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Goodbye Aberration: Physicist Solves 2,000-Year-Old Optical Problem (petapixel.com)
845 points by zafka 17 days ago | hide | past | web | favorite | 207 comments



There are various people speculating on the economic significance of this solution, which, to me, is rather missing the point. It's like measuring the significance of the excavation of Tutankhamen's† tomb by the tourist revenues of museums. The point of economics is that it keeps us alive so we can do math and also think in other ways; the point of consciousness is not to make money.

I don't think there is any economic significance. They found a closed-form formula, not a manufacturing process. They verified that it produces numerically correct results to 12 significant figures, but typical lens grinding is only accurate to about 100 nm; if your lens is 10 mm thick, that's an error in the 5th significant figure of any coordinate. Calculating a numerical solution to the Wasserman–Wolf problem to 5 significant figures is straightforward, and you could probably do it by hand if you didn't have a computer (although that would involve significant economic cost). In fact, it's not that hard to calculate it to 14 significant figures. The achievement is finding a closed-form solution rather than an iterative numerical approximation.

† Or Tutankhaten, as we used to call him.


> The point of economics is that it keeps us alive so we can do math

The most hacker news comment I read all year


It reminded me of a quote from Stephen King [0] "life is not a support system for art; it's the other way around".

[0] beautifully rendered as a short comic here https://zenpencils.com/comic/king/


That's a beautiful comic, thanks for sharing.


I love it, though. Perhaps the more general statement “the point of economics is that it keeps us alive so we can appreciate beauty” would be more universally agreed to.


The specificity gives it a funny edge though. If I were a math student I'd print it and put it on my wall.


That seems like a circular dependency...

Can you do economics without math?


Modern macroeconomists will disagree.


[flagged]


It's well known that Soviet Union didn't really have economics

Huh? https://en.wikipedia.org/wiki/Soviet-type_economic_planning


I notice you don't seem to have dared to say you disagree — would you say that the point of consciousness is to make money?


There's more to life than either math or money, haha. I think a lot of people might fit something in there about friends, family, or community. Spiritual growth maybe?

There's options.


I venture to say that spiritual growth is part of consciousness — that is, it's "math and thinking in other ways", which is of course what I said above, not "math".

As for friends, family, and community, what is it that gives friendship value, if not your conscious experience of the friendship, and your friend's? Could you coherently call something a good friendship if neither friend enjoys it or improves their thinking from it? The same questions generalize to family and community, though in more complex ways.


Although I agree with your main point (it's closer to the truth that the point of economics is so that we can do mathematics, than the other way around), perhaps it can be refined to incorporate the fact that not all experience is cognitive; not all enjoyment is intellectual. The feeling of joy/bliss/whatever from being with friends and family, of doing a job well, of leaving a legacy, of having good health, dignity, curiosity, material comfort, relationships, spiritual growth, having a good “life story” for oneself / serving some higher purpose etc (everything on “Maslow's hierarchy“), are not always cognitive or even conscious in nature, though some (like curiosity) tend to be.

The Greeks used the word eudaimonia for this highest / all-encompassing utility function (the experience that everything is in service of: the “point”); in Indian thought it's called ānanda. But yeah, making money is only a means to it, and not the point. (Even this understanding can get clouded. In Indian thought, “religion” only posits a higher ānanda that can be obtained by experience of the divine, without denying the everyday sorts of joy that resemble it. In Western thought, influenced by the monotheistic religions with their opposition between true and false, the fact that neither money nor comfort is the highest good gets reflected in ideas like “money is the root of all evil” or “Happiness versus Meaning” that tend to vilify them in order to counter our impulses towards them, rather than recognize them as being partial means to some components of happiness. It's fine; whatever works I guess.)

PS: Totally offtopic, but thanks for your transcript of Knuth's Web of Stories interview!


Glad you enjoyed it!


No. But there are things besides money and math. Like photography!


I thought the point of photography was primarily so that you (people, in general, not necessarily just you individually) could look at the photos, i.e., experience them consciously, and secondarily to consciously experience the process of photography itself. But perhaps you have a different reason for considering photography worthwhile, other than the conscious experience of the photographs and of the photography process? Or do you consider photography to be a good-in-itself, even if nobody ever sees the photos or experiences the process of taking them?


> i.e., experience them consciously, and secondarily to consciously experience the process of photography itself. > other than the conscious experience of the photographs and of the photography process?

Is there a way of unconsciously experiencing these things?


None that I'm conscious of.


Hmm, I'm uncomfortable with the conflation of consciousness, awareness, sapience, and cognition.


Also, most crucially, experience — qualia. How would you formulate the relations among these five possibly different things?


Sure, various systems can use photographs to automatically do useful stuff like pea sorting even if a person is never aware of individual photos.


That sounds like a means to making money, not a viable third alternative.


It’s not just cheaper products, self driving cars are another advantage to machines taking and interpreting their own photos.

Extend this to asteroid tracking and it may be the difference between human survival and extinction.


Again, you are offering ways that photography can be a useful means to an end that we previously accept as good for other reasons, not ways that photography can be a good in itself. Suppose the humans were already extinct; would you then consider it a good in itself for machines to be taking and interpreting photos? Would you set up a video camera with a solar panel in a park, endlessly taking 60 photos per second, then deleting them, because photography is good even if nobody looks at it and it produces nothing else outside of itself?

