( 2^(1920 x 1080 x 3 x 8) )!
Just an FYI, this isn't the calculation for an HDTV... Each pixel has 16777216 colors that it can display, each of the three sub-pixels displays 256 levels of saturation. So shouldn't it be ^(1920*1080*16777216) or ^34789235097600? An astronomically higher value, and an even more mind-fuckingly higher value when made a factorial. Not that either number would be able to be written down with out scientific notation. @shivoa made a topic a couple weeks ago dealing with the math of an 8 pixel by 6 pixel "HD" screen, and that topic is really the only reason I even know the number of colors a pixel can display. I'm out of my depth and both of you far outclass me when it comes to math.
EDIT: And now I am second guessing myself, wouldn't it actually be 2^((1980*1020)^16777216)?
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.
@shouldice: Nope, all looks right to me. Unless I've skipped over something I don't think you've made a mistake in any of the numbers in this thread.
@sgtsphynx: As @shouldice says, it's all just about where you put the numbers. You can look at a simple bit (0 or 1) and say how many of them you need, or how many 0-255 (256 values) sub-pixels you need, or how many pixels (made up of 3 sub-pixels for the ~16.7m combinations per pixel) and they're all the same.
2^(24*1920*1080) is conceptually a row of 24 bits with 1920 of these rows and 1080 of these slices (so think of each as the X, Y, Z and the volume of bits) - we're just counting how many 2s we multiply together.
2^8 = 256 (so if we take enough bits for a sub-pixel then we confirm it does add up to the right number of combinations) 2^(8*3) = 2^24 = ~16.7m (so it all adds up for a pixel too).
So if we want to look at that 2^(24*1920*1080) as actually a load of rows of 24 bits then we can do (2^24)^(1920*1080) which is the 16.7m raised to the power of the number of complete pixels. I used the middle point, breaking into a base unit of a sub-pixel of (2^8)^(3 subpixels per pixel * xdim of screen * ydim of screen). But that's all just different ways of writing the same value.
@shouldice: @shivoa: As I said, my math skills are woefully atrophied, haven't taken a math class in 14 years. I got that what I was seeing and what you were saying were essentially the same thing (minus me not needing the 2) once I realized that 2^24 = 256^3. Your way of writing it looks cleaner. Anyway, my mistake, carry on.
8x8 gridsize with monochrome is actually pretty good if you want to expand the idea. For example, you could rent 8 floors of an office building for next 19 billion years and flicker the lights on and off to create the images. That is much more feasible than trying to do it at 1080p resolution, for starters there's very few buildings that have 1080 floors and most of the skyscrapers are built tall and are therefore unsuitable for 16:9 aspect ratio. Not to mention the rent would be astronomical.