shouldice

@sgtsphynx:

It's true that each pixel can show 16,777,216 possible colours, but what I was doing is counting the number of bits in the entire image, and then using the exponent operator with a base of 2.

The exponent operator can be used for this sort of calculation:

[number of states a thing can be in] ^ [number of things] = [number of combinations of states for those things]

For example, a bitstring of length 2 can be put into 2^2 = 4 configurations. They are 00, 01, 10, and 11.

Another example: 6 friends are having drinks. Each person can have coffee, tea, or neither. The number of things (people) is 6, and the number of states is 3 (coffee, tea, nothing), so the total number of possible configurations is:

3 ^ 6 = 729

In the case of the HD picture, we're counting the number of bits in the image, (1920 x 1080 x 24), and using a base of 2, as there are two states a bit can be in, 0 or 1.

2 ^ (1920 * 1080 * 24)

You could do it a different way, using the number of colours: The things we could be counting could be pixels instead of bits, and the number of states could be all the colours a pixel can be:

16777216 ^ (1920 * 1080)

You get the same answer. :D

(This link shows that the two are equivalent by dividing them and seeing that the quotient is 1.)

I wouldn't be surprised if I made a mistake somewhere in there though! Maybe @shivoa can give it a lookover to make sure I didn't get carried away.

-Andrew

I am not a mathematician, but:

Unique arrangements of distinct things are called permutations. To compute the number of permutations, imagine a tree of decisions about the ordering that needs to be made. Let's use an example of 10 frames of animation.

1. You need to decide which of the 10 frames goes first. There are 10 options.
2. For the second frame in the sequence, you only have 9 options to choose from (the first one was already picked, and all the items in the sequence need to be unique). So far, that means there are 90 possible choices for the first two frames.
3. For the third frame in the sequence, you have 8 options remaining.
4. and so on.

For 10 frames, there are 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 = 3628800 possible sequences. This operation is called the factorial, and it's the ! or x! key on your calculator. In our example, 10! = 3628800.

The factorial operator is crazy; crazier than the exponential operator that you use to compute the number of images. For instance, let's compare 2^x and x!.

• 2^64, which is the number of 8 x 8 images that exist, is a nice friendly number around 18,000,000,000,000,000,000.
• 64!, on the other hand, is 130,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which is much, much, much bigger.

All possible sequences, if strung together, would result in an animation 18,000,000,000,000,000,000! frames long. I'm not going to apply that factorial and write it out; just check out the Wolfram Alpha page.

We can do bigger.

The above calculations were for the PANOPTITRON, with it's 8 by 8 monochrome display. It's kind of a joke, to demonstrate that even with such a limited search space, the number of combinations is insane. ... but what if we went back to the HD screen with full colour depth? How many permutations of all the HD images are there?

So to be clear, we'd be talking about every possible image, watched in every possible sequence. That means that every movie ever (as long as it doesn't contain exact frame duplicates) will exist in there (and because no movies are 19 billion years long, each movie would actually appear many, many times. eg: the one sequence where Boogie Nights starts at frame 0, the one sequence where Boogie Nights starts at frame 1, and so on). There would also be, of course, every movie that would ever be, or even could be. Want to watch a century-long long documentary about people having sex with cars? It's in there. What about the same thing but it's set on the moon? Sure. Any possible sequence of images. (Again, as long as there are no exact repeat frames, but with full colour depth that's not going to be a problem.)

In this case, if you strung all the sequences together, the number of total frames would be ( 2^(1920 x 1080 x 3 x 8) )!

This number is so big, it can't be expressed in decimal notation. Let me be clear: I'm not talking about the value of the number, I'm talking about just writing down the number itself. Without scientific notation or anything, if you tried to write this number, even if each digit were a single atom, you'd run out of atoms in the universe. To write it down.

By a long shot.

The traditional "can't even write this down lol not enough atoms" number is the googolplex, and that's 10-to-the-10-to-the-100. Our number is 10-to-the-10-to-the-about-fifteen-million.

So anyway, it would take a while.

Wow, thanks everyone! <3

@amyggen: Cool, done. (link) Thanks for the pointer!

Presenting, PANOPTITRON.

Presenting, PANOPTITRON.

(I don't actually know if this works anywhere other than my computer with my browser.)