# The Beauty and Power of Mathematics | William Tavernetti | TEDxUCDavis

Translator: Suleyman Cengiz

Reviewer: Lisa Thompson 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)