William Tavernetti has a PhD in Applied Mathematics from UC Davis and is currently a lecturer at UC Davis in the department of Mathematics.
Here is the full text of Tavernetti’s talk titled “The Beauty and Power of Mathematics” at TEDxUCDavis conference.
TRANSCRIPT:
Some people look at a cat or a frog, and they think to themselves, “This is beautiful. Nature is 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, 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.
And 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.
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 science is this way: You have the natural sciences – that is physics and chemistry with applications to life sciences, earth sciences and space science. You have the social sciences where you will find things like politics and economics.
And then of course, there’s engineering and technology where you’ll find all your engineering fields: biomedical engineering, chemical engineering, computer engineering, electrical engineering, mechanical engineering, 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 and some, like physics and engineering, the role of mathematics 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 next in the question. It is 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 to this question 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 equal distance 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 through our history. Let’s go through some of that history together.
Around the year 212 BC Archimedes was murdered by 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 cyllinder that can contain that sphere.
Around 1620 Johannes Kepler discovered what he thought was 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 the key connection among 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 the column 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 the 20st 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. Remember we’re trying to uncover connections. Now look on all these formulas, every one of them with pi in it, this number that was born from the geometry of the circle! And look at all of the physical phenomena, how different they all are, and yet they share this common connection to this number – 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 some way the circle plays a part in the derivation of this formula.”
Now, circle is just one geometric form and math is so much more!
And so, too, is the world and connections that exist within it.
Look here, this is an airflow in 2D. Like the cross-section of the wing, and the lines you see here are like the air flowing over and under it. And here, this experiment, this shows a gas, initially compressed by retaining wall to one side of the vessel. And then a hole is made in the wall, and the gas spans to fill the entire vessel, until it reaches a kind of equilibrium.
And here at this example, it shows a metal rod with a source of heat held under it -in this case a flame- and where the flame touches the metal, the point of contact, the heat will heat the rod, and it’ll 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 the 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 the 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 equal 0, that’s Laplacian of u equal 0! And what changes for the mathematician is: u can be potential, u can be chemical concentration, 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 this 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 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 the fifth… 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 1970’s, after Benoît Mandelbrot’s work. And part of the reason for the late bloom of this idea was that it really took the height 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 of the tiny segment of the fractal, magnified so you can see it, and 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 Earth Landsat data on the left. And the 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 became aware of the existence of this type of geometry and they had access to codes, and from within their computer they could make this kind of geometry, they started to make use of it in the unpredictable ways.
Now, this is an aesthetic application of mathematics, but many people study mathematics just because they find it interesting, or aesthetically beautiful. For other people, they want math as the 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.”
