**Mathematics Illuminated**

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*Category: Tutorial*

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A video course for high school teachers and college level instruction;

13 half-hour video programs, online text, course guide, and Web site;

graduate credit available

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. Mathematics also provides us with a

powerful language for uncovering and describing phenomena in the world

around us. The groundbreaking videos, interactive Web exploration,

text materials, and group activities included in Mathematics

Illuminated reveal the secrets and hidden delights of the

ever-evolving world of mathematics.

Produced by Oregon Public Broadcasting. 2008.

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. This unit explores our fascination with

primes, culminating in the million-dollar 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.

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. 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 . as

with the human genome . combinatorics is essential.

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 . and even measurement. This unit explores how

mathematics attempts to understand infinity, including the creative

and intriguing work of Georg Cantor, who initiated the study of

infinity as a number, and the role of infinity in standardized

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. 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.

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. 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

real-world objects such as ferns, mountains, and coastlines.

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 a "group." 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.

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. 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.

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 Fourier analysis, which allows the

disentangling of a complex wave into basic waves called sinusoids, or

sine waves. In this unit we discover how Fourier analysis is used in

creating electronic music and underpins all digital technology.

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. This unit looks at how this branch of mathematics

provides insights into extremely complicated networks such as

ecosystems.

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. This unit shows how

synchronization can be analyzed, studied, and modeled via the

mathematics of differential equations, an outgrowth of calculus, and

the application of these ideas toward understanding the workings of

the heart.

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. This

unit explores the mathematics of chaos, which involves the discovery

of structure in what initially appears to be randomness, and which

imposes limits on predictability.

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