Transcriber: Marcos Vinícius Melo da Silva Reviewer: Emma Gon
Hello. I’m Maury and today I’m going to give you
a brief introduction to the Fibonacci sequence and Phi
as it relates to geometry, mathematics and nature.
And hopefully offer you a shortcut or a template
to a deeper understanding of our world
through the recognition of some very simple patterns and shapes.
These are the Fibonacci numbers.
It’s a series of numbers that goes on forever.
And the way it works is you add the preceding two to get the third.
So zero and one is one; one and one, two;
two and three, five and so on.
This is the golden rectangle. It has dimensions of 1 to 1.618.
1.618 is an abbreviated form of Phi
that is an irrational number that goes on forever similar to Pi.
So we use the Fibonacci sequence to build out this shape.
Let’s see how that works.
One, one, two, three. So on and so forth.
And we get our golden rectangle.
Now nested within there are further golden rectangles
as we move through the shape.
If we draw a curve through those squares, we get a spiral.
The spiral is the most common shape found in nature.
We see it on a grand scale in galaxies
and on a small scale at CERN,
where they shoot particles through superheated liquid nitrogen.
Based on the charge of the particle,
they’ll spin off and create a spiral in a direction.
We see it in pinecones, storms, a fluorescence within this flower.
In fact, we’re covered in spirals.
You may not know this, but this is a sweat pore.
Doesn’t matter how pretty you are, you got these all over you.
Well, we see here in the flower is that nature looks at this circle
and wants to fill this circle in the most efficient way possible.
It does this by moving five number of degrees
around a circle radiating out to fill the void.
If we do that, mathematically to create a template,
place that template on top, we see that it fits almost perfect.
If we’re 7/100 of a degree off, we don’t get the shape.
We get this shape.
If we're 7/100 of a degree in the other direction,
we get this shape, both nice but not found in nature.
It’s very specific what needs to be there.
There are many symmetries in nature.
Corn, for example, has a 180 degree rotational symmetry.
So as corn grows, you see the leaves
come out 180 degrees from the plant.
If you were to look down on corn, the leaves would overlap each other.
A more common symmetry, though, is this one.
The leaves move around the plant at 222.5 degrees.
You can see this in artichokes, celery, succulents and sunflowers.
Now. Why 222.5 degrees?
Well, if we take 360 and rotating around
and we subtract 222.5, we get 137.5.
Well, if we take a rectangle with dimensions 137.5 by 222.5,
we get the golden rectangle again, Phi.
Fibonacci numbers are also found in nature in other ways.
In flowers, any flower has a Phi number of petals.
Three, five, in wild carrot, coffee, blueberry, okra.
Different varieties of daisies have 55 or 89,
and sunflowers have as many as 144 or more.
All Fibonacci numbers, all numbers found in the Fibonacci sequence.
What’s interesting about this celery
is that we see that stocks move around in 222.5 degrees.
But when we take a cross-section, we see something else,
we see a pentagon.
Now, did you pay attention or remember
what the five symmetry petals had in a few slides earlier?
They were all edible.
So it turns out that almost all edible plants have five petals on their flowers.
Now, that is not to say that that all five petal plants are edible.
So, no, no foraging. I'm not going to be responsible for that.
But of the plants that are edible, almost all of them have five petals.
And that makes sense, because if there’s one shape
that personifies the Phi ratio the most,
it’s the pentagon and the pentagram.
So if we take the pentagram
and we break it up based on all its dimensions
and we compare those dimension to one another,
we find that they all represent the five ratio.
It's repeated in multiple ways throughout the shape.
I first became interested in Phi because I don’t know,
I always kind of paid attention to these patterns,
I was kind of into it.
And when I started studying it and looking into it,
I felt that I could see relationships between things at a small scale.
You then understand things at a greater scale.
Now, one of the things that I first remember seeing
were cracks in the sidewalk.
And I want you right now to imagine a crack in the concrete
and what you think that crack looks like.
Now you have that image in your head, here we are.
Do you see order? Do you see chaos?
Do you see any patterns forming?
There actually are very specific patterns that you can see,
when you go out and leave today, you can look and see.
And that is that the cracks are always divisible by three,
actually not just divisible by three, they are three.
They make a “Y” crack.
Now, where else can we see this?
We see it in trees, oftentimes not always.
Ripe fruit when it splits open, lightning, paint chips,
your veins and the capillaries in your body, neurons.
In fact, in biology, it’s very prevalent to see this “Y” split.
Lightning and trees really stood out to me and I started
seeing trees as really just organic lightning growing really, really slow.
So one thing you might notice when you look at this stuff is
it has a fractal like nature.
