The articles main thesis is that an intuitive, artistic understanding of turbulence can help inform mathematicians.
And yet none of the historical evidence that was presented supported this claim. The pure mathematicians made progress and discovered interesting patterns. Those who focused on art and philosophy (Schauberger and Schwenk) contributed nothing to the understanding of turbulence.
I read the article differently, namely, how the approaches of both art
and science can be similar and often complimentary to understanding and
appreciation. At times, looking at another perspective can really help
improve understanding and appreciation, but at other times, an alternate
perspective is merely entertaining. I wasn't expecting the author to
provide strong and specific evidence of "artistic understanding helping
to inform mathematicians." But saying "none" was presented seems a bit
unfair; the evidence presented was by inference and was anecdotal. For
example, both artists and experimental scientists arriving at remarkably
similar ways of "recording data" and/or "making art" from turbulence in
fluids. Even if the artist doesn't fully understand the math, and the
mathematician doesn't fully understand the art, both can often gain a
more enlightened appreciation of each others' work than someone unversed
in both art and math.
A lot of educators agree that studying both art and math are important
to development and are complimentary to both appreciation and understanding.
I guess I didn't read it close enough? I think the inclusion
of art and curly hair was to make the article more readable? As an idiot--I am today, I started to think about curly hair. I have curly hair , I don't know why I have curly hair. I was told curly hair
arises fron asymmetrical follicles? My hair was straight as a child. I hit puberty, and it went to a perm! Now, as I
age it's just wavy? I guess it's genes, and hormones? Or,
I remember hearing someone state homosapians have two types of hair--thick, and thin; thick hair being curlier. Or, maybe my curly hair has something to do with Fibonacci equation? Or, maybe I should exercise tonight? Yea, it's been a bad weekend for me--I hope Tomorrow is better?
I did my GCSE physics project on Reynolds' number, many moons ago. The actual experiment itself was simple, but the hard bit was creating several-meter long hollow glass syringes in order to inject die into the flow tube well into its flow, in order to avoid turbulence where the header emptied into it... ended up dangling borax glass tubes off the roof with a 1kg weight lashed onto the end, and then blasting the centre with a blowtorch - serious fun.
Anyway, long story short, it's a really interesting experiment to run, Reynolds number (for water, at least) can be deduced by a teenager with about £100 of equipment, and the boundary between laminar and turbulent flow is a mysterious beast indeed.
One interesting artefact was inducing turbulent flow in speeds "too slow" for turbulence, by increasing flow to the point where turbulence triggered, then decreasing flow to the laminar realm again, yet retaining the turbulence indefinitely. Interestingly, this low-speed turbulent domain required a smaller throttle on the header tank to maintain the same flow rate - i.e. the turbulence decreases resistance to flow in the tube. Something that pipeline engineers could/should/maybe do employ.
Had to pay for a new floor at the end of it all, as there's no tidy way of running this one.
The author mentions that having an equation that describes a system accomplishes little if the solutions to that equation are unavailable. He kind of downplays quantum mechanics, but I would argue the situation is just as severe as that with turbulence. The Schrödinger equation
iħ(∂Ψ/∂t) = HΨ
is a very simple equation. Yet the solution to this equation is nearly impossible to solve for all but the simplest systems. A lot of approximations to it have been developed, but to numerically converge on the exact solution is an NP-hard problem in most cases (using QMC). And we all know what that means.
There's a famous quote attributed to Max Born: "It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem."
In other words, what if we discover all the equations that govern nature, but they are simply infeasible to solve (and worse, we even mathematically prove that they're intractable)? It's an interesting, yet somewhat depressing idea.
1) What you've written, where H is independent of t is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics. Even with time-dependent hamiltonians we have decent tools for talking about the solutions (Dyson series).
2) Of course for certain values of H, (esp. in continuous space) you can contrive ways of making the eigenvalue problem hard, but you don't have to go as far as quantum mechanics to find difficulty. Just take three bodies under newton's gravitation.
Yes, the quantum n-body problem is exponentially harder in n, but that's a fundamentally different type of "hardness" than the hardness of Navier-Stokes.
1) is reducible to just the Eigenvalue problem for H. Certainly better understood than fluid dynamics.
I feel like that's akin to stating that Fermat's theorem just shows there is no n > 2 such that x^n + y^n = z^n
2) you can contrive ways of making the eigenvalue problem hard
You've got this backward. You can contrive ways of making it easy. Almost all physical problems for any system larger than but the simplest of atoms is incredibly hard.
Maybe I'm just a glutton for punishment, but for me, intractable and even
undecidable problems are, if you pardon the pun, endless fun. Routing
of PCBs and ICs is one of my favorites, and decompliation/disassembly is
another. I'm just totally content with knowing that I'll never be "that
guy" who gets famous for finding "the" perfect solution (especially if
we can already prove a perfect solution doesn't exist). But with enough
effort, I might be one of the many people who come up with one of the
many increasingly better incomplete solutions. A small step in the right
direction, like the approximations of Schrödinger's equation you
mentioned, can be very useful as well as an amazingly fun challenge.
It's a bit like those video games that keep getting tougher but never
end; all the fun is in seeing how far you can get.
I'd risk the optimistic expectation that human ingenuity can work with such a state of affairs.
Here's one example I can think of: if an interesting measure in a phenomenon follows some equation, we could find some solutions to that equation through experimental approach treating the phenomenon as a program and Nature as the compute engine. Very often you can prove statements about relationships among solutions to an equation (e.g. they may form a linear or affine space). You could then use the original experimental solution to generate further solutions using numerically tractable transformations.
The scientists who study turbulence are quite aware of its beauty. Indeed, the Division of Fluid Dynamics of the American Society of Physics has an image contest at their annual meeting -- it's quite competitive.
Their online gallery format could use some modern-design love, but the images are still beautiful:
To art and math, I would add sport: whitewater kayaking and surfing are two sports in which participants directly engage and experience transitions between smooth and turbulent flows in water.
I doubt they will lead anyone to mathematical breakthroughs. But they are both a lot of fun, and can go beyond physical exertion to inspire deep emotion and thought, even spiritual experiences.
Edit to add: A fun book along these lines is "The Wave", which is a pop-science book that contrasts big wave surfers with oceanographers studying rogue waves.
And yet none of the historical evidence that was presented supported this claim. The pure mathematicians made progress and discovered interesting patterns. Those who focused on art and philosophy (Schauberger and Schwenk) contributed nothing to the understanding of turbulence.