Mathematics Illuminated
Geometries Beyond Euclid (#108)
Our first exposure to geometry is that of Euclid, in which all triangles have 180 degrees. However, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherwordly and firmly of this worldâ€”and present the key to understanding the human brain. [28 minutes]
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Series Description: Mathematics Illuminated explores major themes in the field of mathematics, from mankind's earliest study of prime numbers to the cutting edge mathematics used to reveal the shape of the universe.
All Episodes

The Primes (#101)
The properties and patterns of prime numbers have been a source of wonder across cultures for thousands of years. This unit explores our fascination with primes, culminating in the milliondollar puzzle of the Riemann Hypothesis, a possible description of the pattern behind the primes, and the use of the primes as the foundation of modern cryptography. [28 minutes]

Combinatorics Counts (#102)
This unit looks at combinatorics, the mathematics of counting complicated configurations. In an age in which the organization of bits and bytes of data is of paramount importance, combinatorics is essential. [28 minutes]

How Big Is Infinity? (#103)
This unit explores how mathematics attempts to understand infinity, including the intriguing work of Georg Cantor, who initiated the study of infinity as a number, and the role of infinity in standardized measurement. [28 minutes]

Topology's Twists and Turns (#104)
Topology concerns those properties of an object that remain the same even when the object is stretched and squashed. In this unit we investigate topology's seminal relationship to network theory, the study of connectedness, and its critical function in understanding the shape of the universe in which we live. [28 minutes]

Other Dimensions (#105)
This unit explores different aspects of the concept of dimension, what it means to have higher dimensions, and how fractional or "fractal" dimensions may be better for measuring realworld objects such as ferns, mountains, and coastlines. [28 minutes]

The Beauty of Symmetry (#106)
This unit explores Group Theory, the mathematical quantification of symmetry, which is key to understanding how to remove structure from (i.e., shuffle) a deck of cards or to fathom structure in a crystal. [28 minutes]

Making Sense of Randomness (#107)
Probability is the mathematical study of randomness, or events in which the outcome is uncertain. This unit examines probability, tracing its evolution from a way to improve chances at the gaming table to modern applications of understanding traffic flow and financial markets. [27 minutes]

Geometries Beyond Euclid (#108)
Our first exposure to geometry is that of Euclid, in which all triangles have 180 degrees. However, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherwordly and firmly of this worldâ€”and present the key to understanding the human brain. [28 minutes]

Game Theory (#109)
Competition and cooperation can be studied mathematically, an idea that first arose in the analysis of games like chess and checkers, but soon showed its relevance to economics and geopolitical strategy. This unit shows how conflict and strategies can be thought about mathematically, and how doing so can reveal important insights about human and even animal behaviors. [28 minutes]

Harmonious Math (#110)
Sound is the product of airwaves crashing against our eardrums. The mathematical technique for understanding this and other wave phenomena is called the Fourier analysis, which allows the disentangling of a complex wave into basic waves called sinusoids, or sine waves. In this unit we discover how the Fourier analysis is used in creating electronic music and underpins all digital technology. [28 minutes]

Connecting with Networks (#111)
Connections can be physical, as with bridges, or immaterial, as with friendships. Both types of connections can be understood using the same mathematical framework called network theory, or graph theory, which is a way to abstract and quantify the notion of connectivity. This unit looks at how this branch of mathematics provides insights into extremely complicated networks such as ecosystems. [28 minutes]

In Sync (#112)
Systems of synchronization occur throughout the animate and inanimate world. The beating of the human heart, the swaying and near collapse of the Millennium Bridge, the simultaneous flashing of fireflies in Southeast Asia: these phenomena all share the property of spontaneous synchronization. This unit shows how synchronization can be analyzed, studied, and modeled via the mathematics of differential equations and the application of these ideas toward understanding the workings of the heart. [28 minutes]

The Concepts of Chaos (#113)
The flapping of a butterfly's wings over Bermuda causes a rainstorm in Texas. Two sticks start side by side on the surface of a brook, only to follow divergent paths downstream. Both are examples of the phenomenon of chaos, characterized by a widely sensitive dependence of the future on slight changes in a system's initial conditions. This unit explores the mathematics of chaos, which involves the discovery of structure in what initially appears to be random, and imposes limits on predictability. [28 minutes]