17
Aug

Fermat Spirals for Layered 3D Printing | Two Minute Papers #77

Dear Fellow Scholars, this is Two Minute Papers
with Károly Zsolnai-Fehér. What are Hilbert curves? Hilbert curves are
repeating lines that are used to fill a square. Such curves, so far, have enjoyed applications
like drawing zigzag patterns to prevent biting in our tail in a snake game. Or, jokes aside,
it is also useful in, for instance, choosing the right pixels to start tracing rays of
light in light simulations, or to create good strategies in assigning numbers to different
computers in a network. These numbers, by the way, we call IP addresses. These are just a few examples, and they show
quite well how a seemingly innocuous mathematical structure can see applications in the most
mind bending ways imaginable. So here is one more. Actually, two more. Fermat’s spiral is essentially a long line
as a collection of low curvature spirals. These are generated by a remarkably simple
mathematical expression and we can also observe such shapes in mother nature, for instance,
in a sunflower. And the most natural question emerges in the
head of every seasoned Fellow Scholar. Why is that? Why would nature be following mathematics,
or anything to do with what Fermat wrote on a piece of paper once? It has only been relatively recently shown
that as the seeds are growing in the sunflower, they exert forces on each other, therefore
they cannot be arranged in an arbitrary way. We can write up the mathematical equations
to look for a way to maximize the concentration of growth hormones within the plant to make
it as resilient as possible. In the meantime, this force exertion constraint has to be taken
into consideration. If we solve this equation with blood sweat and tears, we may experience
some moments of great peril, but it will be all washed away by the beautiful sight of
this arrangement. This is exactly what we see in nature. And, which happens to be almost
exactly the same as a mind-bendingly simple Fermat spiral pattern. Words fail me to describe
how amazing it is that mother nature is essentially able to find these solutions by herself. Really
cool, isn’t it? If our mind wasn’t blown enough yet, Fermat
spirals can also be used to approximate a number of different shapes with the added
constraint that we start from a given point, take an enormously long journey of low curvature
shapes, and get back to almost exactly where we started. This, again, sounds like an innocuous
little game evoking ill-concealed laughter in the audience as it is presented by as excited
as underpaid mathematicians. However, as always, this is not the case at
all. Researchers have found that if we get a 3D printing machine and create a layered
material exactly like this, the surface will have a higher degree of fairness, be quicker
to print, and will be generally of higher quality than other possible shapes. If we think about it, if we wish to print
a prescribed object, like this cat, there is a stupendously large number of ways to
fill this space with curves that eventually form a cat. And if we do it with Fermat spirals,
it will yield the highest quality print one can do at this point in time. In the paper,
this is demonstrated for a number of shapes of varying complexities. And this is what
research is all about – finding interesting connections between different fields that
are not only beautiful, but also enrich our everyday lives with useful inventions. In the meantime, we have reached our first
milestone on Patreon, and I am really grateful to you Fellow Scholars who are really passionate
about supporting the show. We are growing at an extremely rapid pace and I am really
excited to make even more episodes about these amazing research works. Thanks for watching, and for your generous
support, and I’ll see you next time!

• Two Minute Papers says:

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• Kram1032 says:

Nice stuff although I really wish you'd have gone just a tiny bit more indepth about these here. From this presentation I don't really have any idea what makes these spirals "fermat" as opposed to "archimedes" or "logarithmic" or what not.

It turns out that, for fermat's spiral, the radius grows proportionally to the square root of the number of turns, rather than linearly or exponentially or otherwise. I think the special spacial efficiency comes from precisely this: The length of the line increases linearly (duh) with the radius whereas the area increases to the square of the radius. So if you set it up that the radius increases like the square-root as the number of turns increase, you get a really nice, even fit in terms of area-coverage.

For the reasoning behind plants' affinity to the golden ratio or, indirectly, the fermat spiral, I highly recommend ViHart's mini-series on the topic. She also goes into the pheromone explanation described in the last of these videos:

• Neoshaman Fulgurant says:

math is a linear representation of massively parallel simple interaction, it's like projecting a complex space on a line. Of course nature find optimization with simpler rules than the mathematical description. Can't wait till we move from turing complete machine to UMM paradigm with quantum computing OR memcomputing, mind will be blown. I just can't understand how to program the later though lol, it seems to be configuration driven or at least we lack the right hi level abstraction, it can be argue that a cpu is configuration driven too, do learning to program FPGA help?

• RelatedGiraffe says:

You can also use Hilbert curves to fill up quadtrees and octrees. This can be useful for load balancing when for example solving partial differential equations on multiple computers and you want to get a low amount of communication between computers (which I guess you already know): http://j.teresco.org/research/publications/octpart02/octpart02.pdf

• RelatedGiraffe says:

Also, interesting way to print 3D objects. But aren't those rather Archimedian spirals than Fermat spirals? An Archimedian spiral keeps the same distance between each layer everywhere (which it looks like this fabric also does), while the layers in a Fermat spiral grows closer and closer together the further out from the center you get.

• Zergling says:

That's a cat?

Wait, I see it now.

• Jesse King says:

Amazing video, as always

• The Other Other says:

So will this beat hexagonal filling in the amount of material needed for the same structural integrity?

• Ralf Dietrich says:

I really like ur style.

• Jesse Wilburn says:

keep it up. the videos amaze me everytime

• Anastasia Dunbar says:

Is there a 3D printed comparison video between a one without Fermat spirals and one with.

• 陈龙 says:

at this point in time — it's a redundant expression to be avoid