This is a particularly egregious example of the pop-sci genre "making things sound more mysterious than they really are". How does light choose the fastest path to its destination when light can't "choose"? Well, we could hypothesize light is conscious and is making globally-optimal calculations, or we could just observe that if you work through the wave equations on what waves do when encountering a speed change, that's what happens. It so happens that solution also matches how to get to the eventual point in the shortest time, but that adds an unnecessary concept of "intention to arrive at a given point". Light has no such preference or intention, it just arrives where it arrives.
We could hypothesize dogs are working out calculus problems, or we could just hypothesize that the dog tried it one way, then another, and eventually over the course of its lifetime works out how to get to the ball the most quickly, which in comparison to the difficulty of manipulating all of its dozens of muscles to perform the running optimally in the first place is a frankly insignificant challenge. Or, to put it into a human reference frame, we do not do calculus problems to catch things. We "just" catch them. Neurons are built to solve this problem.
We could hypothesize that the ants have a simple pheromone-based optimization system that naturally mathematically converges on the same solution as Snell's Law when, after all, faced with the same problem, but... uh... actually it's not at all clear to me how the article manages to make this sound mysterious after what is a fairly effective solution to the problem. If nothing else, the ants taking the most optimal path will make more to-and-from trips than other ants, thus laying thicker trails; that plus an initial shotgun approach are adequate to explain.
This is particularly egregious since none of the math here is very complicated, and it is completely unsurprising that when the same problem is given to several optimizing agents in several domains that the same solution pops out. There is not very many bits of information in this problem.
You're definitely correct, this article overhypes the mysteriousness. It's a shame, though, and a missed opportunity to go deeper, because it is a surprising fact that almost all of physics is most elegantly expressed in terms of least action principles, where it really does seem like the universe conspires to minimize some quantity.
In classical physics, the Lagrangian formulation is merely useful for some calculations, and can be fairly viewed as a mathematical trick that is equivalent to using Newton's law directly; similarly in E+M it's a compact and often useful way to express Maxwell's equations, but not particularly fundamental.
When you reach quantum field theory, though, it's all but impossible to do anything without a Lagrangian/action, and theories are almost always described in that form. String theories are the same. General relativity is not always presented that way, but gravitational dynamics are stunningly simple in that form: in a vacuum, nature abhors curvature.
Principles of least X are fascinating and incredibly useful, and the ex-physicist in me believes that they describe the actual universe so concisely that they are truly fundamental in some sense. But you have to really dig in to get any sort of appreciation for that, and I agree that this article didn't go anywhere near deep enough.
Does the universe seem to prefer minimizing some quantity, or do human brains gravitate towards theories that identify quantities that are minimized/maximized?
Couldn't physical theories be expressed in many different ways, maybe with completely different fundamental concepts than mass/energy etc? Where the theories describing nature would not be so clean?
That's the same thing here, though, since physics is a scientifically descriptive endeavor. Our conceptualization of the universe is limited by our perception of it, and thus the whole idea of "what the universe does" is fundamentally entangled in what we see it doing and how we can talk about it.
So that the universe "optimizes" something must be true, given that only one past is observed. It has to have been optimizing for the past that we see. (It is perhaps more remarkable that we can make accurate predictions of the future, but this more or less seems to be a side-effect of optimization requiring some notion of continuity.)
Our descriptions of nature are clean because they are based on observations of the past, which are the exact phenomena the universe appears to be optimized to produce. All other possible results are merely counterfactuals in our brains, and having those encoded in our theories would make them noisy and inaccurate. (And the best evidence for such counterfactuals being 'real' does exactly this by introducing quantum indeterminacy into the theory.)
There is a small problem with your explanation: refraction can happen with single photons, which means that the "minimum path length" rule continues even as the wave formulation breaks down. If you say "the photon travels along all paths at the same time, finally choosing one path to have 'really' gone down," then you have to explain how it was able to move faster than light to collapse down to the single point where you found it on your camera's sensor.