I am fond of the humans and so I would like them to survive, but only because that is a means for them to be conscious, at least in some cases.


Efficiency can turn something from useless to useful, which means the outcome is not inherently useful.

Thus if photography is required and the only thing that makes it possible then it as a means becomes and end to it's self.

Consider, in a pure nitrogen environment animals will suffocate under no great distress. The goal has become separated from the result.


You seem to be unclear on the distinction between means and ends, which is to say, instrumental and terminal values.


A legacy that lives past humanity seems better than one that does not. If something views or does not view that legacy is effectively irrelevant as humans would never know.

PS: As to your final central point, some feel keeping a loved one alive even if they never recover consciousness is a net good.


Higher quality affordable lenses yield more conscious enjoyment of photographs. Hence the question of economic applicability.


Your comment seems to be an attempt to answer my questions, but I can't figure out how it relates to them. Do you have a different reason for considering photography worthwhile, other than the conscious experience of the photographs and of the photography process? Or do you consider photography to be a good-in-itself, even if nobody ever sees the photos or experiences the process of taking them?


Photography can be likened to painting, it's a form of art.


I just like the smell of the chemicals (yes, I know).


> It's like measuring the significance of the excavation of Tutankhamen's† tomb

> † Or Tutankhaten, as we used to call him.

This is a rare case where the name change is original to the foreign subject. Akhenaten, the heretic Pharaoh, threw out the traditional religion of Egypt and replaced it with monotheistic worship of the aten, the sun-disc. This didn't go over well -- after Akhenaten died, the traditional system reestablished itself, various records were "corrected", and Tutankhaten's name was changed, during his own lifetime, to something more traditional. Unlike his father, Tutankhamen didn't have the political power to maintain the heresy.


Yeah, I meant it's what we called him when he was a kid.


>The point of economics is that it keeps us alive so we can do math

Well yes, life continuity is not something to subestimate. Thinking derives from perceiving and acting in the world. I agree that the achievement was finding the _closed-form solution_. But it will have a beautiful impact in wealth creation, now and in the future, no doubt. Can't be seen now and who knows when and to what extent this could disrupt optics for the better. Maybe not too much or maybe four decades from now this make possible to start using devices that we can't recognize now. Holodeck, yeah I'm watching you!


I can't see some of the other comments so maybe someone else has put this out there, but economics Is definitely one of those topics that are interesting and worth thinking about. It's not (just) about money, its about the distribution of resources and the structures that facilitate that distribution. Economic theory certainly can have applications beyond informing world economic policy, not to mention that informing policy is also a good idea.

With respect to this solution, I bet it will be great for astronomy, especially superpowered space telescopes that need to squeeze every bit of accuracy out of their projected lifetime. They might also improve optical telescopes but I'm not sure how much there is to improve before running into physical limits. I can't think of any other big impacts off the top of my head, but I bet there are others.

Not sure how much of an economic impact this result will ultimately have, and I agree that that's missing the point here, but in general I think asking about the potential economic impact of scientific/technological advances is a prudent line of questioning.


Couldn't you use the formula to create a correction algorithm?


A correction algorithm will probably rely more on the shape of the lens than the shape of an 'ideal' lens.

A single image also won't technically contain enough information to even allow you to reconstruct the image, but with sparse reconstruction you might be able to get it done.

In fact I'm fairly sure people have tried to do exactly that, and phone manufacturers seem pretty willing to throw software at the problem as well, although the latter don't really care much for the method as long as it looks nice.


The economy is, and always has been, the society.

I get that your comments are playful, at least in part, but the thing that makes discoveries worthwhile is usually getting them in the hands of mere mortals.


Since they put the paper on the arXiv, any mere mortal with internet access can download and read it, and some of them will find that worthwhile.

The economy is an important part of society, but only part of it.


arXiv works because of a combination of abundance and game theory; it's lovely that we all have access to such a tremendous resource, but it's not as though it gets to live "outside" the economy.


This is phrased as if it were a disagreement with something I have said. Why?


It might be useful for a massive scale gravitational lens in some post-scarcity future :)

It's still incredibly cool that there are solvable problems of this sort still put there.


Next question: give a formula that describes the best lens taking into account manufacturing tolerance on the lens faces.


> The point of economics is that it keeps us alive so we can do math and also think in other ways; the point of consciousness is not to make money.

The point of consciousness absolutely is to make money - or rather, to improve the organism's efficiency and thereby survivability by making it capable of introspection and self-optimisation.


It is true that evolution optimizes organisms' efficiency and survivability. Chemical reactions, similarly, optimize the potential energy of assortments of compounds, and gravity optimizes the potential energy of assortments of masses. Would you therefore say that the point of an iron bridge in air is to rust and collapse?

An individual human consciousness, its genes, and its memes have different goals which sometimes come into conflict. I suggest you choose sides in these conflicts, and not against yourself.


> After months of working on solving the problem, Rafael González recalls, “I remember one morning I was making myself a slice of bread with Nutella, when suddenly, I said out loud: Mothers! It is there!”

> He then ran to his computer and started programming the idea. When he executed the solution and saw that it worked, he says he jumped all over the place. It is unclear whether he finished eating the Nutella bread.

This is my favorite quote from the article. Soon to be the most famous slice of Nutella bread?


Gave me shivers reading it. Beautiful example of the incredible mystery of the human mind.

Poof! And the solution materialized. Of course, all the groundwork had been laid. All the neurons primed. Just incredible.