Not exactly fractals. Fractals are not defined by classical geometry
because when you look at a fractal, right, one becomes two,
two become four, four become eight.
But when you zoom in on it, it just continues on and on and on.
So it can’t be defined by classical geometry.
The definition goes on to say that fractals are especially apparent
in natural forms and phenomenon because the geometric properties
of the physical world are largely abstract.
That word abstract stuck out to me
because abstract is a sense of dislocation.
But the way I understand fractals, they have more order than that.
In fact, the definition of abstract, existing in thought or as an idea,
but not having a physical or contcrete existence.
So, the reason that fractals have this order is
because they can only exist and grow under a very particular growth rate.
Too shy a growth rate, they’ll peter out, if it’s too small.
If the growth rate is too great, they’ll crowd each other out and stop.
There’s only one growth rate that will allow a fractal,
any fractal to go on forever, ever and ever.
That growth rate is Phi.
Phi is also found in hexagons.
We described pentagonal geometry as regenerative growth.
Now we’ll talk about hexagons.
Snowflakes are always hexagons
and you can construct a snowflake by one right triangle.
Johannes Kepler, 400 years ago, is famous for many things,
one of them being the elliptical orbits of the planets.
Well, it turns out during his many discoveries,
he also wrote a book on snowflakes in which he poses the question,
who carved the nucleus before it fell on to six horns of ice?
which is an amazing thing for him to say.
He describes the creation of a snowflake
a lot different than you and I might.
I think the creation of a snowflake,
I think warm and cool air under the right circumstances.
You get a hexagon, you get a snowflake, It floats down, It lands.
He described the formation of a snowflake
as matter being formed through the prism of an octahedral polyhedron.
That’s hard to imagine until you draw it. This is an octahedron.
Now it’s one of the five platonic solids. There’s only five of them.
And what makes them unique is that this one’s made of triangles.
Each of the platonic solids are made of one polygon
like a triangle or a pentagon, and they’re tiled around each other.
And there’s only five of them that you can tile with the exact same shape.
This is an octahedron. It has eight sides.
What he said was, if you connect the vertices of a octahedron
and tilt it in such a way that you create a two dimensional matrix,
a snowflake emerges through it.
What a beautiful way to understand the formation of matter.
So let’s try. Let’s try. Let’s use Kepler’s philosophy.
This is a hexagon or is it?
It’s actually a two dimensional representation
or view of an icosahedron,
which is another one of the platonic solids.
To construct this shape, you need three rectangles.
You take these three rectangles, move them together at 90 degrees,
create the latticework connecting the vertices.
These rectangles have to be a very particular size and ratio.
They are golden rectangles.
So we see the Phi ratio built into regenerative growth
for the pentagon and also the hexagon.
The hexagon is found in nature in many places,
the compound eye of an insect, a turtle shell,
the honeycomb in a beehive, basalt rock formations,
the pattern on a giraffe, the north pole of Saturn,
which is a huge hexagonal cloud formation
about an Earth and a half wide.
If the pentagon can be described as regenerative growth,
what can we describe the hexagon as?
I would say structure.
When nature wants to grow something, it uses the pentagon.
When nature wants to build something, it uses the hexagon.
And there are certain reasons for that.
The hexagon can be tile flat, the pentagon cannot.
But if we understand that in the tangible world where we can see it,
what does that mean for the intangible world to take a leap?
What about your thoughts?
Now I know what it means to have momentum.
I know all of you know what it means to have momentum, to be on a roll.
I know that those are maximum states of efficiency.
I think we would all agree that that’s what that is.
So I know what nature does.
in maximum states and efficiency it uses geometry.
So then, what’s the shape of your thought?
A long time ago, structures were built using the PhI ratio.
The use of Phi ratio to build stuff, it’s never going to go out of style.
What I learn and look at when I look at these images is
that I know something about the people that made them.
I know that it is important to them to reflect nature's pattern.
The utmost importance because these were no small tasks.
What if our infrastructure buildings were built based on the Phi ratio,
based on these natural patterns
that are found in nature, that matter is created from.
Then they would all fit into an existing kind of matrix
and that personifies sustainability to me.
It’s much better than arbitrary rectangles
or squares to fit in a space.
They’re designed through the reflection of nature.
Maybe a long time ago, thousands of years before electricity,
when the only great technology on Earth was mankind
and we had to know about the seasons,
we had to know about the tides and all the different cycles,
we had to know that to survive.
So maybe in those moments we looked inward more
and saw reflected in us a deeper pattern.
I will close with this.
If we use a universal pattern as a foundation
and then take our endeavor, our creativity,
and place it upon that foundation that’s concrete,
then we will always build and exist within an existing framework
of nature’s design, timeless and efficient.
Thank you.