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Pretty much any pairing function (for a discrete universe) or space-filling curve (for a continuous universe) can be the backing track for holograms. If locali

Pretty much any pairing function (for a discrete universe) or space-filling curve (for a continuous universe) can be the backing track for holograms. If locality preservation is required, then one can use locality preserving pairing functions or curves such as the Z-order curve. All that the holographic principle really implies is that there is redundancy in the state of the universe in part due to the symmetries in the laws of the universe. But there's no implication that the simplification from N dimensions to N-1 dimensions is any easier than recovering the information that went into a black hole from its Hawking radiation. The best we'll probably get is existence proofs or constructive proofs about the boundary/bulk relationship. But I doubt we'll actually have any real calculations on what we're dealing with. https://en.wikipedia.org/wiki/Z-order_curve https://en.wikipedia.org/wiki/Z-order_curve https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikipedia.org/wiki/Space-filling_curve https://en.wikipedia.org/wiki/Pairing_function https://en.wikipedia.org/wiki/Pairing_function

peter_d_sherman 4y ago
Very interesting! I did not know about Pairing Functions prior to your post (again, very interesting!) -- it makes me wonder if there are pairing functions for pairing functions -- in other words, recursive pairing functions For lack of a better term... In other words, does there exist a pairing function (or recursive pairing function, again, for lack of a better term!) -- which can collapse not just N dimensions to N-1 dimensions -- but some high dimensional number N -- down to 1 -- or even 0 (like how simultaneously weird, yet interesting, yet, let's not kid ourselves, "weird" <g>, would something like that be?) >"But there's no implication that the simplification from N dimensions to N-1 dimensions is any easier..." Point taken, and it's a very valid point(!), and I don't mean to offend you by saying what I'm going to say next (you're a nice guy and all!) -- but: I don't care about easy! I care about possible! Nothing that is significantly interesting at a given point in time in Physics -- is ever easy(!) But if something is possible -- or even just on the verge of moving from "previously thought impossible" to moving to "highly improbable and extremely super difficult to do, if it could be done at all" but not outright "impossible" -- then sign me up! I want to learn more! So, in conclusion, you are right, you are completely correct(!) -- There is no implication that the simplification from N dimensions to N-1 dimensions is any easier than recovering the information that went into a black hole from its Hawking radiation! But... wouldn't it be interesting to do some research around that idea? ? I mean, the absolute worst (the absolute worst!) that could happen in undertaking such research -- is that other questions, other entirely different areas of future research -- might be discovered in that undertaking! It would be sort of like if someone wanted to win the Nobel Prize in Physics for research in some complex subject matter, couldn't win the Nobel Prize in Physics for that research because the problems were so complex that they couldn't be solved -- but as a result of that research, new research questions in all sorts of other diverse areas of science became apparent, as a result of those questions, new research into those areas was performed, new breakthroughs were had, and the guy who originally wanted to win the Nobel Prize in Physics -- wound up winning the Nobel Prize in Biology(!) -- or got some other great reward in life (invented something interesting and became a Billionaire?) -- as a result! In other words, the scientific research equivalent -- of a startup "pivot"... Anyway, some interesting links, and you are right about it being extremely difficult to recover information that went into a black hole via its Hawking radiation... But if someone had already accomplished that... then it would be no fun -- to try to figure out how to do it... In other words, if someone already did it... then it wouldn't be a challenge anymore, would it? I mean, proving Fermat's Last Theorem -- is no longer challenging anymore, now that it has been proven, is it? (Thanks a lot, Andrew Wiles! <g>)
- goldenkey 4y ago
  I like your enthusiasm! Yes, you can recursively apply pairing functions to generate N -> 1 dimensional pairing functions. See https://en.wikipedia.org/wiki/Fueter%E2%80%93P%C3%B3lya_theorem#Higher_dimensions https://en.wikipedia.org/wiki/Fueter%E2%80%93P%C3%B3lya_theo... These pairing functions can be used to sample from configuration/state spaces that would be hard to sample from otherwise -- for example, stars and bars problems: https://churchofthought.org/blog/2019/07/24/an-ordered-variant-of-the-cantor-pairing-function/ https://churchofthought.org/blog/2019/07/24/an-ordered-varia...

peter_d_sherman 4y ago

Very interesting! I did not know about Pairing Functions prior to your post (again, very interesting!) -- it makes me wonder if there are pairing functions for pairing functions -- in other words, recursive pairing functions For lack of a better term... In other words, does there exist a pairing function (or recursive pairing function, again, for lack of a better term!) -- which can collapse not just N dimensions to N-1 dimensions -- but some high dimensional number N -- down to 1 -- or even 0 (like how simultaneously weird, yet interesting, yet, let's not kid ourselves, "weird" <g>, would something like that be?) >"But there's no implication that the simplification from N dimensions to N-1 dimensions is any easier..." Point taken, and it's a very valid point(!), and I don't mean to offend you by saying what I'm going to say next (you're a nice guy and all!) -- but: I don't care about easy! I care about possible! Nothing that is significantly interesting at a given point in time in Physics -- is ever easy(!) But if something is possible -- or even just on the verge of moving from "previously thought impossible" to moving to "highly improbable and extremely super difficult to do, if it could be done at all" but not outright "impossible" -- then sign me up! I want to learn more! So, in conclusion, you are right, you are completely correct(!) -- There is no implication that the simplification from N dimensions to N-1 dimensions is any easier than recovering the information that went into a black hole from its Hawking radiation! But... wouldn't it be interesting to do some research around that idea? ? I mean, the absolute worst (the absolute worst!) that could happen in undertaking such research -- is that other questions, other entirely different areas of future research -- might be discovered in that undertaking! It would be sort of like if someone wanted to win the Nobel Prize in Physics for research in some complex subject matter, couldn't win the Nobel Prize in Physics for that research because the problems were so complex that they couldn't be solved -- but as a result of that research, new research questions in all sorts of other diverse areas of science became apparent, as a result of those questions, new research into those areas was performed, new breakthroughs were had, and the guy who originally wanted to win the Nobel Prize in Physics -- wound up winning the Nobel Prize in Biology(!) -- or got some other great reward in life (invented something interesting and became a Billionaire?) -- as a result! In other words, the scientific research equivalent -- of a startup "pivot"... Anyway, some interesting links, and you are right about it being extremely difficult to recover information that went into a black hole via its Hawking radiation... But if someone had already accomplished that... then it would be no fun -- to try to figure out how to do it... In other words, if someone already did it... then it wouldn't be a challenge anymore, would it? I mean, proving Fermat's Last Theorem -- is no longer challenging anymore, now that it has been proven, is it? (Thanks a lot, Andrew Wiles! <g>)

goldenkey 4y ago
I like your enthusiasm! Yes, you can recursively apply pairing functions to generate N -> 1 dimensional pairing functions. See https://en.wikipedia.org/wiki/Fueter%E2%80%93P%C3%B3lya_theorem#Higher_dimensions https://en.wikipedia.org/wiki/Fueter%E2%80%93P%C3%B3lya_theo... These pairing functions can be used to sample from configuration/state spaces that would be hard to sample from otherwise -- for example, stars and bars problems: https://churchofthought.org/blog/2019/07/24/an-ordered-variant-of-the-cantor-pairing-function/ https://churchofthought.org/blog/2019/07/24/an-ordered-varia...