And that formula. Damn.


Isn’t that how most of us are solving complex problems? Building context and then walking away from the desk for a walk, or perhaps while we’re in the shower in the morning or at night, not focused on the problem but our brain working on it in the background until it is brought forward to our consciousness?


There are many similar stories. Famously Tesla's recollection of himself "inventing the inductive motor" while walking in some garden in Hungary, and the discovery of the Benzene ring.


I often do this when I need a good solution to a problem:

Work hard on it for a while, then leave it for one-two days. More often then not this give a way better end result.


That was my thought. Why is it so damn complex? You’d think the solution would be simpler.


Sometimes complicated things are complicated.


the full equation for general relativity would take many pages to print, but Einstein notation makes it much shorter looking. This was not the case here.


Probably looks a lot simpler in A-normal form, with intermediates factored out.


> You’d think the solution would be simpler.

Why would I think that?


It is. Some parts are repeated, the one which caught my eye was (zₐ’(rₐ)²+1)


I mean, if it was simpler it wouldn't have taken 2000 years.


I don't see the correlation here. Is every equation henceforth discovered guaranteed to be more complex than any before it? I hope not.


That formula is just a Mathematica dump. If you read the paper, which is unfortunately paywalled, you'll see the result is the solution to a complex system of equations, which are far easier to digest than the result you see there.


This seems to be a version of it: https://arxiv.org/abs/1811.03792


> Poof! And the solution

...snapped into focus


mind-blown by that formula

amazing what people think of


It wasn't thought of, it was calculated using a computer algebra system. Such systems are very powerful, but tend to be fairly bad at writing things nicely and simplifying, so they often produce insanely big and complicated formulas.


This raises the perennial question of whether product placement is a legitimate source of funding for fundamental scientific research.

After placing an ice-cube in my delicious glass of refreshing coca-cola brand carbonated beverage, I yelled Eureka! and jumped out of the bath.

Is that really how it happened? Is it not inconsistent with the pursuit of truth?


It's really natural, on first mention, to refer to something using a member of the degenerate set of its most unique identifier, right? From that point on, to avoid repetition, mix in antecedents or generic names.

Brands exploit this mechanic in language so that any time a, thereto unknown, good has to be addressed, that identification become advertising. It shifts from being a unique identification of an object/item/thing to a conjuring of the ethos/identity that contextualizes that good as being different from all the others (un)like it.

I'm sure the dude was just having some hazelnut spread and that's how they recalls it but us, like them, are getting hacked. Now if you'll excuse me, the rey--, I mean tinfoil, is starting to get itchy, I need to switch it out.


Hazelnut spread is one of those things that is much more commonly referred to as the brand name. No one says "Reynolds wrap" they say "aluminum foil".

No one says "hazelnut spread", they say Nutella.

No one says "resealable sandwich bags", they say ziploc bags.


Most people would just say "sandwich" unless the spread is somehow critical to the tale. I know it's a quote, but it sounds most odd to me as most people I know don't really brand drop in conversation. Coke is probably the big exception there...

TIL what ziploc bags are. Just called "sandwich bags" here. Cling film isn't "sarran wrap" here either. :)


There used to be a distinction in the US: Saran Wrap was made of Saran and was much less oxygen-permeable than other cling films as a result. In a development that represents some kind of evidence about trademark law, Saran Wrap in the US is no longer made of Saran, due to concerns about plasticizers leaching into food; but it is still sold as "Saran Wrap". Here in Argentina, I can still get cling film made of PVDC, just not Saran-brand PVDC.


> Coke is probably the big exception there...

Even then, it may not be. There are plenty of places where Coke has become the generic name for pop, and doesn't necessarily refer to the brand.


> No one says "resealable sandwich bags", they say ziploc bags.

Everyone says resealable bags. I've never even heard the word ziploc. In New Zealand.


In the US, they’re always referred to as ziploc bags. If you called them something else you’d get some confused expressions.


Do they also not use hyperbole in New Zealand?

Perhaps no one in New Zealand, but New Zealand has the population of a small US state.


Nutella though is a sugar-and-palm/vegetable-oil spread, which - probably accidentally - contains traces of hazelnuts, powdered milk and cocoa.


I don't understand the downvoting of the parent here, which is factually correct and worth noting. It's incredibly unhealthy and this is just down to marketing that anybody eats it. You can make really nice chocolate spreads with your own hazelnuts


The needlessly bad-faith "probably accidentally" is factually inaccurate.


Just to clarify: it wasn't bad-faith, I was just (trying to be) sarcastic, as most of the store-bought food nowadays "may contain traces of nuts and/or eggs and/or celery" etc.


Aha, do note that sarcasm mostly goes down badly on hn, because literal people, and jokes historically frowned upon.

Nothing in that comment is factually correct or worth noting. Do you realize what butter, Marmelade, Margarine or vegetable spreads are mostly made of?


>It's incredibly unhealthy and this is just down to marketing that anybody eats it.

Because a spoonful of palm oil and sugar with hazelnut flavoring is tasty?


I think 13% hazelnuts, 8.7% powdered milk and 7.4% cocoa[0], are quite a lot more than "traces". Sure it could be better, but then you could say that about 90% of products in the average supermarket.

[0]Just went into the kitchen to check


> No one says "hazelnut spread", they say Nutella.

For one thing, Nutella is chocolate flavored. No one would ever even consider calling it "hazelnut spread".



