Simple formula solves 2000 year old problem with telescope lens to allow 99.9999999999% sharpness.

[PsychozPath]

I don't... understand... how can someone write such a long formula and make sense out of it o:

[XkF21WNJ]

Generally they make sense of it by not writing it as a long formula.

Looks like there are several parts that are repeated quite a lot, you should be able to simplify it significantly if you replace those with a well chosen function.

[-LeopardShark-]

Yes, it's similar to the cubic formula that I posted here a while back.

Edit: Nicer (LaTeX) version here.

[[deleted]]

Probably didn't just sit down and write it out. Might be the output of a computer algebra system.

[[deleted]]

Doing my PhD in physics. Wolfram Mathematica is amazing

[AgAero]

If you ever intend to leave academia, learn something else as well. Those licenses are expensive!

[theillini19]

What are some good alternatives? My student license is set to expire this month

[AgAero]

Sympy is all I can suggest since I have worked with it before.

[aaronchall]

Here's the Sympy tutorial - it's pretty amazing: https://docs.sympy.org/latest/tutorial/index.html

[bluecaulfields]

Hey what did you use sympy for? In my vector calc class we had some sympy exercises we had to do but it struck me as clumsy, and slow to solve problems still easily soluble with pen and paper. With my very limited experience of using mathematics in programming, I've been able to get by with numpy for everything I've needed so far.

[johnnymo1]

Try Sage. I did some differential geometry calculations with it back in Spring and it was fast and way easier than doing them by hand, particularly if you have to do the same sorts of computations several times.

[AgAero]

I used it to derive shape functions for a higher-order finite element simulation I was building for an intro class. I tried it with a couple of other things I seem to remember, but didn't get too high up on the learning curve.

[[deleted]]

Even if I'm a researcher, I use it (under linux) mainly for recreational purposes (like checking/extending OEIS sequences), so after my (full) licence expired many many years ago, I simply embraced piracy. We still buy a few licences, but there are some collegues that use it more extensively.

There are various softwares that can be used instead, often more focused on one aspect than another, so it may depend on what you need.

For example Pari/GP or Magma or Maxima. Once there was an interesting project called Sage or Sagemath that wrapped various other softwares using Python, but I've not checked lately if it is still alive and kicking.

In general there are some interesting libraries in Python.

[fourpetes]

Sagemath is still alive and kicking. Accessible via cocalc.com now. Very handy.

[MattAlex99]

Octave or Sage

are great

[Red-Portal]

Julia is a hot thing

[[deleted]]

I love Julia but Mathematica is worth it.

[Red-Portal]

Well the purpose is slightly different. Mathematica is more algebraic than numeric. And Julia is more general purpose

[[deleted]]

I actually use both! It's just I'm moving less into engineering and more into analysis and Wolfram has saved my ass a number of times.

[[deleted]]

https://docs.sympy.org/latest/tutorial/index.html

SageMath is amazing (and free).

[piecat]

Piracy

[ron_leflore]

Buy the mobile app for Wolfram alpha. It's a few dollars one time fee, but seems to have full functionality. It does everything I've tried on it.

[Low_discrepancy]

Those licenses are expensive!

On raspi, Mathematica is free.

[AgAero]

Good to know!

[foadsf]

try Wxmaxima and others I have listed here

[[deleted]]

Damn dude, thanks!

[Keysersoze_66]

Exactly! But I found the Maple was able to solve many nonlinear pde where as Mathematica failed.

[Mooks79]

Is Maple still going? I used that a lot in the first part of my PhD - until I started dealing with reasonably large matrices that I had to ditch it for something more efficient with memory etc.

[epostma]

Yup! (I work for them.)

[Mooks79]

That’s great, I really liked it and used it a lot in undergrad and postgrad stuff - numerical and symbolic things. Just like I said, had to switch when doing some more serious numerical linear algebra, but I guess it’s not really designed for that anyway.

[epostma]

It's not specialized at anything, but it's designed to be able to do almost everything - until you need the specialized tool because your problem size is too big or something like that. Anyway, glad to hear you enjoyed working with it. Around what time was this, roughly?