There is indeed a mystery here. In fact, since the only option I know of that eliminates spooky FTL signalng and spooky hidden (unobservable) mechanics has spooky alternate universes, I feel safe in concluding that at least one spooky thing has to be happening, even though we don't know which one it is.
Thinking about photons is bad for you, especially if you are doing quantum-optics. It is almost always better to think about quantum fields, and in this case the quantumness adds nothing to the physics.
The "photon" is just a quantum of excitation of the optical medium (in this case the EM compled with the electromechanical state of matter in it). These exictations obey wave equations, just as classical waves do, and hence Snell's law.
It's true that you can also (almost) think of photons as point-particles travelling over every possible path in time-space and then do infinite-dimensional path-integrals. That point of view is important for a physicist to understand, but it is not needed to explain refraction.
Huygens explained refraction using classical wave velocities only. Though I agree there is a similarity between his notion wave-sources everywhere and path integrals. (Which is odd since path-integrals are a very particle oriented way of thinking).
What Huygens' explanation leaves slightly mysterious, is why (beyond a mathematical coincidence) this result just happens to also give the minimum-time result. Path integrals explain that very nicely.
You're describing quantum mechanics, but the phenomena here are classical. There is no mystery with Fermat's law, even if you do not accept wave optics and want a theory of light particles. At each instant in time a light particle can determine how its trajectory will bend based on purely local information. The bending of light is determined by the local variation in the index of refraction. Snell's law is what you get if the index of refraction jumps discontinuously. This is analogous to how in classical mechanics the trajectory of a particle is determined by the local variation in the potential energy function (or more generally, the Lagrangian), and how a particle's trajectory has a kink if the potential energy has a discontinuous jump. The particle bounces back if the jump is too high, which corresponds to reflection of light.
Well, here's why I wanted to bring up QM: the principle of least time is really most at home in geometric optics where beams of light are arrows and everything is steady-state. In that view, things are indeed nonlocal: I can alter the arrows drawn for a beam of light going from A to B by doing something over at point C. Wave optics would understand this as interference, but if we insist that we want to keep the arrows then we have to introduce the path integrals.
Essentially, geometric optics lets us draw arrows and assert refraction, QM lets us draw arrows and derive refraction, but wave optics doesn't really involve arrows at all.
Things aren't nonlocal in geometrical optics though. Something happens at C at the instant in time when the particle is at C. You can start a ray at any point in any direction, and just follow a straight line until you hit something. Then you determine what happens next based on local information at that point, and you continue. Ordinary tennis balls have the same property that the path from A to B can be altered by putting a wall at C.
By the way, wave optics is already QM in some sense. Maxwell's equations are the analogue of the Schrödinger equation for light. Geometrical optics is the analogue of classical mechanics.
Look at Bohm's interpretation of quantum mechanics and subsequent formulations for quantum field theory.
Hidden variables are possible for quantum mechanics, they just necessitate that the Universe is fundamentally non-local, offering a possible explanation for quantum entanglement results.
Von Neumann's rejection of hidden variables was not an absolute one but a conditional one; if the universe is local, then hidden variables cannot be invoked to explain the weirdness of the quantum world. Non locality opens the door to hidden variables.
AIUI photons are an observational phenomenon, the photon isn't travelling any path, the probability distribution is occupying all possible paths, the observation collapses the wavefunction and makes it so we can find the photon in a general locality.
I think the physics of QM is too far displaced from our natural ideas of how mechanical things work to be able to reason about in the way we would like.
> We could hypothesize dogs are working out calculus problems, or we could just hypothesize that the dog tried it one way, [...] Or, to put it into a human reference frame, we do not do calculus problems to catch things. We "just" catch them. Neurons are built to solve this problem.
But the point remains that you are solving a calculus problem.
I mean, I agree that some of this is over-hyping things, but I think it is actually a really important realization that a lot of people are lacking, often because they are stuck in some completely nonsensical notion of dualism, that your body does a lot of computation.