Yes, I'm aware that it's made with hazelnuts. But in order for someone to refer to it as a "hazelnut spread", hazelnuts would have to be the dominant flavor. They're barely there at all.

(Compare the consistency of nutella to the consistency of peanut butter.)


Since we are splitting hairs, the container list the product as; "Hazelnut Spread with Cocoa". Quite prominently, in fact.

Why would your definition be more valid than the manufacturers?


We are discussing in the context of what people use in everyday language. In the U.S., the use of "hazelnut spread" over Nutella is exceedingly rare. Even when dealing with an off-brand, non-nutella "hazelnut spread".

It's just a writing device, in this case. The contrast of mundane detail and the high calling. Used all the time, e.g. NYT's style has pretty much devolved to the formula “begin with the details, then introduce the central topic.”


>> It's really natural, when first mentioning it, to call something using by a member of the degenerate set of its most unique identifier, right?

Huh? ‘degenerate set’ being what, the brand placement? I likes me some Semiotics, though not sure what this passage refers to.

In the context of a photography site, brand mentions must be a hazard. The author has no problem name dropping an Ancient Greek, Newton, Leibniz, Huygens. Why not Nutella? Rather than increasing the cognitive gap between the auditor (us) and the subject (2000 year old problem), Nutella seems to take the shining brilliance of those luminaries down a notch and in line with the form of writing—a neutral density filter? Haha. Or maybe it’s just a laugh and some color.


I don't know any Semiotics! But, in an effort to drive the conversation, I can be more precise with what I'm trying to say.

I'm working with this informal idea that there is always a relationship between the audience and the speaker. That relationship define a shared knowledge space.

If I am speaking to a group of my friends and need to refer to my brother the set of identifiers that uniquely conveys his identify to my friends might be {brother, Joe, Joe Blow, brother-man, etc}. Any of those identifiers coming from to me specify exactly one person, my brother. The degeneracy is that they equally identify him to the audience.

If you were not that close to me, and I say "brother", you might wonder if I have more than one brother. If you did not know me at all I might say Dr. Joe Blow to identify him and why he might be relavent in some area of expertise. As the relationship between speaker and and audience grows more distant the set size decreases. Inversely, the context-free uniqueness of the identification provided by the remaining identifiers has to be greater, with proper names maintaining greater uniqueness (sorry to the John Smiths of the world!) than nicknames and so forth.

To return to my point, the scientist in question called it Nutella because he does not have a deep relationship with his audience and that is the most unique way of identifying it. I don't believe he was doing anything atypical or nefarious. If he knew the audience well, he could have just said "breakfast", and they would know exactly what he meant because he has the same thing for breakfast most mornings.

The problem is that brands, unlike most proper names(obviously famous people are an exception), have a greater ethos associated with them. If my brother was named Joe Blow or Jim Deal, my unique referencing of him doesn't do much to color the statement around it. This is contrast to how much extra you get when you call a car a Ferrari vs a grand tourer. My point is that advertisers recognize this linguistic norm and exploit it, this is what brand identity is. As such, just the natural mention of a good becomes advertising.


There is a lot to unpack in your response, but I guess it’s your use of the term ‘degenerate’ which confuses me. It’s not common in the Semiotic cannon, and so you say it’s not something you’ve studied.

From your original comment: “...contextualizes that good as being different from all the others (un)like it.”

This is textbook Semiotics. Meaning is not derived from the designation (‘apple’ is the red fruit with the thin skin and firm ...eh flesh. Rather, Apple is not a banana; not a kiwi; not a tomato; not beef; ad infinitum.


It's totally common when talking about quantum states that have identical energies! Which is not helpful at all in this conversation. It's just one of the words in my everyday tool bag and I used it without thinking too deeply.

All that said I just order an intro book on Semiotics. Interest definitely peeked.


Under the US system, at that point the term is now generic, no longer trademark-able, and anybody can use it.

In other words, Brands by definition must be distinct from regular language.


> This raises the perennial question ...

Not really, no. There's no scientific funding riding on your reading this article in a photography magazine.

Specificity is almost always recommended in any creative writing from song lyrics to infotainment like this. It enhances humor and mental imagery, which enhance general enjoyment. That's all that's happening here. Hanlon's razor is your friend.


Do we know the brand of the chocolate that melted when the microwave was discovered? Maybe it's not that important.


Indeed. That is why I like the Nikon product placement a lot better because it just highlights what kind of a problem they create with the decision to have only one card slot.


My eurekas are generally in front of the mirror while using my gillet razor. I do not think the brand of the razor has any influence.


What truth?


I wonder that programming language he used. Many ideas are lost because we don't have something to write it on. That's the good thing about the so-called 'scripting languages' that IMO reduce the latency between idea and its materialization.


It was Mathematica, they provide the code in the paper.


If you're really curious, maybe ask?

Shouldn't be too hard to figure out appropriate contact details. :)


Reminds me of that story about Archimedes.

For anyone unfamiliar: supposedly, while bathing; he suddenly realized the principals of buoyancy. He then jumped out of the bath and ran down the street -- without clothes -- yelling "Eureka!" (I have found it). Not sure if this is true or just a legend, but I doubt we'll ever find out what happened to that Nutella bread either...


Probably should include what problem Archimedes has been stuck on until his realization of the principles of buoyancy gave him a solution.