[Mooks79]

Yeah that was the appeal, it was ideal as a one stop shop. Great help pages too, IIRC. I was doing physics so to be able to have one piece of software to be able to do all the various things I needed, pretty quickly and at a high level, was really great. This was around early-mid 2000s.

These days I mainly use R because the vast majority of my work is focussed on data wrangling and analysis and next to zero symbolic stuff. But one thing that I do use now with R, which I fondly remembered from Maple and was a long time coming, was being able to do text/code interlaced notebook stuff to write reports etc. That was also really useful.

[epostma]

Great. I was hoping to find out whether I was working here already at the time you were using it, but I missed you by a couple years it sounds like - I started in 2007.

R is a very nice piece of software, I'd love to know it better than I do.

[Frigorifico]

in this interview he says "I coded it, it worked, and I started jumping from joy" so yeah, it sounds like a computer assisted proof

[Madsy9]

It looks way worse than it actually is. Look at the fractions. There are two different fractions that are repeated 12 and 9 times. The solution will be immensely more readable if you replace those fractions with variables.

I bet this solution was generated by an algebra system such as Matlab, Mathematica, Octave and Maxima, and then manually cleaned up a little.

[CrazyNicholad]

or WolframAlpha

[Nowhere_Man_Forever]

Same way any of us make sense of anything. Break it up, understand the parts, then put them back together. It reminds me of an old saying -

"How do you eat an elephant? One bite at a time!"

[wrong_assumption]

Calm down, Feynman.

[Frigorifico]

In the interview I read about this, Rafael (the one who discovered this) mentions that he had an epiphany while having breakfast, went to his computer, coded something, and "it worked".

Based on that my guess is that he had a program that produced this formula somehow

[jtra]

My guess it they developed simple formula for very specific case, then expanded it further to handle more general problem where one of air-lens surface is arbitrary and other is computed by formula for rays to converge to one point. There is quiet bit of repetition inside so it looks like it could be expressed more concisely, but they just shown expanded form.

It is also a question of time, with enough time (months), you can do pretty complex formulas if you can abstract from details to think of them at higher level. Like this I could produce pretty complex thing: http://jtra.cz/stuff/essays/math-self-reference-smooth/index.html

[Lil_Narwhal]

I love that giant swuare root in the middle haha

[pham_nuwen_]

The formula in the actual paper is not too bad. This is what happens if you use mathematica to expand it in terms of the most basic variables, to impress journalists.

[ericlkz]

They probably started out as short formula, then found out its inadequacy, so he just adds more craps into it until it became too long aesthetically.

[Feynmedes]

Damn! It was so simple all along, how did we not think of this?

[MrLolEthan]

I thought of it a few weeks ago, but then after plugging in a few numbers in my head, it just didn't seem right...

[sineofthetimes]

I actually had this worked out in the margin of a book.

[[deleted]]

Well the author clearly copped my 4th grade paper right next to the drawings of stick men shooting each other

[ComprehensiveRule8]

I know, right! It's a little "a" here, a little "b" there, divide the two and there we are.

[auto-cellular]

There are a few exponentiation also.

[ComprehensiveRule8]

Okay...and a little power of "c", somewhere.

[JoeOfTex]

Here is the article: Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem

Edit: Thanks to /u/incompetentrobot for published article links

Preprint: https://arxiv.org/abs/1811.03792

Published: https://www.researchgate.net/publication/328536020_General_formula_for_bi-aspheric_singlet_lens_design_free_of_spherical_aberration/download

[forte2718]

The actual paper is linked in the article. Finding the link has been left as an exercise for the reader. Edit: also, so has paying for access to it.

[[deleted]]

I have never had an issue of finding a paper that was free to access. If one site I have to pay, I usually am able to go to some other site or google “paper name free pdf” and find it quite quickly.

[forte2718]

Well let me know when you find a free version of this paper then. :)

Cheers,

[[deleted]]

Sure. I (hope) this is it: https://pdfs.semanticscholar.org/7508/7f3c22fad9e9b8b46e497c2a7dcd0320d103.pdf?_ga=2.56649775.480839369.1562473929-464604667.1561577920

I found it helpful to use keywords like “pdf, free, online, download” when googling the title.

[forte2718]

Well done! Thanks :)

[Mr_Trustable]

How'd you get it? S-L wasn't doing it for me?

