To Save Drowning People, Ask “What Would Light Do?”
nautil.us
nautil.us
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.
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?
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.)
But there's only one min/max isn't there? Otherwise all values may be permitted, and we'd have entropy.
The vast body of physics may concern behavior simple enough to be expressed, but much of reality behaves in complex ways?
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.
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.
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.
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.
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.
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.
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.
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.
This is a really cool example of plants using math too.
It's plenty bizarre that electricity follows the path of least resistance, light the shortest path, etc.
If you have a 1000 Ω path and a 10 Ω path (in parallel), current will go through both paths. However, much more current will go through the 10 Ω path.
The current through each path ends up being the voltage applied across the path, divided by its resistance.
"Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong."
- H L Mencken
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.
I suggest you think a little more before sanctimoniously talking down to people. I was making a supportive reply, in agreement with what you wrote.
> at the very least you should make sure it is actually applicable to the topic at hand.
It is perfectly applicable. Light "being conscious and making decisions" is a neat explanation, but wrong.
Why is this mysterious?
This is the model of Nautilus, Quartz, and a bunch of other online publications. It's fun to read so what's wrong with it?
(I haven't read the OP because it sounds like things I already know.)
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.
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.
I could see the argument that there is a fine line between leaving a bit of mystery and just plain lazy writing.
When the reader is interested by the mystery and really goes to find more on his own, isn't that wonderful?
Perhaps it is fun for people who like being misled?
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 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.
[1]: https://d3chnh8fr629l6.cloudfront.net/2904_b8b9c74ac526fffbe...
[2]: https://webcache.googleusercontent.com/search?q=cache:l0DmOI...
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.
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.
See e.g. https://en.wikipedia.org/wiki/History_of_variational_princip...
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.
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.
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.
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.
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?
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
=>
X = Xb + Xw
cos(Ob) = Yb / b
sin(Ob) = Xb / b
cos(Ow) = Yw / w
sin(Ow) = Xw / w
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.