Mathematics Illuminated is a 13-part multimedia learning resource for adult learners and high school teachers in math and other disciplines. The series 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. Rather than a series of problems to be solved, mathematics is presented as play we engage in to answer deep questions that are relevant in our world today.
1. The Primes—The properties and patterns of prime numbers — whole numbers that are divisible only by themselves and one — have been a source of wonder across cultures for thousands of years, and the study of prime numbers is fundamental to mathematics.
2. Combinatorics Counts—Counting is an act of organization, a listing of a collection of things in an orderly fashion. Sometimes it’s easy; for instance counting people in a room. But listing all the possible seating arrangements of those people around a circular table is more challenging.
3. How Big Is Infinity?—Throughout the ages, the notion of infinity has been a source of mystery and paradox, a philosophical question to ponder. As a mathematical concept, infinity is at the heart of calculus, the notion of irrational numbers — even measurement.
4. Topology’s Twists and Turns—Topology, known as “rubber sheet math,” is a field of mathematics that concerns those properties of an object that remain the same even when the object is stretched and squashed.
5. Other Dimensions—The conventional notion of dimension consists of three degrees of freedom: length, width, and height, each of which is a quantity that can be measured independently of the others. Many mathematical objects, however, require more — potentially many more — than just three numbers to describe them.
6. The Beauty of Symmetry—In mathematics, symmetry has more than just a visual or geometric quality. Mathematicians comprehend symmetries as motions — motions whose interactions and overall structure give rise to an important mathematical concept called “group.”
7. Making Sense of Randomness—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.
8. Geometries Beyond Euclid—Our first exposure to geometry is that of Euclid, in which all triangles have 180 degrees. As it turns out, triangles can have more or less than 180 degrees. This unit explores these curved spaces that are at once otherwordly and firmly of this world — and present the key to understanding the human brain.
9. Game Theory—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.
10. Harmonious Math—All 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.
11. Connecting with Networks—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.
12. In Sync—Systems of synchronization occur throughout the animate and inanimate world. The regular beating of the human heart, the swaying and near collapse of the Millennium Bridge, the simultaneous flashing of gangs of fireflies in Southeast Asia: these varied phenomena all share the property of spontaneous synchronization.
13. The Concepts of Chaos—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.
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