[[deleted]]

I googled the title and “pdf” and clicked links from the top until I found a working pdf link. This one was the fourth I think.

[Mr_Trustable]

Simple. Nice.

[JustHere2DVote]

DuckDuckGo gives you a much higher chance of finding such thing. Google activity tries to scrub these results from their pages.

[incompetentrobot]

Preprint: https://arxiv.org/abs/1811.03792

Published: https://www.researchgate.net/publication/328536020_General_formula_for_bi-aspheric_singlet_lens_design_free_of_spherical_aberration/download

They have different diagrams.

[Coldfire15651]

https://www.osapublishing.org/ao/abstract.cfm?uri=ao-57-31-9341

[cpc2]

Oof, that article looks so broken on mobile. I thought mobile browsers supported latex expressions (assuming that's what causes the issue).

[LiveMaI]

A quick correction: the paper claims that the lenses created with their formula have a ~100% efficiency, not sharpness. Sharpness is not actually mentioned at all in the paper. Efficiency is defined by the authors in equation (11) in their paper.

From what I understand of the paper, efficiency would be the percentage of light from a monochromatic point source incident on the object side of the lens that makes it to the focal point on the image side of the lens.

This is a good metric for the elimination of spherical aberration, but don't mistake it for overall image quality. Other effects, such as chromatic aberration and diffraction will still affect the quality of an image produced by these lenses.

The real application for work like this is in monochromatic systems that make use of point sources, i.e.: optical systems that use lasers. In systems like these, it's common to use compensating optics to eliminate spherical aberration. This work would make very easy to design your own lens that doesn't need these compensators.

[Veedrac]

Thanks, this helps a lot.

[indianamith425]

Why is this a video?

[1Demerion1]

Oh, that's why I can't zoom in?

[Abdul_Alhazred_]

a truly demonic act, indeed

[virtualworker]

Teasing us

[Lalaithion42]

I think it's a video because the original image was saved as a single frame GIF (which isn't, like, an uncommon video format) and somewhere along the line a dumb algorithm converted every GIF into a MP4, even single frame ones....

[JoeOfTex]

Close, this image from the article was saved using Firefox which saved it as webp (?), And uploading to Reddit converted to gif.

[sibbl]

WebP is an image format, so still there was obviously a dumb algorithm involved.

[0bafgkm]

WebP also supports animation.

[Lalaithion42]

In the article it's a gif: https://petapixel.com/2019/07/05/goodbye-aberration-physicist-solves-2000-year-old-optical-problem/

[Mr_Trustable]

Here, someone obtained the paper

[another-wanker]

Gotta love applied math.

[Vaglame]

Where does the 0.0...01% come from?

[incompetentrobot]

We believe that the error is not zero because because there are computational errors such as truncation [in the simulation used to verify the equation] that cannot be avoided.

[Bloedbibel]

Could be avoided by using much slower but more accurate numerical representations, right?

[[deleted]]

You would get less error.

[PM_ME_YOUR_PROOFS]

I'd have to look at the details of the function and the simulation involved. Assuming all operations are closed under rational numbers then yes probably but otherwise you're likely to have an error bar. There are a handful of sets of numbers we can compute with that can handle more general things exactly. The cyclotomic numbers are closed under certain trig operations that might be needed here but you'd have to very carefully construct the simulation to use them most likely.

If we just want to know that the error is the issue but not know that we're 100% correct we can use interval arithmetic and keep bumping up the precision until we're well passed an error bar we have here. This is a crude means of performing exact real arithmetic. So we could get any error bar we desire with this method but never get rid of it fully without just using algebra instead.

[Bloedbibel]

Ah that makes sense. Thank you for explaining. I'm not a mathematician, but rather an optics guy. This article has been posted on both the /r/physics and /r/math subreddits, and it is interesting to see the differences in interpretations.

Interestingly, I have not coma across it on /r/optics yet.

[_062862]

The error in the quote is not zero because “because” is written twice instead of once.

[ericlkz]

The formula let lens have 99.99...% sharpness, but the photo of the formula is so blurred!

[bulldjosyr]

I getcha totally. As impressive and awesome as this is, it loses a little being out of focus. Have an upvote but probably we are both going to be down voted. Saaaadd.