Just because you are not consciously manipulating numbers does not mean you aren't computing the trajectory of your arm and the necessary activation of muscles to execute that trajectory. It's not that thinking is a prerequisite of computation, but rather computation is a prerequisite of thinking. Your arm doesn't catch a ball by thinking, but rather your arm catches the ball by computation--just as your brain thinks by computation.
It's been found that to catch a fly ball (as pros do) your brain has to solve (approximately but accurately) a third-degree calculus equation, and does.
Yeah it's almost akin to asking "If I put hot coffee in a thermos, the thermos keeps it hot, but if I put cold lemonade in there, the thermos keeps it cold. How does the thermos know the difference? Well because quantum, that's how!"
Let me put it this way: never have I heard an explanation in terms of classical mechanics. In my intuition, the billiard ball model of physics makes perfect sense. But a current taking the path of least resistance really seems like it "knows" the best way. And it's amazing.
> Well, we could hypothesize light is conscious and is making globally-optimal calculations, or we could just observe that if you work through the wave equations on what waves do when encountering a speed change, that's what happens
"Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong."
Well, quantum mechanical explanations for why light refracts may be in some sense wrong, since we know QM still hasn't been harmonized with relativity very well, but "neat" and to some extent "plausible", definitely not, at least by the person on the street's standard.
I suggest you think a bit more before you fling a quote at somebody as if it's the last word on a topic; at the very least you should make sure it is actually applicable to the topic at hand.
The last bit is what got me too---pheromones are pretty simple to understand, really. you drop 2 ants, one walks 30 cm and another walks 10 cm. the one that walked 10 cm will be back sooner, can signal where he went, and others can follow his (fresher) pheromones.
On the one hand, I like that they're doing old physics. There is a tendency to do only cutting edge physics such as modern quantum mechanics research, particle physics, even string theory. There is little chance to convey an accurate understanding of this to a layperson. Every part of physics was at the cutting edge at one point in time. The physics that is now hip will be old in 100 years, and it will be no longer interesting for science journalists. Celestial mechanics, classical optics, classical electrodynamics were at the cutting edge some time ago, and they're still super interesting. They also have the advantage of being thoroughly understood, so there's a chance to convey genuine understanding. In that sense, this article is great: it truly explains Fermat's principle as a physics college student would learn it. It would be even greater if it had more physics and less metaphysics. On the other hand, would that have reached the front page of HN? Probably not.
That this model exists is OK; that it dominates pop science is a problem. I wish there were a lot more books like the one by Feynman that I linked to in another comment, which focus on real understanding. We can't all be Feynman, but we can try to do better.
(I haven't read the OP because it sounds like things I already know.)
I think it's the same reaction that police or lawyers have with watching crime procedurals, or why software engineers get disgusted by how "hackers" are portrayed in most popular media.
However, I think this sort of simplified shorthand, with a bit of innocent embellishment (the whole mystery aspect of how nature does it) keeps people interested.
It could be something that gets someone interested in discovering more. I think any simplification could be viewed as misleading, but I think these sorts of articles really do more to educate than if they were to only be clicking celebrity gossip or worthless listicles.
They don't educate anything. They generate crackpots who spread more pseudoscience. True science can be explained simply and clearly with good understanding and good communication (think Terrence Tao's blog) without dumbing it down.
Thank you for the response, and I will Google Terrence Tao for the link, but maybe adding a link will help others?
I don't disagree with you, but honestly, Id rather argue science with a crackpot that only has 50% of the info than the crackpot who refuses to, and can't understand anything above 0%.
It's like literacy. I'd far rather live in a world where most people have a 50% literacy rate rather than in a world where 0.5% has a genius level of literacy and the rest are illiterate.
Sure it's a weird, cold calculation, but in the pragmatic part, I'm happy to accept the compromise.
This was my thought. I think the article reads almost like a children's Did You Know? kind of book: it presents the problem and just enough background information to understand it, then leaves the mystery for the reader to go and find out more.
