The beauty and power of mathematics | William Tavernetti | TEDxUCDavis

William Tavernetti has a PhD in Applied Mathematics from UC Davis and is currently a lecturer at UC Davis in the department of Mathematics. William also works as a teacher fellow for the Introduction to Engineering Mechanics Cluster at the California State Summer School for Mathematics and Science (COSMOS). Prior to graduate school, William worked as a Junior Reliability Engineer for Valador Inc., supporting the NASA Altair lunar lander. William also holds a BA in Philosophy and a BS in Mathematical Sciences from UC Santa Barbara.

This talk was given at a TEDx event using the TED conference format but independently organized by a local community. Learn more at http://ted.com/tedx

Some people look at a cat or a frog,
and they think to themselves,

"This is beautiful, nature's masterpiece.

I want to understand that more deeply."

This way lies the life sciences,
biology for example.

Other people, they pick up an example

like a roiling, boiling Sun,
like our star,

and they think to themselves,

"That's fascinating.
I want to understand that better."

This is physics.

Other people, they see an airplane,
they want to build it,

optimize its flight performance,

build machines to explore
all the universe.

This is engineering.

There is another group of people

that rather than try to pick up
particular examples,

they study ideas and truth at its source.

These are mathematicians.

(Laughter)

When we look deeply at nature and really
try to understand it, this is science,

and, of course, the scientific method.

Now, one way to divide
the sciences is this way:

you have the natural sciences,
that's physics and chemistry

with applications to life sciences,
earth sciences, and space science.

You have the social sciences,

where you'll find things
like politics and economics.

There's engineering and technology,

where you'll find
all your engineering fields:

biomedical, chemical,
computer, electrical,

mechanical and nuclear engineering.

And all the applications of technology:

biotechnology, communications,
infrastructure, and all of that.

And last, but certainly
not least, is the humanities,

where you'll find things
like philosophy, art, and music.

Now, math does show up
in all of these disciplines;

in some, like physics and engineering,
its role is quite pronounced and obvious,

while in others, like in art and music,

the role of mathematics is definitely
somewhat more specialized

and usually secondary.

Nevertheless, math is everywhere,

and for that reason, math is especially
good at making connections.

How? How does math make connections?

This is an excellent question.

It's actually a question
that's not easy to answer.

I think, for us now, in our time together,

the best we can do is get a sense
of what the answer might look like

by examining some of the connections
that mathematics can make

through the lens
of some mathematical ideas.

Now, math is, of course, numbers,

and perhaps the most famous number
of all is the number pi.

Pi was discovered because it represents
a geometric property of the circle:

it is the ratio of the circumference
of every circle to its diameter,

but nowhere in the world
is anything a circle.

Circle is a kind of pure,
mathematical idea,

a construction from geometry that says,

"You fix the center point,

and then you take all points that
are equidistant from that center point."

In two dimensions,
this construction produces a circle,

and in three dimensions,
the same construction produces a sphere.

But nowhere in the universe
is anything circle or sphere.

This is a perfect, pure, mathematical
idea, and this world that we live in

is imperfect, rough, atomized, moving,
and everything is slightly askew.

Nevertheless, the number pi

has been astonishingly useful
to us throughout history.

Let's go through
some of that history together.

Around the year 212 BC, Archimedes
was murdered by a Roman soldier.

His dying words were,
"Do not disturb my circles."

He wanted his favorite discovery
put on his tomb.

It's shown here.

It says, basically,
that the surface area of the sphere

is equal to the surface area
of the smallest open cylinder

that can contain that sphere.

Around 1620, Johannes Kepler discovered

what he thought of as a harmony
of planetary motion.

Isaac Newton would later
build on this work.

Shown here is Kepler's celebrated
third law of planetary motion.

From 1600 to 1700, Christiaan Huygens,
Galileo Galilei, and Isaac Newton

were early pioneers,
studying the pendulum,

shown here as a formula -
T for the period of the pendulum,

which tells us something about
how long it takes to swing back and forth.

The greatest mathematician
of the 18th century, Leonhard Euler,

is responsible for
discovering this formula:

e to the i theta equals
cosine theta plus i sine theta.

This formula provides a key connection

between algebra,
geometry, and trigonometry.

In the special case when theta equals pi,

it produces a relationship between
arguably the five most important constants

in all of mathematics:

e to the i pi plus one equals 0.

Some people have called this the most
beautiful formula in all of mathematics.

Leonhard Euler was also an engineer
of some repute, and this formula for F -

the applied buckling force
that a column, as shown in the cartoon,

will buckle under such an applied force -

is shown here.

The greatest mathematician
of the 19th century, Carl Friedrich Gauss,

usually gets the credit
for his work on what we call today

the standard normal distribution.

A staggering amount of real-world data
is distributed this way,

according to what you might know
as the bell curve of probability.

And our tour of history
ends in the 20th century

with Albert Einstein
and his famous theory of relativity.

Shown here are Einstein's field equations.

The difficulty to understand these
equations is not to be underestimated.

Now, that was too fast, I know;
that's a lot of information.

There's no exam,
no midterm, so just relax.

(Laughter)

Remember we're trying
to uncover connections.

Now, look at all of these formulas,
every one of them with pi in it,

this number that is born
from the geometry of the circle.

Look at all of the physical phenomena,
how different they all are,

and yet they share this common connection
to this geometric number from the circle.