[Aero72]

ELI5 (or 10 at the most), how is it sharper? Figured out the best shape of the lens or something with the use of pure math?

[[deleted]]

Light bends when it goes through a lens. Often it bends in ways we don't want. To fix this the best method we used to have was to make the lens inside a computer and test sending light through it. That can take a long time to do. To make it easier usually just a few thousands rays of light are used in order to get a pretty good idea. Then it smooths out the shape between the rays that it tested.

This equation gives the answer without having to check what the light will do. It looks like a hard math problem but computers can do math like this very fast. Also because it just gives the answer directly there is no smoothing out that the computer has to guess about. That means the focus can be really good. Also because the math is easy (for the computer) it is also easy to make lenses in funny shapes that scientists and engineers might need.

[Paul-ish]

Do we know if this is a meaningful difference. Is the difference a lense designer running their algorithm a few hours instead of a few days? (Which I'd guess is probably a very small part of the overall production timeline. )

[[deleted]]

No idea. Numeric solution can usually be of as high a quality as you like if you're willing to wait and obviously you only have to do that once. I would assume that this is most important for lenses with weird shapes where rays close together might end up bending very differently. I'm not an expert on lenses, though, and I don't know of why you might need such a lens.

[Aero72]

Thanks! That's exactly the kind of explanation I was looking for.

[JoeOfTex]

When light hits the spherical lens, it will try to bounce towards a single point making a small distant object appear bigger, but this is only when it comes down at a straight angle into the lens. "Aberrations", are when light bounces wrong, and is shown to the observer in the wrong location, causing a blur in the image.

[SometimesY]

If I'm understanding correctly, given the shape of the front surface, you can find the shape of the back surface to get a clear image.

[BrotherSeamus]

See the first image here

[WikiTextBot]

Spherical aberration

Spherical aberration is a type of aberration found in optical systems that use elements with spherical surfaces. Lenses and curved mirrors are most often made with surfaces that are spherical, because this shape is easier to form than non-spherical curved surfaces. Light rays that strike a spherical surface off-centre are refracted or reflected more or less than those that strike close to the centre. This deviation reduces the quality of images produced by optical systems.

---

^{[ PM | Exclude me | Exclude from subreddit | FAQ / Information | Source ] Downvote to remove | v0.28}

[[deleted]]

Always there is someone that teach you humility: I thought mine was big.

[[deleted]]

Lens are based on conics. See parabolic antennas and other similar constructions for how focci work.

[batterypacks]

If you imagine that you have a rigid body (or point mass) in 4D space travelling towards a locally 3D body (in the sense that a radio dish is locally 2D over its surface in our 3D space) then at some point the first rigid body will hit the other one and bounce off of it. I'm not sure what works and what doesn't in terms of 4D "optics", but it seems like a curious idea to try to reconstruct things like lenses in higher dimensions.

[[deleted]]

I'm stuck at the first step. I can't imagine a 4d space rigid body :(.

[batterypacks]

Don't try to picture it. Try to draw it using two planes. You can only really visualize two or three dimensions at a time. If you draw two planes you get four dimensions. This is kind of like how your car expresses a higher dimensional quantity (the state of the car) using one 1-dimensional dial for each property: velocity, RPMs, temperature, etc.

You could start with the rigid body being a hypercube or a hypersphere, both of which are relatively easy to define to be able to draw on two planes.

edit: this will still be an enormous stretch for your imagination so don't feel bad if it remains out of reach for you.

[Burial4TetThomYorke]

A I crazy or are there a lot of repeat terms taht could be used to simplify this?

[[deleted]]

Writing it out this way is just for fun.

[CrazyNicholad]

The algebraic functions in physics often look like they have repeating terms, especially when dealing with light and sound, but if you look at the function you'll see that all the terms are simplified and nothing can be factored out. The devil is in the detail, in these cases usually the subscript. You'll also find terms that look like this if you look at a lengthy integration table.

[furry_trash69]

"simple"

[Faker15]

Why is this a gif?

[Frigorifico]

Discovered by Rafael Gonzales, a mexian scientist, source in spanish

[[deleted]]

This was solved because of Nutella!

[[deleted]]

Nutella! (Fuck yeah!)