I could see the argument that there is a fine line between leaving a bit of mystery and just plain lazy writing.
So weird that the moment I read the title, I started thinking of DeathNote, and the morality of saving drowning people from the perspective of Light Yagami. :D
I'm glad I'm not the only one! I thought this was going to be an article that was a philosophical/thought experiment, but nope. The author meant actual light. Pretty disappointed.
Weeb reporting for duty. Since it's not a heart attack there's a good chance he isn't the one who wrote the name down. So being the perfect role-model student extraordinaire that he is, he'd dive in and save them...to further throw anyone off his trail, of course.
For the dog optimising when to jump in the water, I think it's fairly easy to see how it does it. It's constantly optimising as it runs. It doesn't "choose" the path and then run it.
It's just repeating the same question as it moves, "is it worth running along the shore a bit more or should I just jump in now?"
This would be more interesting if, instead of throwing the ball while standing at the water's edge, the human stood further back up the beach as in the lifeguard illustration. [1]
This would require the dog to decide on the path to take at the start, or run in a path that wasn't a straight line. Anecdotally, when performing this experiment in the past, my dog typically runs directly towards the ball along the straight-line path Lifeguard->A, then turns somewhat along the edge of the beach to plunge in somewhere between A and C.
I'm also curious if there is a confounding factor in the dog's motivation. Pennings states [2] that:
> By the look in Elvis’seyes and his elevated excitement level, it seems clear that his objective is to retrieve itas quickly as possible rather than, say, to minimize his expenditure of energy.
There is obviously a value that the dog ascribes to the expeditious retrieval of the ball, but when the ball is not present there's also a value to be had in simply running joyously on the sand, and, of course, there's little better than being in the water. We may therefore need to calculate the ideal path based on a static value (perhaps 100 dog-utilitons) for retrieving the ball, plus 2 du/m running in the sand, and 12 du/m when swimming in the water. This complicates the mathematics somewhat.
Finally, the author's method of running ahead with a tape measure may work when measuring the performance of a short-legged subject, but I would be unable to use these methods to measure the performance of my larger and faster (Newfoundland) dog. Aerial drone videography may be more suitable instead.
Dogs are originally forest creatures. Whatever we bred into them we did initially because they were forest creatures.
A few tens of thousands of years haven’t erased and in fact probably reinforce their instincts to solve pathing problems. Ravines and thickets must be circumscribed. Hillocks too, depending. Getting wet in the winter is better avoided so follow the stream and try to ford it later. It’s only preferable to going hungry. Play for time and min-max.
I assumed it was a matter of the dog desiring to go on a straight line directly towards the ball, and knowing that getting as close as possible by land will use less energy-
but as he proceeds along land, his desire to head directly towards the ball increases due to anticipation, until eventually the force of anticipation grows larger than the resistance to excess effort.
I bet as a puppy, he went in a straight line and ignored exertion, and as an elderly dog he'll minimize swimming time/effort, and as a middle-aged dog he's splitting the difference.
Through this short story I also learned that Fermat's principle is just one example of a variational principle. I'm not an expert, but if I understand it correctly, many physical laws can be restated as variational principles.
It's interesting to note how light has become more and more "special" over the past century. It used to be that light was just another entity, similar to sound. That the speed of light was just another constant, like the speed of sound.
But at some point, the speed of light became so special that it became a constant from all reference frames, regardless of how fast the observer herself is traveling. And now we believe that light is so special, that it can be viewed to have agency. That it "decides which is the shortest time, or the extreme one... smell the nearby paths, and check them against each other... and chooses that path"
As someone who's knowledge of physics maxed out at Newton's laws, it's interesting to see how today's perspective of the universe is so different and "light-centric" as compared to centuries past.
