Redefining the impossible
In the comments on my last post, I mentioned that My favourite example of spectacular theorem failure is flight: in a perfectly inviscid fluid, it's impossible (no viscosity means no starting vortex which means no circulation around the wing), but it is possible in a fluid with viscosity > 0, no matter how small.
It's a rather beautiful result, I think, and should serve as a handy warning against people who want to convince you of something using their mathematical model whose assumptions are almost true. But (in the form stated above) it's not entirely correct.
The first thing to note is that I meant to say "heavier-than-air flight". As far as my dim recollection of fluid dynamics can tell me, there's no obstacle to flying a hot air balloon or an airship in a perfectly inviscid medium. However, propellors and fans are essentially sets of wings joined at the hip, so you might have some trouble propelling your craft anywhere other than where the wind was blowing¹. Which leads us to what I really want to talk about...
It turns out that there is a small wasp, Encarsia formosa, which makes use of an entirely different method of flight that does not require a starting vortex. It takes advantage of a hidden assumption of the impossibility proof: that the topology of the surrounding medium does not change. E. formosa, when in hovering flight, briefly touches its wingtips together at the apex: this changes the topology of the surrounding air, and gets around the prohibition. And indeed (see the papers referenced in that hastily-edited Wikipedia article) this method of flight would, apparently, work in a fully inviscid medium.
This sort of thing is surprisingly common, I find. Quite a few times over the last few months, I have encountered some problem that seems impossible to solve; typically of the form "I want to do both X and Y simultaneously, but X precludes Y because...". Several times I've gone as far as constructing a semi-formal impossibility proof, and a couple of times I've presented said proof to my boss as an explanation for my lack of success. And then, a few hours or days later, I've realised that while it may indeed be impossible to do what I've been trying to do, it is possible to achieve the desired effect in some other way: my impossibility proof contained some hidden assumption about the form of the solution, and by subverting that assumption we can proceed to solve the true, more general, problem.
So I think the lesson here (which seems almost comically trite, now I come to write it down explicitly), is that even if what you're trying to do is definitely, provably, mathematically impossible, you shouldn't necessarily give up straight away. Rather, you should attempt to redefine your notion of success, and see if you can achieve that instead.
¹ I suppose you could power your craft with a ramjet, but that leads to a chicken/egg problem. Engineers: am I right in thinking that conventional jet engines rely on a fan-like compressor to get started, and thus wouldn't work in a fully inviscid fluid?
It's a rather beautiful result, I think, and should serve as a handy warning against people who want to convince you of something using their mathematical model whose assumptions are almost true. But (in the form stated above) it's not entirely correct.
The first thing to note is that I meant to say "heavier-than-air flight". As far as my dim recollection of fluid dynamics can tell me, there's no obstacle to flying a hot air balloon or an airship in a perfectly inviscid medium. However, propellors and fans are essentially sets of wings joined at the hip, so you might have some trouble propelling your craft anywhere other than where the wind was blowing¹. Which leads us to what I really want to talk about...
It turns out that there is a small wasp, Encarsia formosa, which makes use of an entirely different method of flight that does not require a starting vortex. It takes advantage of a hidden assumption of the impossibility proof: that the topology of the surrounding medium does not change. E. formosa, when in hovering flight, briefly touches its wingtips together at the apex: this changes the topology of the surrounding air, and gets around the prohibition. And indeed (see the papers referenced in that hastily-edited Wikipedia article) this method of flight would, apparently, work in a fully inviscid medium.
This sort of thing is surprisingly common, I find. Quite a few times over the last few months, I have encountered some problem that seems impossible to solve; typically of the form "I want to do both X and Y simultaneously, but X precludes Y because...". Several times I've gone as far as constructing a semi-formal impossibility proof, and a couple of times I've presented said proof to my boss as an explanation for my lack of success. And then, a few hours or days later, I've realised that while it may indeed be impossible to do what I've been trying to do, it is possible to achieve the desired effect in some other way: my impossibility proof contained some hidden assumption about the form of the solution, and by subverting that assumption we can proceed to solve the true, more general, problem.
So I think the lesson here (which seems almost comically trite, now I come to write it down explicitly), is that even if what you're trying to do is definitely, provably, mathematically impossible, you shouldn't necessarily give up straight away. Rather, you should attempt to redefine your notion of success, and see if you can achieve that instead.
¹ I suppose you could power your craft with a ramjet, but that leads to a chicken/egg problem. Engineers: am I right in thinking that conventional jet engines rely on a fan-like compressor to get started, and thus wouldn't work in a fully inviscid fluid?