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First, I am not a math specialist, I am asking a question that seems to have a very clear answer to me but it seems wrong, so I am hoping to learn why. Here goes... Long ago I read a description of Maxwell's Demon in a Douglas Hofstadter book (I think) and ever since learning about digital images I've been thinking about what I read, but using digital images instead of the video tapes that were used in the book.

I'm using a permutation formula I found for $N$ possibilities at each of $r$ locations: $N^r$. If I allow only 256 colors (so $N = 256$), which produces "OK" fidelity, and an image size of, say, 256 x 256 pixels ($r = 256^2$) then there are $256^{256^2}$ possible images that may exist. That is a very large number (Wolfram Alpha tells me there are 157,827 digits) but to me it seems very, very small because:

That set contains every possible image, from every angle, at every time (past, present and future) of every real and imaginary thing that may be represented in an image. That includes images of soundtracks, sound wave forms, musical scores, film frames, and digitized text. It includes printouts for any 3D printer models as well as the plans for every device ever invented throughout all of time. As if that isn't enough, the images themselves may be combined to create higher resolution versions of all of the same images plus other images that were too large to fit in the 256 x 256 pixel frame!

I hope I've explained this idea well enough. Most of the people I've tried to talk with about it just get glassy-eyed or tell me it is nonsense. If it is not nonsense, is this written about somewhere so that I could read an expert version and share it with my friends? Also, if it is not nonsense, how can all of time and space be so "small?" (smile)

EDIT what melts my brain is that all possible real and imaginary images across all of space and time, can seemingly be represented by a finite number of images.

If I increase the color depth from 8 to 24 bits, the number of digits roughly triples and then increasing the images to 1,000 pixels on a side gives a number with over 7 million digits. Insanely huge, but still finite and most will still be static/noise (those those might be a smaller part of a larger image, too.)

Steve Valliere
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    It is enormously large. 157,000 digits is a lot! Try writing out 157,000 digits! – Qiaochu Yuan Jan 09 '21 at 00:10
  • well, that number is the number of all possible moderate quality photos in a given space and time. That number is not infinity and I don't think its reasonable anyone would say it is. – fleablood Jan 09 '21 at 00:11
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    The number of atoms in the universe is estimated to be about $10^{80}$, and this is unimaginably larger than that. If each atom in the universe was itself an entire universe with $10^{80}$ atoms, that would make $10^{160}$ atoms. To get $10^{150000}$ atoms, each atom in our universe would have to be a universe of atoms that were universes of atoms that were universes that were ... about a dozen universes deep. – Steve Kass Jan 09 '21 at 00:20
  • If we limit ourselves to the physical universe there is no such thing as infinity. And a number with $157,827$ digits is immense. Considering it a billion times larger than an number with $157,818$ digits and that is a billion times larger than a number with $157,809$ digits and so on, a number that large is certainly large enough to include everything in the physical universe it not surprising. – fleablood Jan 09 '21 at 00:20
  • Spiked Math did a comic very similar to this talking about a finite box of movies. "In this box of videos is The Lion King along with every movie ever created... and a video of you watching yourself watching The Lion King is in the box. There is even a video of you watching a video of yourself watching a video of yourself watching a video of yourself watching a video of The Lion King." – JMoravitz Jan 09 '21 at 00:24
  • You might be interested in this https://en.wikipedia.org/wiki/The_Sand_Reckoner In Archemedes day, it was said the number of grains in a beach was too immense to count or concieve. It was infinite as it were. Archimedes figured it had to be finite and found bounds. And further figured if the solar system was filled with sand that would be have a finite bound and only about $10^{80}$. You have stumbled across another such bounding problem. You are astounded it has a bound but if you think of it, all concepts must. And actually your bound is HUGE. – fleablood Jan 09 '21 at 00:26
  • Not just astounded, but somewhat disappointed that there is, in fact, an upper bound to imagination! – Steve Valliere Jan 09 '21 at 00:28
  • "EDIT what melts my brain is that all possible real and imaginary images across all of space and time, can seemingly be represented by a finite number of images." i can imagine places that are infinite. But because my brain is finite I can't imagine it in detail. And as there are only finitely number of beings in the universe with finite brains the result is finite. ... Everything finite in finite bounds must have some boundaries. – fleablood Jan 09 '21 at 00:31
  • I'm reminded very much of Haruhi Suzumiya's monologue from the anime of the same name. The truth she decided was to (quite literally) make her life interesting. – JMoravitz Jan 09 '21 at 00:32
  • But what would infinity mean in practical contexts. We can, in theory, make things as small as we like (far smaller than we can in actuality with pesky Plank's constant) and make these beast numbers as large as we want and there is an infinite potential but anything actually stated in physical bounds would be finite. You should be glad it's as high as it is. – fleablood Jan 09 '21 at 00:37
  • Also consider that the vast, vast majority of those images look like static. There are far fewer images that actually look like something. – subrosar Jan 09 '21 at 00:53

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