fractals they're gorgeous seemingly
simple and yet infinitely complex but
they're not just pretty to look at what
if we could harness the mathematics of
these fascinating patterns and use them
to totally transform our electronics but
first we need to cover what fractals are
a fractal is a self-similar shape so if
you zoom into it you will see that same
shape again
take a fern each little frond is a
miniature replica of the larger plant or
take Romanesco broccoli a snowflake or
even the blood vessels in your lungs
each smaller part looks like a miniature
version of the whole but the thing about
fractals in nature is that they're not
mathematically perfect the mini version
of the fern leaf may differ slightly
from the whole plus if you zoom in
enough you'll hit the cellular molecular
at atomic levels that don't adhere to
the pattern so there's only so far you
can go before the fractal breaks down
that's in contrast to pure fractals that
are mathematically infinite these are
the kind used to generate those screen
savers you've probably seen the most
famous being the Mandelbrot set this
visualization is the result of a
relatively simple math formula that
includes something called recursion
which applies the same mathematical step
to each further iteration of the shape
like Georg Cantor did when exploring the
beginnings of fractal math back in the
1880s you take a line and you divide it
in half take each product of that
division and apply that same previous
step and on and on and on into infinity
a recursive pattern but let's look at
some more complex visuals shall we take
the Koch snowflake we have two triangles
we lay them on top of one another to
make a star we've taken each line of
that original triangle and transformed
it into four identical lines each
one-third the size of the original that
action can be expressed as a
mathematical formula now we repeat that
process and this is the recursive part
remember with every resulting triangle
and so on and so on until infinity then
every time you zoom in on a tip of the
snowflake you get something that looks
exactly like the step before it so the
mind-blowing thing about the Koch
snowflake is that the size of the object
stays the same the snowflake isn't
expanding as we perform this calculation
at least not significantly but it's
perimeter is getting longer in fact the
theoretical Koch snowflake has an
infinite perimeter but finite area which
is so cool things like this are too
complex for regular geometry you
couldn't take a ruler and go around and
measure the length of each side of each
triangle on the Koch snowflake you would
literally never finish which is why we
have fractal mathematics that do work in
this space the resulting number from
that recursive equation is actually its
fractal dimensionality the Koch
snowflake doesn't exist in one dimension
and it also isn't quite two-dimensional
it's fractionally dimensional and has
approximately one point two six fractal
dimensions it's weird I had literally no
idea that was possible all right enough
with the hypotheticals what about
real-world applications for instance
we've applied the Koch example to the
modern antenna regular antennas are
designed to receive a certain range of
signal types and are physically
constructed to then match that
wavelength but if we use some clever
fractal thinking we can pack more shapes
into a smaller device making an antenna
that is not only compact but can receive
many more kinds of signals we use the 3d
version of this kind of antenna in our
cell phones fractal math helps us
describe and model natural phenomena to
to make computer simulations of our
circulatory and nervous systems or
natural watersheds or brain waves to
better understand how all of these work
done fractal patterns can exist even on
the molecular level with really exciting
results a new study has shown that
researchers have been able to make unsee
a fractal out of electrons specifically
Buster Pinsky triangle to understand how
this happened we have to go back to the
dimensionality of fractals electrons
behave differently based on what
configuration they're in because of the
way their charges pull on each other
they're in one dimension configured
linearly or in two dimensions making up
a sheet or in three dimensions in
something like a cube their interactions
with one another and their resulting
behavior differs based on these
configurations this team used a scanning
tunneling microscope to place carbon
monoxide molecules very carefully on a
background of copper the copper allows
the electrons in these particles to
behave very freely so the measurements
they took of the resulting shape both
spatial and in electronic wavefunction
showed that when in the
shape the electrons behaved as if they
were in 1.58 dimensions the fractional
dimension expected of a sierpinski
triangle Pini sierpinski triangle
fractals just twenty nanometers big the
implications of this brand-new research
have yet to be fully explored but we
know how useful fractals are on larger
scales so taking advantage of fractal
geometries on such tiny scales could
radically change the efficiency and
versatility of our electronics and it
already has research has shown that if
we employ fractal circuit elements you
can get a much larger output signal
compared to the input signal than you
can with traditional circuitry or if we
want to go even further out there
another team recently used fractal wire
patterns to create stretchable
electronics which you could wear on your
skin okay and just one last one because
it's so cool the neurons in our brain
are fractal which curly limits the
applications of neural implants but with
fractal electronics the bioelectric
interface is something that we could do
a lot more with just like the fractal
pattern itself the possibilities here
seem endless and we just have to dive in
to see how best we can apply them to
meet our needs
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mind-blowing science thanks for watching