So, when you see a formula
and you see pi in it,

you might think to yourself,

"Maybe, somehow, someway,

the circle plays a part
in the derivation of this formula."

Now, a circle is just one geometric form,
and math is so much more,

and so, too, is the world
and the connections that exist within it.

Look here; this is an airfoil in 2D -
like the cross section of a wing.

And the lines you see are
like the air flowing over and under it.

And here, this experiment shows a gas,

initially compressed by a retaining wall
into one side of a vessel.

Then a hole is made in the retaining wall,

and the gas expands
to fill the entire vessel

until it reaches a kind of equilibrium.

In this example, it shows a metal rod
with a source of heat held under it,

in this case a flame.

Where the flame contacts the metal,

the heat will heat the rod
and distribute along the rod

until it reaches
a kind of thermal equilibrium.

And it would not do, it would not do
to have all of this science

without

electricity making an appearance.

Shown here are the potential lines
in an electric field,

which give us the paths
that electrons will take

going from positive to negative charge.

Now, all of these examples
are very different to our five senses -

so different, in fact, that in science,
we give them all a different name.

That's potential flow,

Fick's law of chemical
concentration diffusion,

Fourier's law of heat conduction,
and Ohm's law of electrical conductance.

But in another kind of way, in a math
kind of way, they're all very similar -

so similar, in fact, that in mathematics,
we give them all the same name:

Laplace's equation.

That's not triangle u equals 0,
that's Laplacian of u equals 0.

What changes for the mathematician

is u can be potential,
and u can be chemical concentration,

and u can be heat and many
other physical quantities

that this equation can be used
to describe from nature.

You see, in mathematics, not only do we
have numbers and we have geometry,

but we also have equations, and when we
compare the equations of things,

this gives us yet another way
in which things can be connected.

Now, the connection between
all of these science problems is calculus,

and you should see
that calculus is essential

and foundational
to modern computational science.

Now, we've seen something about numbers
and geometry and equations.

But let's put it all together,

because that's math.

Let's see an application of mathematics.

I want us to go
through a construction here.

This is what we'll call
the first generation.

And this, the second generation.

Look at the pattern, what happens
to positive and negative space.

And then the third generation.

And so you see a pattern start to develop.

Now, in your mind,

decide what the fourth generation
should look like.

Is this your expectation?

And then the fifth generation.

And then the fifth generation.

And then - there we go.

And then the fifth generation
and dot dot dot forever.

That's the fractal.

The pattern never terminates.

It never completes.

There's no end to the complexity,

no smallest part
of this geometric structure.

In fact, this is a famous fractal,
a Sierpinski triangle.

The fractal has never even
been constructed in all of human history.

It's never been completed; it can't be
completed; it never terminates.

When you see a fractal with your mind,
you never see all of it,

you only get the sense of it.

Now, appreciation of fractals
really took off in the 1970s,

after Benoît Mandelbrot's work.

And part of the reason
for the late bloom of this idea

was that it really took
the aid of the modern computer

to properly compute and visualize

this type of tremendous
geometric complexity.

Shown here at the top
is the famous Mandelbrot fractal.

And notice, there is a zoom up
of the tiny segment of the fractal,

magnified so you can see it.

Just look at the complexity
of that region.

If we zoom in there and magnify, no matter
how much we zoom into the fractal,

the complexity will never diminish.

This is not an easy thing to understand.

This geometry is so complicated,

it is unclear if it has any equivalent
in the natural world.

And yet, once people became aware
of the existence of this kind of object,

they started to see examples of it
in applications everywhere.

This is a kind of
Baader-Meinhof phenomenon,

where your mind becomes
primed for knowledge.

And then when you go out

and look after learning it,
you start to see it everywhere.

People started to see fractals
in the geometry of landscapes

and coastlines, like this of Sark,
which is in the English Channel.

People found uses for fractals
in signal and image compression,

and they even saw fractals in the snowfall
deposits on mountain ridges,

like this Google Landsat data on the left
and a fractal that I made on the right

to mimic the same type of structure
and geometric complexity.

Fractals even show up in the geometry
of the snowflakes themselves

and in a staggering number
of biological forms.

There are also notable uses
of fractals in human creative space,

like music and art,

where once people become aware
of the existence of this type of geometry

and they had access to codes

and with their computer they
could make this kind of geometry,

they started to make use of it
in unpredictable ways.

Now, this is an aesthetic
application of mathematics,

but many people study mathematics
just because they find it interesting

or aesthetically beautiful.

Other people want math as a hard skill:
they want to be an engineer,

they want to predict the weather,
they want to go to space.

There is no wrong reason to learn.

Now, you see, mathematics
is like a vast ocean of ideas,

the source of truth.

And today, we took one cup

and walked to the water's edge
and dipped it in the water.

And in our cup was one number, pi;

one geometric form, the circle;

and one equation, Laplace's equation.

And just look at the breathtaking scope
of ideas that we were able to consider.

And finally, in fractals,

we just glimpsed the faintest hint
of an idea about geometric complexity

that expands our experience
of what is possible.

You see,

the power of mathematics is that it
is useful in so many different ways,

and that is the beauty
of learning mathematics.

And to me, this is the meaning
in the words of Galileo:

"If I were again beginning my studies,
I would follow the advice of Plato

and start with Mathematics."

Thank you.

(Applause)