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 Romanesque Oh, 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 and 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 Mandel Broad set.

This visualization is the result of a relatively simple math formula that includes something called Riker JH, in which applies the same mathematical step, each further iteration of the shape like George Cantor did when exploring the beginnings of fractal math back in the 18 eighties.

You take a line and you divide it in half, take each product of that division and apply that same previous step on on and on and on into infinity of a curse.

A pattern.

But let's look at some more complex visuals.

Shall we take the Coke 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 1/3 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 Coke 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 its perimeter is getting longer.

In fact, the theoretical Coke 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 cook 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 it's fractal dimensionality.

The Coke snowflake doesn't exist in one dimension, and it also isn't quite two dimensional.

It's for fractionally dimensional and has approximately 1.26 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 Coke example, too, of the modern antenna.

Regular antennas are designed to receive a certain range of signal types under 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 three D version of this kind of antenna in our cell phones.

Fractal math helps us describe and model natural phenomena to to make computers and relations of our circulatory and nervous systems or natural watersheds or brain waves.

To better understand how all of these work on 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 on see a fractal out of electrons, specifically Mr Pinsky Triangle.

To understand how this happened, we have to go back to the dimensionality of fractals.

Electrons behave differently based on what configuration there in because of the way their charges pull on each other.

There 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 Elektronik wave function showed that went in this shape.

The electrons behaved as if they were in 1.58 dimensions.

The fractional dimension expected of a serpent ski triangle.

Teeny sir Pinsky Triangle fractals just 20 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, then 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, just one last one, because it's so cool.

The neurons in our brain are fractal, which currently limits the applications of neural implants.

But with fractal electronics, the bio electric 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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Thanks for watching.