Powers + How Many Films Could Possibly Be Made

Rainbows give an inaccurate representation of how many colours the human eye can see, but if I were to be asked this question I would say the nursery rhyme; count the colours; say seven, maybe plus one as white is the presence of all other colours, whereas black is the absence of any; state everything else are just shades of them; get on with my day. Thankfully I wasn’t alive in 1931 and worked at the CIE, The International Commission on Illumination, although I heard their parties were pretty ‘lit’.

Thank you, I’m here all week. They did scientific research into the colours that the human eye can see and came up with the CIE 1931 XYZ Colour Space and CIE 1931 Chromaticity Diagram (inventive names), the latter of which is shown below. For example, white and grey have the same chromaticity but different brightness. These are both different ways of showing the approximately 2.4 million colours they calculated, so a few more than 7 or 8, but I was pretty close.

I prefer this more basic version of it though, not only because it implies that the answer is closer to my 7/8 colours, but that it would make the numbers involved in getting an answer to this question a heck of a lot smaller.

The next things we must consider are the number of pixels on the screen and its framerate. I know that cinema screens are only just beginning to move to into 8K (7680x4320 pixels) which is 33,177,600 pixels, so let’s go with that quality, just to be safe. Next, we would have to use a camera with a framerate (individual frames per second) which is the same as (or greater than) the framerate at which humans can process images. Studies performed by scientists on the United States Air Force in which pilots sat in a dark room and were flashed images for a very small amount of time to assess whether they could notice the image, concluded that humans can identify up to 220 fps (frames per second). It will not be so high nor so clear for everybody, but we must use the highest framerate to ‘be sure’. Now we can easily work out the total number of frames in, let’s say, a 2-hour period. 220 x 60 x 60 x 2 = 1,584,000 frames per 2-hour film. Now we have our numbers set up for the picture of our possible films, now for some audio.

A CD samples music at 44.1 kilohertz, where each ‘bit’ of information can be a 1 or a 0. This is similar to the first blog on power sets, where our set of possibilities is {0,1} instead of {the superhero is in the film, the superhero is not in the film}. But as there are two elements in each set, the maths works in the same way. In a 2-hour film, the audio file at this quality would contain 5,064,000,000 bits, each one being a 1 or a 0. This means that to find the possibilities of audio tracks that are 2-hours long we would raise 2 to this power. Therefore 2^{5,064,000,000} = 1.10256 x 10^{1,524,415,898} possibilities. I cannot begin to describe how big this number is. That is 110256… followed by 1,524,415,893 zeros, for anyone struggling to get their head round the standard form. In comparison, there are approximately 1 x 10⁸⁰ atoms in the universe. Our possibilities are massive admittedly, yet finite.

Next, we find the amount of 2-hour silent movies there could be. We would then multiply these two astronomically large numbers together to get the number of possible 2-hour films there could be. That’s it. Imagine the first frame of the film, in the bottom right hand corner, there is a pixel. Now there are 2.4 million possible colours that it could be (that the human eye can see, we are also assuming that the camera and screen can capture and display that many colours which may not be true, but always go with the larger estimation to be safe). The pixel next to that also has 2.4 million possibilities, so there are 2,400,000² possibilities between these two pixels alone, in the first 220^th of a second of the film. You can see how the numbers are going to get huge rather quickly. If you repeat these possibilities for the first frame of the film, it is worked out by (2.4 x 10⁶)^33,177,600, which if you have noticed, is the number of possible pictures with 8k resolution. Now we use the power of powers, not mind control, literally ‘powers’ again. To see how many ways that the 1,584,000 frames in our film can be arranged using any of these p = (2.4 x 10⁶)^33,177,600 pictures, we again raise the amount of pictures p, to the number of frames. If you can’t follow the calculations, good. It means you have better things to do than me, which isn’t difficult admittedly. The total number of silent films that are two hours long, T_s, is

                                           T_s = P^1,584,000 = 2.456 × 10^{335,301,272,844,929}

To 4 significant figures. We multiply this by the possible 2-hour audio files we found earlier, 1.10256 x 10^{1,524,415,898}, to get the possible two hour films that could be made with every possible 2-hour audio file in the background. You think of it, it is in there... Every possible film about dogs narrated by Morgan Freeman wearing a poncho, in there. Every possible film about how your life will go is in there. A 90 minute football match where England bring football home from Russia followed by 30 minutes of hysteria, Sunday 4pm BBC One, but conveniently also in there. Its all in there! This number is

T_{A =} 3.15225 × 10^{335,302,797,260,430}

If I wanted to type this number out in full on a word document, I would type 315225 and then hit the ‘0’ key and hold it down… for over quarter of a million years! 284,601 years to be precise. If I aged like David Beckham and managed to live this long and then wanted to print this number out, it would take 87.3 billion sheets of paper. If I cut down and turned to paper, every tree in every park in London, I still wouldn’t have enough paper to print out the possibilities and this is a number that every second I hold down the ‘0’ button, I would be making approximately 10,000,000,000,000,000,000,000,000,000,000,000,000 larger. So, there you have it, the amount of possible films under the given conditions above is 3.15225 × 10^{335,302,797,260,430}. Maybe a fact you can pull out in the kitchen at a party, if you never wanted to be invited back again. Thanks for reading.

Luke Bennett