Maths of Spam

3264 Spam Cans – Art Wants to be Free is an innovative, cutting edge public debut from digital artist Colin Colorful. Colorful launched his career with this work, a 21st century response (perhaps a contemporary upgrade) to Andy Warhol's 32 Campbells Soup Cans. Warhol's original was displayed at a Los Angeles gallery in 1962 and is considered one of the most influential works in Pop Art.

3264 Spam Cans – Art Wants to be Free is much more than just a pleasing aesthetic image. 3264 Spam Cans has, literally, trillions upon trillions of potential unique versions of this artwork; far more that the mere ten million versions planned for the distribution in the next few weeks. To understand just how many artworks there are, we need to examine for a moment the underlying code that generates the image, and the math behind it.

3264 Spam Cans has a thirty-two square grid which comprises the composed image. If, for the sake of an argument, there were only two potential variations of the image (there are more – just how many will be discussed in a moment) then the potential number of unique artworks would be:

232 = 4, 294, 967, 296

Obviously, this is already a significantly higher total than the number of this artworks derivations to be distributed across the internet and in fact is approximately equal number of adults on earth. However, as merely glancing at the work will make clear, there are many more than just two cycling images. In fact, sixty-four images make up the building blocks of the work, with variations in angles and color. If every one of these versions of the artwork were to be viewed there would be:

3264 = 6.277101735 x 1057 (approx.) versions of the work

This number is many quintillions times more than the current estimate of number of stars in the entire universe, which is approximately 7 x 1022

Compounding this figure, which is already well beyond even astronomical terms, the eight-by-four grid that is visible is only a representation of a more complex function of the program running in the background. To explain this, we'll start with how the array of 32 Spam Cans is randomized:

When a viewer looks at 3264 Spam Cans, each of the thirty-two squares are rendered at a very slightly different time, depending largely on the computational speed of the host server, the viewers personal computer and the speed of the Internet connection between them. This is important; as each square on the grid is requested, the script calls up the Unix servers time code in micro-time. This is a numerical string variable which looks like this:

1234390717.02127900

This string is a measure of seconds since the clock was started on January 1st, 1970, and resolves to tens of nanoseconds – thus, the exact time that each 'seed' is generated is unique to begin with. This seed is then multiplied by a prime number to increase its uniqueness, then utterly randomized again. Of course, all computer-generated randomizations are based on complicated maths and the time code – this multi-level random approach leads to more complete randomization than a single random string generator ever could. Coincidently, the time code above translates to 19 minutes and 16 seconds past 10 pm on the 11th of February, 2009.

However, the program does not merely generate thirty-two seeds. Though extremely unlikely, thirty-two random seeds could lead to symmetrical patters in the data and thus the generated image. To compensate for this, the source code calls up not one but three potential seeds for each square in the grid, and then, through a process where each seed is compared with its diametrically opposite seed, the program decides which images to show so that the visual pattern is as distinct and non-repetitive as possible. This does mean, of course, that our formula needs to be revised. The actual number of potential combinations of this artwork are best expressed as:

⅓ x 6496 = 8.2443646715 x 10172 (approx.)

For a tongue twister, just try saying it. It begins “eighty-two sexquinquagintillion” and only gets more complicated from there.

Okay, so just how big is that number?

Bigger than comprehensible. In fact it's a larger number of artworks than there are atoms in the universe... squared.

Of course, this number is an absolute impossibility, as there literally are not enough particles in the universe to actually count to this number: an exciting example of imagination overcoming the bounds of reality.

Current estimates put the number of atoms in the observable universe at somewhere in the realm of 1079 to 1081 (an exact figure is almost indeterminable, due to all the problems associated with counting atoms – which is exactly like counting grains of sand in a desert only trillions of times more complicated).

So, this means that for every atom in the universe, there are millions and millions of versions of 3264 Spam Cans. To put this in perspective, the average grain of salt contains somewhere in the order of 1.2 x 1018 (that's 1, 200, 000, 000, 000, 000, 000 or one quintillion, two hundred quadrillion) atoms. 3264 Spam Cans, on the other hand, contains many more than a googol variations (it refers to a vast number: one followed by a hundred zeros, most easily represented as 1 x 10100), the name the search engine Google was based on.

In fact, 3264 Spam Cans contains so many derivations that, even if the number of atoms in the universe were squared (that is, multiplied by itself), this immense number (somewhere in the order of 10160) still doesn't reach the dramatic total outlined above. That means that, for every atom in the universe, 3264 Spam Cans has that many variations over again. So, what does this mean?

If you were to sit and watch 3264 Spam Cans refresh once per second, no human lifetime would be sufficient to see all the derivations. In fact, humanity's collective lifetime would not be enough to see even one-percent of one-percent of the variations that could occur. Some particularly intelligent (and geeky) readers might have realized already that it is utterly impossible to see all the potential variations, not only due to their extraordinary number, but due to the way they are generated. As the baseline random variable, the source code relies on Unix time. This has been counting for almost four decades now, and is well past a billion. Thus, to see every possible variation, in addition to needing to be immortal, one would have to start by traveling back in time to 1969...

Ignoring the requirement of time travel for a moment: if the artwork were refreshing one per second, to actually see all the variations possible (in other words, to statistically guarantee that you would eventually see the same pattern twice) you would need to watch the artwork unfold for approximately:

2.613142 x 10165 years...

which is

1.906916662 x 1055 x the age of the universe

If you're game – and, let's face it, who isn't? – I'd suggest brewing a really, really big pot of coffee, and getting to it. Not a second to spare...