The King had order a gold crown, and after receiving it suspected that the goldsmith may have replaced some of the gold with cheaper silver. The King asked Archimedes to figure out if the crown was pure gold.

Archimedes could do that by figuring out its density, but to do that he needed to get the weight of the crown and the volume of the crown. Weight was easy, but he had no idea how to get the volume, other than melting it down and reshaping it into a shape whose volume he could calculate. Naturally, the King wanted Archimedes to answer the question without destroying the crown.

His realization that the volume of water displaced by an object that is more dense than water is equal to the volume of the object gave him a way to get the volume of the crown without harming it.


and the crown? or is the crown akin to the Nutella bread.


Brain was doing the computing so that he can eat more Nutella bread. Bummer, he just left the Nutella bread and went programming the solution.

Maybe it should have convinced him to procrastinate instead. Now, I know where all my inefficiency is coming from.


> Mothers! It is there!

Since no one has clearly pointed it out, "¡Madres!" is a multipurpose Mexican expression that rarely has to do with actual motherhood. "Holy crap!" is a good analogy in this case.

https://www.urbandictionary.com/define.php?term=madre https://www.spanishdict.com/translate/madre


Brouwer had similar inspiration from a cup of coffee. In his case, the intuition was that if you stir coffee, there's a "part" of it that stays fixed.

In math terms, f:X->X where X is compact and convex, and f is continuous, has a fixed point (exists x_0 in X, f(x_0)=x_0)


The article says Nutella bread, but I wonder if it was toast. Which I prefer. And the time waiting for the toast allowed long enough for his mind to wonder to the solution.


¡Madres!


I'm not sure I really follow how this actually will result in better lenses. Is the precision of the numerical solution really the problem, or is the problem the actual precise manufacturing of the lens elements?

I have a strong suspicion that the numerical solution is as precise as needed and the limiting factor is the manufacturing, but I would be interested to hear how this results in something new.


Your suspicion is justified. The news here is the discovery of an analytical formula - which is "precise," sure, but whether using it directly in an actual computation could yield a more accurate and practically useful result is a good question, due to the complexity of the formula.


Your suspicion is right, as far as manufacturing goes.

The real boon of this discovery is in designing lenses. For example, the authors have already used it to design a single-lens telescope for the first time [1]. I don't know much about the forefront of optics, but at least in my field having analytical models that you quickly iterate on and use to prime intuition are extremely useful.

[1] https://arxiv.org/abs/1903.11129


I think the problem was being unable to solve a generalized version of the equation. Our old methods only made it economically viable to make "good enough" lenses and now this method makes it economically viable to make "perfect" lenses.


The paper is from Nov 2018, and is easy to find online. It’s a pretty amazing result; the examples they give in the paper are for some reasonable shaped primary surfaces and some really neat “exotic” stuff. The solution actually admits aberrant free solutions for any geometry that doesn’t include self-crossing rays. So, some of the ‘weird’ examples include a Bessel-function cross section, negative index of refraction, and lenses that focus at negative infinity.

The author’s have a paper from March 2019 (in arXiv) that uses the result to build some novel telescope geometries.


[0] https://arxiv.org/abs/1811.03792 "Rigorous analytical formula for freeform singlet lens design free of spherical aberration"

[1] https://arxiv.org/abs/1903.11129 "Single lens telescope"


Hmmm..... I have a 1/3rd hand-ground telescope mirror lying around... maybe I should see if I can apply some principles here...


> and lenses that focus at negative infinity.

Any reasonable chance of of implications for projective geometry?


Slightly related: One thing I've learnt in Physics is that there are still a huge number of practical and fundamental problems to be solved. You might expect everything easy will already be done by now but that's just not true. Hence we still manage to write so many papers! In fact many Physicists have too many papers they want to write and not enough time to write them. This was a huge surprise for me coming in to my PhD. I had naively assumed that physicists were all scrounging for any morsel of content to publish!!

Of course it depends on the field. It's a requirement here in the UK to publish a paper to get your PhD (at least I've been led to believe that), and many particle physics students really struggle to manage to fulfill this. On the other hand gravitational wave detectors have a huge amount of unexplored science as there are so many things that we still do not fully understand about the detectors themselves.


I agree with this big time. One of my favorite Feynman quotes is that science is an "expanding frontier of ignorance". There is always somewhere to go and something on the other side, that may change everything and invalidate deeply held convictions forever, all it takes is the guts to, again quoting Feynman, "be willing to stick your neck out".


Oh, I'm sure there are lots of things to discover, but I think this underestimates what it takes to actually know enough to find the frontier and decide on a good problem to solve.


While Feynman is correct on expanding the frontier of ignorance Asimov is also correct about the relativity of wrong (that we come asymptotically closer to the truth). This is why people think most of physics is solved, because in the context of what most people need it is extremely accurate.


> It's a requirement here in the UK to publish a paper to get your PhD

Categorically false. I did my physics PhD in the UK and while I did publish a paper, it was not a requirement and I have plenty of friends who did not publish a paper and still got their physics PhD.


Just yesterday I was watching a video about machining flat surfaces, and it was mentioned that the reason why two very flat surfaces stick together isn't thoroughly understood yet.

We can build things that deal with unimaginable precision, create new elements, compute with quantum effects, but we don't know why flat things stick together?


I always thought this was just air pressure, and their extreme flatness just didn't allow for air to come in between the two objects to counteract it.