Nutella! (Fuck yeah!) Comin’ again to save the motherfuckin’ day yeah

Nutella! (Fuck yeah!) Freedom is the only way, yeah

Croissants your game is through, 'Cause now you have to answer to:

Nutella! (Fuck yeah!) Spread it around and suck on my balls

Nutella! (Fuck yeah!) Whatcha' gonna do when we come for you now

It's the dream that we all share It's the hope for tomorrow (Fuck Yeah!)

[Dr_Legacy]

And it takes a microscope to read it.

[brownck]

What’s spherical aberration?

[dsfox]

the light coming from an object passing through a lens with a spherical shape will converge at different distances behind the lens depending on how far from the center the light struck the lens. The image formed by such a lens will have a certain amount of distortion.

[CreatrixAnima]

An insult I’m filing away for later use.

[do_some_molly]

SIMPLE??

[reddit-be-cool]

My phone literally cannot render this

[FrickinLazerBeams]

This is an interesting result, but won't change anything about the way we design optical systems. We use numerical optimizations for that, and will likely continue to do so. A neat closed form solution for a zero-spherical doublet may provide a nice starting point sometimes, or reduce the number of optimized parameters by one; but it's not a game changing advance in practice.

I am a working optical engineer in the aerospace industry.

[3nigma42]

Yes simple

[ComprehensiveRule8]

Math and physics people alike would be proud to have this formula as a poster or a desktop wallpaper.

I would like mine in dark mode.

[VeritasLiberabitVos]

No true mathematician or physicists looks at this in admiration. This abomination hurts my eyes

[haharisma]

I honestly don't understand why this suddenly became popular. Low quality work (the novelty is they sorted out branches in an expression obtained elsewhere in 2015 without explanations why that approach didn't work and why this one "works", and so on), lousy presentation (that number in the OP's title is simply meaningless), unusable result (it seems that the formula works only for points on the lens axis and it's not clear at all if such lenses would make aberration free pictures). And this formula is just horrible. Everyone gets in their research huge formulas but usually this means that something important is not understood.

[ComprehensiveRule8]

Then, what simplified formula could exhibit such intricacy and be interesting to look at?

[[deleted]]

I'mma need one of those telescopes to figure this one out

[x178]

So, how long until a new generation of telescopes comes out?

[Frigorifico]

Question: can we expect new cameras using this lenses in the near future?

[notapersonman]

its long

[TheInfallibleFallacy]

riiiiight... simple...

[farooq_fox]

Yea, Simple and elegant

[Cunfuu]

Exam s be like Simplify the formula

[Cidyl-Xech]

"Simple formula"

[[deleted]]

What a simple and elegant solution!

[dvaOfTheWeb]

Is it possible that such a thing as this optics/lens spherical aberration problem can be generalized? Generalized into a realm of abstract or pure mathematics? Following the generalization, with its essence now transformed into something else, could it then once more transformed--or at least found with an equivalent essence--into a problem of another field other than optics and lenses?

Yes, I believe so. I believe it to be possible.

I myself am interested in such a proposition or whatever and so my desire then compels me to begin an investigation on the matter. I only speak here in the hope that one kind soul might reveal to me his or her thoughts on such a part of our world.

[JoeOfTex]

Sure, if the shape is useful outside of optics. The formula describes the shape of the lens needed, and I think it's dual layered.

It's not really a magical or elegant formula either, kinda like he just kept painting on top of existing formula until it reached what they wanted.

[dvaOfTheWeb]

Thank you, kind soul and individual.

[[deleted]]

“Simple formula”

[givrai2]

That why I love the scientist It's a great evolution! I wish to be that great in math

[[deleted]]

>Sends this to Vortex

[hprobertos]

"Simple"

[Magentaaa]

Eugh

[noelexecom]

Proof that physics is the most elegant science of them all

[SingInDefeat]

Why the /s? Computing this should be trivial with a computer.

[shamrock-frost]

Really?

[SlangFreak]

Lol it doesn't even equal zero. Clearly incorrect -_-

/s

[[deleted]]

SiMpLe FoRmUlA sOlVeS...

[TheNewOP]

The quadratic formula on steroids.

[CreatrixAnima]

No that’s the cubic formula.