". (It took about 2 seconds to find on YouTube!)

Anyway, Tyson puts it very well and I think it's a great video. It makes me look forward to the new Cosmos sequel he'll be hosting. But this is my blog, after all, and if you think I'm going to get through a whole post without making it all about me, me, me... well, you'd be overestimating my self-control. =P

So what is the fact about the universe that I find the most astounding? Well, I'm glad you asked.


Like a lot of kids, I went through the phase when superlatives were the coolest things ever. Dinosaurs, the biggest land creatures ever! Jupiter, so huge the Great Red Spot could swallow the Earth whole and still have room for afters! Absolute zero, where everything just stops! Any kid who's been through this is surely aware of the stupidly huge number googol (one followed by a hundred zeroes) and its bigger brother, googolplex (one followed by a googol zeroes). Like the fact that we're all star stuff, or the continent of Pangaea, googol and googolplex are quite familiar faces to anyone who was a kid like me.

Which makes the hidden power of googolplex all the more astounding.

Googolplex is so freaking big that it's literally impossible to write out in long form - the number of zeroes wouldn't fit in the universe. ...Well, let me put a finer point on that: it's impossible to write out in base ten; just declare a "base googolplex" by fiat and you can write it out as 1. That would be sort of unfair, though, wouldn't it? But it's good that I brought up bases (or radices, for the verbose), since the subject is an essential part of the astounding fact I'm on my way to imparting.

Your computer, for instance, uses base two (more popularly known as binary). The number we sensible people would write as 255 is 11111111 in binary. But even programmers like me, who "talk" to computers on a routine basis, don't often handle a lot of binary. No, base sixteen, "hexadecimal", is the most common way programmers look at raw data. Since hexadecimal needs 16 distinct symbols, and our damn ten-fingered ancestors didn't have the decency to furnish us with that many numerals, we have to borrow a few letters that don't really want to be there (much like your sister when playing Rock Band 3), and 255 looks like FF. (The saddest cases amongst us, when we encounter "fffffff" in forum posts and YouTube comments, will obsessively convert to decimal numbers. We (*koff* they, I mean they) also get bonus points if we remember to factorial when it's exclaimed.)

So any data on a computer can be expressed in good ol' decimal, as well. It's not convenient for programmers, for a number of reasons, but it can be done. So, technically, any string of data - a program, a file, even (of course) a song or video - is just a really big number; this is just the corollary of the trivial observation that any number can be stored as a string of data on a computer.

So, what kind of program would googol be? No, it wouldn't be Google's source code (seriously, dude, that's a terrible guess). One followed by a hundred zeroes doesn't represent enough bytes of data even to store this blog post! Even a hexadecimal number with a hundred zeroes (which would represent a significantly greater value than googol, considering hexadecimal 100 is equal to decimal 256) could only represent about 50 bytes.

But what about a googolplex? It's got a googol zeroes! I mean, that's got to be enough digits to fit a program in, right? Well, I hope so - even if we considered all the storage devices ever manufactured by humanity altogether, we wouldn't even come within ten orders of magnitude of needing a googol zeroes to represent the data. Remember, we can't even write a googol zeroes in the universe.

So, googolplex is big. Yawn. Anyone can look at a 1TB hard drive on their desk (or Radio Shack if they're unlucky enough to lack one) and be briefly titillated at the thought of how many floppy disks it replaces. (Hint: more than you can carry.) Big numbers can be reduced to mewling kittens simply by giving them names; hell, even infinity can be tamed by giving it a name and a wiggly little symbol. But the thing about a number, when thought of as a string of data, is that by counting down from it to 0 (or by counting up from 0 to it, which makes a bit more intuitive sense when imagining the process) systematically traverses every permutation of data that can possibly be represented by the number of zeroes it has. Try it; count to 100 - there are only 100 possible 2-digit numbers (counting 00, of course).

So, let's count to googolplex. 1, 2, 3, 4... Well, let's just imagine we kept going. By time we finish, we'll have traveled through every possible permutation of data capable of being represented by the intermediate numbers. Sure, at first we'll have "boring" numbers like 8, 13, and 42. But by the time we reach 10,000, we'll have every 4-digit PIN ever used. And by 1,000,000,000,000,000,000,000 we'll have every possible 8-digit alphanumeric password.