To put it simply, two clean flat surfaces stick together for the same reason each of the two objects is held together in the first place: weak and strong nuclear forces and electromagnetism. However the specifics of this might vary greatly depending on the material and other circumstances.


I believe that even if humanity were given a book that perfectly describes physics on the smallest levels, there'll still be hundreds of years of work to find useful approximations and abstractions.

That's of course nevermind the effort involved in grokking the book. Especially, if particles are more subdivided than we think/know they are.

Think of it as knowing the rules of the game vs developing strategies for it.


Are there fields like this in computer science where many papers are yet to be writen?


ML


To clarify: the innovation isn't that they design better lenses. It's that given the first surface of a lens, they have an analytical formula for the lens's second surface so that the lens doesn't exhibit spherical aberration. Traditionally, numerical methods are used to compute this second surface instead of an analytical formula. Also, as a practical matter, designers often use several spherical lenses instead of a few aspherics because the latter are harder to manufacture/more expensive.


It's an interesting mathematical result, but note that gives a solution for the problem of the spherical aberration, but real lens have also chromatic aberration. I.E. the speed of light for each color inside the glass is slightly different, so the value n (refraction index) in the equation is different, so in the equation you get a different surface for each color. In the real lens you must pick one surface, so the effect is that one color is perfectly focused and the other colors are not focused and you get some rainbow-like effects.

The solution is to use combination of a few lens of different glasses, to compensate the differences. It's not easy to design these kind of system, because they must compensante also for other types of aberrations.

More details:

https://en.wikipedia.org/wiki/Chromatic_aberration

https://kenrockwell.com/tech/lenstech.htm


wouldn’t this be simply stacking SA free lenses, with the remaining aberrations being alignment related?


It is SA free only for one color, not for every color.


This is a seriously awesome piece of work. The mathematics in the paper are pretty dense but if you're ok with vector calculus its not too bad.

The implication here is that you can build lenses that have a consistent focal plane across their entire surface. A pair of glasses made this way would have crystal clear vision regardless of your 'look' direction (as an example). It also suggests that you could build lenses for lasers that were much more consistent, so a laser projector could project an in focus image from edge to edge rather than "fuzzing out" at the corners.

If there is a top prize in optics these guys clearly deserve it.


That would have pretty amazing applications (well everywhere) but in VR definitely, my Rift S is an amazing piece of kit but aberration is still noticeable on the outside edge (when aiming down a gun sight in Pavlov for example).


Is this the kind of thing that will have an effect on optics in ray tracers?

"Real world" simulation optics I'm meaning (eg for material design/prototyping), more than gamer oriented. ;)


Expect to see an implementation included in subsequent releases of Code V and Zemax.


Could this also be significant for super large lenses like giant telescopes?


Those tend to be mirrors instead I think



And to anyone who is averse to reading dense, long papers: it's only 4 pages long, and a quarter is pictures. It's heavy on the maths, because of course it has to be, but there isn't a single integral in the entire thing. It's remarkably accessible to anyone who took an introductory calculus course.


Or on the arXiv: https://arxiv.org/abs/1811.03792 . It also looks like the first author wrote up a local article which seems to have substantially, umm, inspired the linked write up: https://transferencia.tec.mx/en/2019/02/21/eureka-they-find-... .


The article talks about the corners of an image being less sharp than the center. That’s not spherical aberration; that’s off-axis aberration. Spherical aberration causes the image to be less sharp at larger apertures.

I wouldn’t be terribly surprised if this result helps indirectly with off-axis aberration, though — a closed-form solution to the spherical aberration problem may make it easier to optimize for lens shapes that minimize other aberrations.


What’s your background? In the photography field this is known as spherical aberration indeed: https://en.m.wikipedia.org/wiki/Spherical_aberration


Physics, but not optics. Also, I read and understood the paper. The Wikipedia link seems to agree with me too. Look at the diagram there, and compare it to the diagram for coma [0].

Spherical aberration causes the single point in the center of the object to focus to a different place depending on where the ray hits the lens. This means that the image of that single point is not fully sharp. If you restrict what parts of the lens can be hit by rays from that point (by closing the iris, for example), you reduce the area in the focal plane that the rays hit.

[0] https://en.m.wikipedia.org/wiki/Coma_(optics)


This was extremely well written. Not being a photographer myself, the mathematical discussion and subsequent impact to optics/photography are very accessible. Kudos to the author!


> In this equation we describe how the shape of the second aspherical surface of the given lens should be given a first surface, which is provided by the user, as well as the object-image distance [...] (emphasis mine)

I didn't check out the paper in detail, sounds like the shape of the correction surface depends on the distance to what you're imaging? Does it do so in a non-trivial way? If so, this sounds a bit more like an academic victory, no?


I used to be a photographer and have stared at the lens group diagrams from many a lens manufacturer, so take this with a grain of salt: In a lens there are multiple lens groups, it's often that the aberration correction is its own distinct group, with fixed input and output groups flanking it. I'd imagine that the distance to the projection surface, in this case the next lens group, can be fixed and thus distinct from the distance to your ultimate subject. A fun fact is that Helmholtz reciprocity says that regardless which direction light passes through an optical system you can reverse the distortion and reconstruct the input signal.


So it's useful if distance to your object changes predictably enough, like in astronomy, satellite imaging and many other kinds of space missions.


Well in astronomy you usually want your telescope to be able to observe different targets. So it depends on the details of how the correction surface has to change when the image-object distance.

