Fresnel Integral
en.wikipedia.org
en.wikipedia.org
Edit: their application to smooth path generation is pretty interesting: https://en.m.wikipedia.org/wiki/Track_transition_curve
Posting random Wikipedia links is about as interesting as posting random dictionary entries. I'd much rather see a short intro or at least comment on the article rather than just a link.
The plot may or may not be pretty, but that's not why it matters in engineering and physics.
>[tanek]: Why, though? Just because it’s pretty?
Isn't that reason enough? <g>
Usually in Math/Science/Physics -- if something is pretty -- then usually it has a lot of purposes/functions around it that are employed by Nature in various ways... One example of this might be Phi, AKA The Golden Ratio: https://www.goldennumber.net/
Both beautiful and purposeful!
But back to Fresnel Integrals!
>[sidlls]: They have applications all over physics.
Indeed they do!
>[exmadscientist]: Strongly agreed. Why are these more interesing than the Airy functions?
Well, I don't know! <g>
I do know that prior to your comment, I didn't know what Airy functions were!
So, here's a link for the folks back home, just tuning in!:
https://en.wikipedia.org/wiki/Airy_function
Note the similar features (at least in part!) of the graph of Fresnel Integrals (https://en.wikipedia.org/wiki/File:Fresnel_Integrals_(Unnorm...) and Airy Functions (https://en.wikipedia.org/wiki/File:Airy_Functions.svg).
It's sort of like, if you took one graph and flipped it over 180 degrees, and cut out the part where the two functions start converging or start diverging, and ignore the units of measure, then the two graphs almost look similar...
So maybe they're related in some way... maybe there's something there...
Airy functions -- seem like they would be very interesting to study alongside Fresnel Integrals!
So, thank you for suggesting them!
The rest of my comments are for everyone in general -- so back to the Fresnel Integrals!
OK, so here's what my intuition tells me...
We've all seen a hologram right?
OK, so a hologram is basically a 2D space -- masquerading as a 3D space.
First question (discovered easily by inverting/reversing the above idea):
Is it possible to have a 3D space -- which is actually a 2D space, in disguise?
?
Sort of like a "reverse-hologram" -- for lack of better terminology...
In other words, is it possible, for an actual, real, 3D space in reality -- to have similar characteristics to a flat, 2D hologram?
?
Well, I for one don't know! <g>
But what I do know is that IF such a thing is possible, IF it is (again, this is highly speculative!) -- then IF so -- then on the 2D holographic plane -- ALL of the 3D information -- must exist!
So now the question is (again, IF such a thing is possible, which is highly speculative!) -- How/where is that INFORMATION -- which is in the 3D view of reality -- encoded into the 2D plane, which is a basically hologram of that view of reality?
The answer for that question, is simply: "I don't know!!!"
But Fresnel Integrals (and Airy Functions!) -- seem like they might have something to offer here -- some way of encoding extra (depth!) information -- into what is basically a 2D frame!
Now IF that's true (and I should point out that we're beyond speculative at this point -- we're in "super-crackpot-they've-gone-to-plaid" land now, just for reference! <g>), then Fresnel Integrals do not just generically have applications all over physics -- but they're also intimately, intimately related to such things as the double-slit experiment, gravity ("A track transition curve, or spiral easement, is a mathematically-calculated curve on a section of highway, or railroad track, in which a straight section changes into a curve. It is designed to prevent sudden changes in lateral (or centripetal) acceleration." -- well, this is information, and this information encoded in a 2D image would determine gravity (gravity = acceleration)), and of course, diffraction -- which is sort of like the measure of how fuzzy distant objects in a 2D image are, depending upon where the camera lens is focused...
So, that's why Fresnel Integrals (and Airy Functions!) -- might be interesting for Future Physicists (and people who are not physicists -- but just like "pretty" things! <g>. I should point out that I am in the latter camp! <g>)
Anyway, since you asked! <g>
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
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>)
See 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-varia...
> So maybe they're related in some way... maybe there's something there...
The two Airy functions, like the two Fresnel integrals, the Bessel functions, sine and cosine, and other pairs of popular functions, are the real and imaginary part of a very smooth complex function of one real variable, with the typical appearance of perfectly interleaved oscillations with the same approximate period and a delay of 1/4 period between the two functions.
While sine and cosine represent a perfect specimen of this behaviour, other functions can have peculiarities: Fresnel integrals oscillate around the asymptotes rather than around 0, Airy functions cease to be approximately periodic around infinity, and so on.
Some excellent comments!
I'm going to highlight them and add to them a little bit -- I hope that's OK with you [my modifications will be in brackets!]:
>The two Airy functions, like the two Fresnel integrals, the Bessel functions, sine and cosine, and other pairs of popular functions,
are the real and imaginary part of a very smooth complex function of one real variable
, with the typical appearance of
perfectly interleaved oscillations
with the
same approximate period
and a delay of 1/4 period [well, some period that doesn't throw the two functions completely out of sync -- at least not immediately!] between the two functions.
While
sine and cosine represent a perfect specimen of this behaviour
[Indeed! A very important and excellent point! (Also cosine is the perfect derivative of sine, negative sine is the perfect derivative of cosine, negative cosine is the perfect derivative of negative sine, and sine is the perfect derivative of negative cosine! In other words, derive or integrate one of those functions 4 times -- and you're back to the original function you started from!]
, other functions can [and do!] have peculiarities [are not quite as mathematically perfect as sin/cos!]: Fresnel integrals oscillate around the asymptotes rather than around 0, Airy functions cease to be approximately periodic [cease to oscillate] around infinity, and so on."
PDS: This leads me to wanting to research the following idea:
Do sin and cos (and by extension, -sin and -cos) -- have a simple common function (or variable, complex or otherwise, as you alluded to) -- from which both (and by later extension, all) of them can be derived?
Or are sin and cos (and by extension, -sin and -cos) sort of "mathematical absolutes" (can't be derived from anything except themselves?) for lack of a better term?
?
Anyway, excellent post!
Tightly wound spirals, like those formed by the Airy functions and the Fresnel integrals, have almost the same shape as exact repetitions of a circle; graphs end up looking very similar.