There are other particles that move at the speed of light (gluons, gravitons), so in that sense, light is not special. Classical mechanics can be formulated in Lagrangian form such that the path that a particle takes is the path of least action (for light, action is proportional to time). Don't put too much weight on light having agency because this description of how particles move is mathematically equivalent to a system of differential equations, in which the infinitesimally future position of a particle is determined by local information. Classical mechanics is just an approximation of quantum mechanics anyway. Quantum mechanics clarifies why particles move according to the principle of least action: the paths close to the path of least action have large probability amplitudes.
Just triggered the thought that we should be trying to exploit the property of light to pursue the shortest path as a means to calculate hard problems. No idea how this would actually work... but I would gander someone already thought of this and it didn't pan out.
I was a little disappointed by the lack of explanation on how light refracts. It references Snell's law and mentions how light travels slower in water, but it did not go into how it's an effect of parts of the wavefront changing velocity at slightly different times as a result of the light entering the medium at an angle (or at least that being a simple way of understanding refraction.)
However, it was cool to see them reference "Do dogs know calculus?" I actually saw Pennings give a presentation on it in high school because my calc teacher was one of his students. He started out the talk by telling us that his dog knew calculus. He then proceeded to ask his dog the derivative of x^3. The dog did nothing, and he told us that dogs don't know calculus.
Interesting math story problem, but probably not much use for real lifeguards.
Is there any observational evidence that lifeguards choose paths this way, rather than the direct line or shortest water path?
Perhaps they do some optimization, particularly for very oblique angles, but in real "shoreline" situations, I expect wave and current action will dominate the optimization, rather than the delta between running and swimming speed.
Guidelines for lifeguards talk about avoiding water entry if possible and definitely avoiding direct contact from the victim if at all possible (https://www.sauvetage.qc.ca/en/lifeguarding/rescue-technique...) but don't really address the run-vs-walk choices.
Funny that both top level comments about current got down-voted, while a fifth-level comment about electrical current go up-voted.
Seriously, as someone going surfing tomorrow morning, current was my first though on this. And rips. The catching the rip will save more time than any of this.
I figure that animals and humans are very good at assessing the amount of “work” that a given path requires and that we are likewise very sensitive to minimizing the work function, due to evolutionary training.
Yes. (Or at least we have a good model.) One description is that a photon takes all paths simultaneously and the paths that are not the shortest in time end up mostly cancelling out. The light's behavior is consistent with locality.
> When light travels from one place to another, it always takes the path of least time.
Refraction is seen in other kinds of waves and media, like sound or even waves on the surface of a liquid (which refract when traversing from a deeper to shallower place).
Is this "least time" principle true for all kinds of waves, or just light?
Huygens–Fresnel principle applies to waves in general.
Fermat's principle follows mathematically from Huygens' principle at the limit of small wavelength.
Does anyone know how to solve the tennis-ball formula?
All I can muster is that the distance on the beach (b) divided by the speed on the beach (Vb) + the distance on water (w) divided by the speed on water (Vw) should be minimized. So I should find the minimum of b/Vb + w/Vw , but I have no clue as how.
You got the first equation right, in expressing time-taken in terms of b and w.
Now you need the second equation, which expresses b in terms of w (or vice-versa). Hard to describe it here over text, but I believe using the following set of equations, it should be possible to express b in terms of w.
------------------
Yb = vertical-distance from start-point to water (constant)
Yw = vertical-distance from water to end-point (constant)
Xb = horizontal-distance between start and point-of-contact with water
Xw = horizontal-distance between end and point-of-contact with water
X = horizontal-distance between start/end points (constant)
Ob = angle taken when running towards the water
Ow = angle taken when swimming towards the end point
You need to draw some triangles and relate b to w. Assuming a still, straight shore, the answer will depend on your distance to the waterline, the ball's distance to the waterline, and the distance parallel to the shore separating your starting spot and the ball's spot.
On a graph, plot b/Vb, with b on the horizontal axis and b/Vb on the vertical axis. This plot shows the time you spend on the beach if you run to b on the beach.
Add to this graph a plot of w/Vw. You'll have to express w as a function of b to do this. This plot shows the time you spend in the water if you run to b on the beach and then swim to the ball.
