A way to refine our worldview is to become more precise in describing what we see. Each of the classical theories of numbers, geometry, topology, fractals, and probability offer tools.
Numbers accompany us throughout our lives and play a fundamental role in the realm of mathematics. By counting and quantifying, we understand our world with more refinement.
To a mathematician, numbers have their own personalities. This lecture explores the ways they have been used and understood—and have captivated humankind—through the ages.
We see how a hidden order of numbers actually underlies much of nature's beauty, and explore the remarkable insights available in the pattern known as Fibonacci numbers.
A potent method for discovering new insights is to isolate and examine patterns—a process that leads us to the most pleasing proportion in art and architecture: the Golden Mean.
We investigate our newly honed sense of mathematical aesthetics to see how it illuminates the structure behind aesthetically pleasing art and architecture to arrive at a new appreciation for what is known as the Golden Rectangle.
Why, exactly, is the Golden Rectangle so visually appealing? A surprising property may hold the answer.
The Pythagorean Theorem perhaps best represents all of mathematics, and we examine some of its most elegant proofs, along with the alluring relationship between the conic section and the ellipse.
Symmetry and regularity lie at the heart of classical beauty. The five regular, or Platonic, solids embody not only elegant symmetry but also an elegant duality in their nature.
For millennia, the five Platonic solids inspired thinkers with a mystical allure. Kepler mistakenly thought they explained the orbits of the then-known planets. But planets aren't involved, as we see when we discover why there are only five Platonic solids.
Though the fourth dimension lies beyond our daily experience, understanding is within our reach, and we can visualize and explore it by using our knowledge of familiar realms and arguing by analogy.
We consider the geometry of the fourth dimension, beginning with artistic works inspired by dimension, then building and visualizing our own four-dimensional cube.
Must every surface have two sides? Surprisingly, the answer is no. We explore a remarkable surface known as a Möbius band.
Though a single-sided surface with no edge at all cannot be constructed entirely in three-dimensional space, it can be effectively described and modeled, as illustrated by the elegant surface of the Klein bottle.
Even the act of folding a piece of paper can be the gateway to a rich trove of nuance, introducing us to the idea of fractals and showing how patterns and structure can emerge from seemingly unpredictable "randomness."
Our simple paper-folding sequence leads us not only to the secrets of the dragon curve fractal, but to an example of the classic computational theory of automata developed by Alan Turing, the father of modern computing.
What does it mean to speak of an infinitely detailed image? We look at what is possible by repeating a simple process infinitely and then reasoning about the result, producing images that illustrate the ideas of self-similarity and symmetry.
We examine how chance, with some simple rules, leads us to an infinitely intricate world of fractals, which quite possibly overlaps with our own physical world.
The uncertain and unknown are not forbidding territories into which we dare not tread. Instead, they can be organized and understood as we construct a means to measure the possibilities for an undetermined future.
The game show Let's Make a Deal® involved a question of chance that surprises people to this day, and leads us to an exploration of probability and the ways we measure it.
This lecture shows us how to put a number to the possibilities of the unknowable future as it examines the quantitative measure known as expected value and how it can be used.
Coincidences and random behavior do occur, often with predictable frequency. This lecture takes a look at randomness and how the principles of probability help us to understand it better.
Coincidences are so striking because any particular one is extremely improbable. But what is even more improbable is that no coincidences will occur. We examine why.
This final lecture looks at 10 "lessons for life" that can be drawn from a range of mathematical themes and concepts.