If it's just a mild deformation and offset, then yeah I guess technology similar to adaptive mirrors could work well.


Distances are usually treated as infinity, at least in astrophotography. I'd be surprised if this secondary lens was able to distinguish between light coming from 1ly and 1000ly


Doh, of course. Not sure what I was thinking.


Does this mean there’s hope for those of who had mild vertical diplopia (double vision) before and after lasik surgery? http://lee2see.com/double-vision-vs-ghosty-image/


I’m questioning my sense of humor because everyone here seemed to have enjoyed the tone of the article, while I was quite put off by it.


It's a good article if you enjoy jokes and references about photographers and how they behave. It's not a good article if you want a minimum of fluff and just want to understand what the discovery is and why it might be significant.


I usually enjoy jokes intermingled with info, those didn’t hit the right note for me, for some reason.


Not every person finds every joke to be funny.


While I usually enjoy a good amount of fluff, I must say I, too, was underwhelmed by the amount of detail and/or substance reported. The formula makes no sense to me, and was apparently just thrown in in this form for a gag.


I also didn't like it, the title calls him a physicist and in the article it says mathematician, they did an experiment with 500 samples and got how many 9s, the presumption that Newton was the greatest scientist (von Neumann?), the comparison between such theoretical luminaries and someone who did a numerical optimisation of someone else's theoretical work, ... it reminds me of A Beautiful Mind, where the real accomplishments are obscured (and here greatly exaggerated) purely to tell a story to lay-people. More attention to the facts please, less attention to the sandwich.


Hopefully this will increase the sweet spot in the VR headsets.


VR headset lens already require software chromatic abberration correction -- I'm curious what constraints if any would limit the application of this new result for VR spherical aberrations. Would love some thoughts from an optics expert on this. I also wonder if this new knowledge could contribute to a software corrector.


Hopefully it leads to software-based spherical aberration correction too, which may mean better perceived quality with the same lenses, or even thinner/cheaper lenses over time.

Or maybe they'll find a way to apply it to make better lenses that need less software-based corrections. They struggle a lot not just with clarity at the edges but god rays and glare too.


Not an optical expert, just also interested in optics for VR/AR :)


Doubtful that this paper is the first analytic solution. There's a light artist who exhibited lenses like this at NYC's Museum of Math several years ago. They turned flashlight beams into pictures. At the artist's talk he passed around a wavy lens and mentioned that one of the surfaces was solved in closed form. Looked like second-to-bottom lens on the page http://zintaglio.com/lens.html


I voted for your comment because it's interesting but I don't agree that the example you linked is the same problem as solved in this post. But that is an opinion because I'm not informed enough to know for sure. Meanwhile I did dig up a paper related to mapping radiance with a shaped lens like the artist you linked. It's a cool topic! https://arxiv.org/pdf/1701.03076.pdf


I wish there was a bit of insight into his mental process as he realised the solution.

Like the description of Einstein sitting watching a clock and imagining what it would be like to move towards it as a ray of light


Will this have any implications for astronomy?


I don't see any figures comparing the error of the formula with the previous error from numerical methods. While mathematically the discovery seems significant, it isn't clear to me this will actually improve lenses.


From another comment in this thread:

"The author’s have a paper from March 2019 (in arXiv) that uses the result to build some novel telescope geometries."


Note that "build" doesn't mean they built a physical telescope, they came up with lense shapes.

There are some serious problems when building large telescope lenses; if they get too heavy, they start to deform under their own weight, and non-spherical shapes are hard and expensive to produce.

It would be awesome if this analytical solution inspired improvement in real applications, but I'm cautions about that.


Not at the large scale - there aren't really any refracting telescopes left (based on lenses) when it comes to serious research, we replaced them with reflecting telescopes instead (based on mirrors).

There's a few notable exceptions, like the Lick Observatory, but in terms of impacting the state or progress of astronomy, the research from this article is unlikely to have an impact outside of small scale labs and amateur astronomy.

That's not to say the benefits there won't be appreciable for those who rely on optical telescopes, but you won't see this research make a difference for ground based arrays like NRAO, VLA, or in space telescopes.


Brilliant!

Two thoughts:

1) The size of the equation is rather large. Nature tends to like simple equations, I wonder if this equation could be restated in a simpler format and not lose its efficacy. I also wonder if there are still simpler versions that trade off efficacy for degrees of error.

2) If the equation cannot be reduced, then this equation bears looking at from an information preservation perspective, that is, to people who study how the universe preserves information under various conditions, this is one of the equations they should add to their list of equations to study...

But, overall a brilliant, laudable, commendable work with great applications and benefit to society!

Well done!


What wenT unsaid wasn’t anything about expensive camera gear, but that phone thingy we hold in our pocket. THAT device will get much better at photography in very quick order, and the price change will be largely immaterial.

I fully believe this will be the death of all but professional cameras in 5 years or less. (Yes, I know cameras are already in sharp decline, but I’m talking true death - bankruptcies and firms leaving the market).


I fully believe

Why? What makes you think this will be easier to implement than existing methods?


Gut feel. To me, photography is now a software game. If you can put a reasonably cheap lens in front of a sensor and apply enough computing horsepower, I believe that the old one lens and one sensor will be like the mainframe of cameras. They will still be around, but smaller, modular ones put together via software will dominate.