You are trying to minimize the sum of these two plots.
You can see from the plots that as b increases, beach time goes up, and water time goes down. As b decreases, beach time goes down and water time goes up.
If b is less than the minimizing point, water time changes more than beach time as b changes, so you can lower the total by moving b in the direction that makes water time go down. That will make beach time go up, but since water time changes more than beach time, it's a net win.
When b is more than the minimizing point, beach time changes more than water time as b changes, so you can lower the total time by moving b in the direction that makes beach time go down. The will make water time go up, but since beach time changes more than water time, it's a net win.
The minimizing point is the point where the rate beach time is changing and the rate water time is changing match, so that you cannot lower the time by nudging b in either direction.
What you could then do is think about how to calculate the rate of change of a function. Say you have a function, f(x), and you want to know how fast it changes as x changes. To help with intuition, let's say f(x) is the position of a car at time x. The rate of change of position is velocity, so we are asking how you find the velocity at a given point x if you know position as a function of x. If the car was moving at a constant velocity, this would be easy. Just note the position at x, f(x), and then note the position a little bit later, say at x+t, f(x+t). Then the distance traveled is f(x+t)-f(x), in time t, so the velocity is (f(x+t)-f(x))/t.
If the car is not moving at constant velocity that just gives you the average velocity during the time t. If the car isn't accelerating too much, the smaller t the closer the average velocity over an interval t should be to the instantaneous velocity at t.
For example, suppose f(x) = x^2. Then f(x+t) = x^2 + 2xt + t^2. The average velocity is 2x + t. It's pretty obvious that as we make t smaller and smaller, this gets closer and closer to 2x, and so we can reasonably conclude that the instantaneous velocity of something whose position at time x is f(x) is 2x.
Applying those same ideas to the functions involved in the ball problem, you could figure out the rates of change, and find the b that makes the beach time and the water time rates of change match. Unlike the nice x^2 car, though, the algebra would be messy and error prone.
After you have done that, you might realize that this operation of take an average over smaller and smaller intervals and seeing if that converges to some fixed value as the interval goes to zero is quite useful, and start studying it. You'd then discover that while computing it directly can be a royal pain in the ass, it turns out that if your function is built up of other functions by adding, subtracting, multiplying, dividing, or applying functions to other functions, and you know how to do this operation on the component functions, then there are simple rules that you can follow mechanically do figure it out on the combined function.
At that point, you've pretty much invented half of calculus as it was understood back in the days of Newton and Leibniz, and are ready to tackle all kinds of minimization and maximization problems in physics, engineering, finance, and elsewhere, plus all kinds of problems involve rates of change, such as bacteria growth, velocity of water coming from a hole in the bottom of a leaky bucket, and rocket motion.
This is all definitely math and physics interesting but a real beach has waves, currents and rocks. The lifeguard has spent weeks or months staring at it for hours on end. Probably has a good idea best way to save someone. Otherwise continue.
We could hypothesize dogs are working out calculus problems, or we could just hypothesize that the dog tried it one way, then another, and eventually over the course of its lifetime works out how to get to the ball the most quickly, which in comparison to the difficulty of manipulating all of its dozens of muscles to perform the running optimally in the first place is a frankly insignificant challenge. Or, to put it into a human reference frame, we do not do calculus problems to catch things. We "just" catch them. Neurons are built to solve this problem.
We could hypothesize that the ants have a simple pheromone-based optimization system that naturally mathematically converges on the same solution as Snell's Law when, after all, faced with the same problem, but... uh... actually it's not at all clear to me how the article manages to make this sound mysterious after what is a fairly effective solution to the problem. If nothing else, the ants taking the most optimal path will make more to-and-from trips than other ants, thus laying thicker trails; that plus an initial shotgun approach are adequate to explain.
This is particularly egregious since none of the math here is very complicated, and it is completely unsurprising that when the same problem is given to several optimizing agents in several domains that the same solution pops out. There is not very many bits of information in this problem.