Does that number seem large? Yeah? Does it fit in the universe? Okay, then. Just to keep things in perspective. We've got a long way to go.

Before "long", we've got the digital representation of every novel ever written. Or that will be written. Or even could be written. And every possible combination of the first half of one with the second half of another. Also, every novel possible with increasingly bizarre typos. Not to mention the overwhelming proportion of total nonsense. (In fact, we'll get to this milestone much quicker if we don't use ASCII encoding, which wastes a whole byte per letter. That's 256 combinations where only about three dozen are really needed!)

Lost amongst those novels? Your autobiography. Detailed and accurate to a degree even you wouldn't be able to achieve. Also, your autobiography if you'd lived a thousand years ago, or a thousand hence. Alternate yous that commit unspeakable atrocities or perform amazing feats. Accounts of your journeys to Mars and beyond.

You'd have the Library of Babel on steroids.

The concept is so mind-numbing, I've known people to flat-out disbelieve it upon their first encounter with it. But it's ineluctable; it's even demonstrable to an extent (the 4-digit PINs are an example). It's simply a fascinating fact about large numbers.

But we're still not done. That's just text; we've got so much further to go. Before long, we've got mp3's of every possible song. Digital video of all possible movies. Hell, digital video of everything that's ever happened in history. Always, remember, with a disproportionate heap of total garble, of course.

Long before we hit a googolplex, we'd have danced throughout the total cultural output of humanity. Indeed, of all the sentient species in the universe. Embedded in the ever-ticking stream of numbers would be videos of alien worlds. Real ones - and fake ones. Alpha Centauri Idol? It's in there.

Let that sink in. If you're like me, you probably can't. You could literally spend the rest of your lifetime - and our universe's - and you'd still never be able to list the things that you'd create simply by counting to googolplex. All that did, does, will, or can exist. Lurking beneath the surface of a humble number - a number most of us meet as children, throwing our arms wide and saying, "I bet it's this big!", and quickly forget, relegating it to the bin of cool science-y superlatives with the blue whales and blue giants.

Of course, you could never actually undertake this extraordinary performance. Any computer that could display the results wouldn't fit in the universe (more so if you wanted to store them!) and sifting the Harry Potters from the Graxfasczzaxses would be impossible.

But that's not the point. The point - the astounding fact, that promised astounding fact - is that this journey, this parade of numbers, can be easily imagined. Like infinity, we've tamed it with a name - googolplex. And though none of us will ever see the destination, we can all of us embark on it effortlessly. 1, 2, 3, 4...

One simple rule, maybe the simplest possible rule: Add One. Two little symbols: "+1". And from it? Everything, with the full force of what those ten letters can possibly convey.

Is it any wonder that, knowing a fact like that, another little rule like "copy me" could have created us?

And after all of this? That's just googolplex's zeroes. The amount it represents is huger still - after all, "100" only has two zeroes. And it's not even the biggest number humans have had the chutzpah to name. There are numbers mathematicians kick around that are so much larger than googolplex that the conventional notation that makes short work of it - 10^(10^100) - can't express them. Like googolplex's zeroes, these numbers would have too many exponents to fit in the universe.

...610, 611, 612, 613, 614, 615, 616...


Going back to Neil Tyson's fact for a moment, Starts With A Bang! posted about it, too, and at the head of the post was a cool quotation from Richard Feynman:

Is no one inspired by our present picture of the Universe? This value of science remains unsung by singers, you are reduced to hearing not a song or poem, but an evening lecture about it. This is not yet a scientific age.

Well, sadly Feynman is dead, but I like to think we're now living in the dawn of that appreciative age he dreamed of. This blog's very title is a fragment of poetry I wrote, inspired by the picture of the universe revealed by science. In fact, my "About Me" box says I'm a songwriter (though it's something I never really post about, since I'm in a stage of my life where it's taking a backseat to programming), and well over half of my songs are inspired by science. It's without a doubt my greatest muse. Maybe I'll stop playing Sonic long enough to post some eventually. =P

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