The fact that there is an analytical solution to the problem is a really nice achievement, but I have to wonder: solving this problem numerically has surely been within reach of computers for a long time, and should therefore have made actually manufacturing such lenses somewhat straightforward.

What am i missing?


CAD modeling is very slow when you need to use numerical simulation.


I don't follow.

I assume that if numerically solving for a correcting shape was burdensome you would model an element using a first surface and a 'dummy' second surface that your software just treats as magically correcting. ... then when you are happy with your design, you go ahead and compute the actual shape for that second surface... or similar.

I would expect a closed form solution to more useful for meta analysis of the problem-- looking at its behaviors (and especially derivatives) in various external cases may suggest interesting and novel optical system designs. Like, "oh, foo changes cubically with with an infinite focal length that means if we could make lenses with cherry flavored unobtanium we could bounce a gravitation particle beam off the main deflector dish!".



"It is unclear whether he finished eating the Nutella bread."

This article is alright.


Trying to figure out what voice you’re writing that in.

I hope you mean alright alright, and not just _alright_ because I loved that line!


I read it as "This article is 'alright alright alright'".


It is a most excellent line.


Man, that is a long formula! Not something that would pop into my head while spreading nutella.


Unrelated, but seems like optics use a different Arxiv format than ML papers, which are always double column.


Nobel-price worthy, you think? But perhaps only if they manage to derive a more beautiful formula...


Looking at the picture... does this mean we'll all be using mustache-shaped lenses someday?


Awesome article, wild equation


Don't such things always seem a bit obvious in retrospect?

/joke


I would love/hate to read the LaTex that generated that equation.


Does it solve chromatic aberration, or just spherical aberration?


Just spherical.


Is the resulting lens rotationally symmetric?


The formula computes ZbRa; z axis vs radius. So yes, I believe this is symmetric around the center.


Hopefully video game makers will now finally stop putting that stupid chromatic aberration effect in their games.


Proud to see it was solved using Mathematica :)


G Mga


12 nines? Yawn. Let me know when they get to 15. j/k

That's some serious precision. I wonder if this would have been possible to calculate on a Pentium?[0] After last week, cloud service providers would be happy for 3 nines.

[0]https://en.wikipedia.org/wiki/Pentium_FDIV_bug


I have not checked the actual paper, but according to the article itself:

> Moreover, the solution involved aspherical elements, which are harder to manufacture in a precise way and are thus more costly.

> Their findings were published in the article General Formula for Bi-Aspheric Singlet Lens Design Free of Spherical Aberration, in the journal Applied Optics.

I’m no expert in optics, but that does not sound like the kind of advancement that cheapens lenses, even if the math work is mathematically relevant


It also doesn't sound like the kind of advancement that makes camera lenses play mp3 files, but then: neither of those two things are even remotely relevant to the article?

These people solved optical abberation. Even if their work leads to lenses that are twice as expensive as they are today, people whose research, or jobs, rely on perfect optics will be more than happy to pay what it costs to work with those.

Anyone else can wait until one or more companies figures out how to make the manufacturing process cost effective at scale.


Currently abberation is dealt with by including multiple lenses. Even if the lens costs twice as much, it's still a wine from a price point.

But yes, only for applications that care about abberation.


> It also doesn't sound like the kind of advancement that makes camera lenses play mp3 files, but then: neither of those two things are even remotely relevant to the article?

How is it not? The article actually makes that claim:

  It would help make better and cheaper to manufacture optical systems in all areas, be it telescopes, microscopes, and everything in between.


Haven't read the paper, not an expert... but you still feel qualified to dismiss it?


I have not read the original paper, I'm not an expert in optics, but the article clearly makes an argument that seems contradictory:

- Current solution is expensive because it uses aspherical elements.

- New solution promises cost reduction.

- New solution uses aspherical elements as well.

Either there's some hidden detail that the article fails to mention, or it's claims are overstated. If you actually know about optics, I'd be glad to know which is the true or if I'm missing anything here, otherwise I would appreciate to keep your snarkiness to yourself.


Aspheric lenses are frequently molded and these molds are used to make vast quantities of precision lenses. We've been molding lenses in quantity since Kodak perfected the process in the 70's. Yes, grinding aspherical lenses is more costly (yet still possible,) but there are numerous applications for which molded lenses are entirely sufficient.


Something like this could be an advance for optical systems where plastic moulded lenses are suitable. One lens could replace several elements in systems with monochromatic light. Think optical drives


I laughed when they tried to translate "Madres!", but a pissed off Latina mother running at you waving her sandal in the air like a medieval warrior swinging a truncheon is definitely an "Oh shit" moment.


My understanding is that centuries ago in Latin America "madre" became an euphemism for "madame of a brothel", while actual moms are called "mamá". In Spain it carries no such connotation.


Mamá means mom, madre means mother. You use either depending on the context. (I'm only talking about the Mexican use)

Madre is used more often in phrases, such as "Madre de dios!", Or "La madre putria," or "Pinche tu madre", or it's used as a formal to refer to one's mother. It's similar in English, such as "On my mother's side", "Mother fucker", "Mother of God!", etc.

In Mexico, "tu madre" can be a general insult, probably shortened from "pinche tu madre", though in English we have a similar vulgar retort of "your mother". As a weird turn on the phrase, "de puta madre" is slang for a good thing.

I'm not aware of an association with brothels in Mexico, but certainly in some Latin American country it could have.


In México the word "Madrota" does carry that "madame of a brothel" connotation




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