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!
