The Beauty and Power of Mathematics: William Tavernetti (Transcript)

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.

William Tavernetti – TEDx Talk 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.

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