Mathematics Illuminated
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Category: Tutorial
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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