The lions sing and the hills take flight.
The moon by day and the sun by night.
Blind woman, deaf man, jackdaw fool.
Let the Lord of Chaos rule.
                                - Robert Jordan

The world is in turmoil. Pick up a newspaper, turn on the television, look out your window and there it is. Chaos reigns. It’s even in your house, on your stove, in your lungs. Can one escape such universal pandemonium?

Whirlpools of chaos surround us every day, whether we know it or not. And while its effects on our lives are great or small, chaos is a necessity sometimes beyond which we can comprehend. In James Gleick’s Chaos: Making a New Science, we are introduced to the most profound paradigm shift in science since relativity and quantum mechanics. However, what makes this paradigm shift so interesting is that it’s a wonder it never happened any sooner.

I can remember going to my high school physics class and learning about friction and air resistance as stumbling blocks to knowledge. While interesting in and of themselves, friction and air resistance seemed (to me at least) to be the bane of physics. Nice, neat solutions were desired when calculating, say, the movement of a pendulum or the work done on pushing a box up a slope. Accordingly, friction and air resistance were ignored.

This tradition of ignoring real world properties led to the reification of an imaginary universe. In their attempt to generalize the world, scientists were missing the trees for the forest. However, it also worked both ways. As science progressed into the latter twentieth century, specialization of fields led to reductionism and the missing of the forest for the trees. With the introduction of chaos, however, scientists finally glimpsed the forest and its trees simultaneously.

Chaos, traces the origins of chaos theory from its primary roots of the 1960s up until the time of its publication in 1987. It weaves a story of discovery from scientists and mathematicians who never knew each other as they each stumbled upon something incredibly new (or old, depending on your perspective). From disciplines as diverse as meteorology, biology, economics, physiology, and ecology came scientists who all had unknowingly (and independently) discovered ordered disorder. As time passed and the scientists began sharing notes, new ideas would emerge that could explain this entirely new field of study called chaos and these ideas would have sexy names like strange attractors, fractals, bifurcations, self-similarity, and the butterfly effect.

"Mathematics is the language of Nature," says Sean Gullette as Max Cohen in the critically acclaimed film Pi. Until reading Chaos, I knew this to be true but not the extent to which it was. As a student of mathematics and science, I understood the dynamic relationships between matter and energy and how they could be described with mathematics. Or, at least, I thought I did. Like most scientists until the coming of chaos theory, I assumed that the equations that described the world were enough to know how the world is. Even though I was aware that certain variables had been ignored to come at the equations (e.g., friction), I still thought that there was an assured amount of predictability in the universe.

Chaos shows why this is not so. All variables must be considered, even though the system in which they operate outputs results that appear random. But what we attribute to randomness is not as random as we might think. When we flip a quarter, we say there is a 50% chance that it will land on the head side or the tale side. However, if we were able to measure the velocity, acceleration, force, angle of inclination, and air resistance, we could predict the side it lands on with high precision. But even then, it would not be exact. Imperfections on the quarter itself, on the hand that flips, and the surface it lands on all affect the outcome of the flip. Until chaos theory, it was thought that small changes in a system lead to small changes in results. But as it turns out, sensitive dependence on initial conditions causes systems to generate unpredictable results, although in surprisingly predictable ways.

Chaos theory says these results are non-random, although apparently or effectively random. Before chaos theory, it was generally thought that all systems with two or more variables converge to some steady state: termination (like a marble rolling around in a bowl) or repeating oscillation (like a pendulum). Depending on the initial conditions, the system progresses into one of these states.

The assumption has always been, though, that if the system does not progress into one steady state, then it ends up in another. What chaos theorists discovered, though, was that systems could oscillate but never repeat, even in very simple systems. This finding revolutionized the scientific worldview.

Meteorologists had long been searching for an oscillating pattern to improve weather predictions. So to did NYSE stock analysts, ecologists studying animal populations, seismologists observing earthquakes, and a number of other fields where dynamic systems were analyzed. Chaos theory unified these various fields in the name of mathematics.

The unification of the sciences is what most impresses me about chaos. It has long been my view that the various subjects we teach in school should be taught holistically. That is, to understand mathematics, you must also understand science and history; to understand literature, you must understand history and philosophy. Bringing these various fields together helps the student to realize that the topics covered in school are part of a greater whole and should not be compartmentalized every sixty minutes.

With chaos theory, the educator can stimulate student understanding of mathematics’ place in the world. Students are endlessly asking "When will I ever use this stuff in the real world?" The answer, of course, is that they probably won’t. However, instead of getting students to solve problems for the purpose of making the grade, chaos theory can be used to facilitate the interrelatedness of various topics. By offering a glimpse into how mathematics is associated with a wide variety of disciplines can foster an appreciation not just for mathematics but for each subject it applies to.

While Chaos is a great book of information, it is not informative in how to teach mathematics or present the ideas it covers to a high school setting. Instead, it offers an introduction to an exciting branch of science and mathematics where educators can gain plenty of useful knowledge to bring into the classroom. For this reason also, Chaos is not a book that attempts to help one do mathematics in a particular or innovative way, but it helps give perspective of the field and provides a new way to think about mathematics.

Overall, Gleick’s development of mathematics in Chaos is not rigorous but that is not his intent. The purpose is to educate and inform the public about the budding discipline. However, many advanced concepts are introduced in the text that laymen would be unfamiliar with. Gleick, though, is able to describe and explain these ideas in a clear-cut way. This does not mean, though, that Gleick waters down the information or presents only cursory views of concepts. On the contrary, a lot of the understanding of the ideas has to be thought out by the reader, a quality I like in a book.

Chaos surrounds us. It teeters in the form of fractals on the edges of your newspaper, it spins between the frequencies of your television, it roils in the clouds outside your window, it’s there in the boiling waters of your stove, and it’s there in the alveoli of your lungs. That is how I see the world now after reading Chaos. We are part of a grand universe, which is at its core chaotic. Chaos is everywhere, permeating all scales and dimensions, even fractional dimensions. Chaos is a mind-opening, thought-provoking book that challenges the reader’s preconceptions about the world. Once finished, it leaves an indelible impression of unity, understanding and appreciation of the Cosmos.