Monday, January 20, 2014

Euclid's Window

Book Review : Euclid’s Window
Author : Leonard Mlodinow
My Rating : 4 out of 5

The complete title of the book is “Euclid's Window: The Story of Geometry from Parallel Lines to Hyperspace”.

Geometry is a very special subject. It’s one of the oldest branches of knowledge - we started studying it to solve real life problems related to land and measurements. There is an even more important reason for calling it a special subject. That reason is Euclid, one of the most revered figures in Mathematics. He created an edifice for Geometry, whose schematic has been used to organize all mathematical knowledge.

His approach is irresistibly beautiful. At the core are the axioms, the self evident truths about the topic. Using these, and basic laws of logical reasoning, we prove simple theorems. Then we use these simple theorems to prove more complex theorems, and so on. This approach is at the foundation of modern mathematics.

Naturally, the book starts with Euclid and the Greek mathematicians. We proceed to Descartes who gave us the first main enhancement, what we now call Cartesian Geometry. If you have read many books on popular science/math, then some of the material so far will be a repetition for you. It’s still well written.

Then the author moves on to non-Euclidean geometry and the book becomes very interesting. Let me take a detour and give an idea as to what is non-Euclidean geometry.

Note that, in Euclid’s scheme, everything is built on a few trivial self-evident truths, called axioms. There is no way to prove them, but everything else is proven using them. If you change an axiom, you get a very different theory. This is not a random act. The only reason to change an axiom would be if it doesn’t feel as a self-evident truth.

Euclid’s 5th and last axiom roughly (very roughly) states that in a plane, 2 parallel lines do not meet if extended forever. It is not trivial as first 4 axioms. Compare it with the absolute simplicity of the 4th axiom which states that “all right angles are equal to one another”. The 5th axiom, often called as “The Parallel Postulate” has never appeared to be self-evident to mathematicians. Its history and surrounding controversy is a big topic in itself.

The Parallel Postulate reflects our intuitive idea about space. We mentally extend two parallel lines on a plain paper to infinity, and feel that they will never meet each other. But how do we know that space will indeed behave this way at astronomical distances? This is an important and interesting topic that ties mathematics with physics at a fundamental level. Author Leonard Mlodinow does a great job at explaining this in a very accessible and entertaining way. What he is telling is more a story of our understanding of space than a story of Geometry, but that’s a minor complaint against the title of the book.

The author explains how it was not easy challenging Euclid. Gauss did not publish any of his ideas because of the fear of backlash. Others were bolder, and persisted. Eventually these ideas were accepted as valid for Mathematics. But is their any real value in these competing geometries? Or is this just a mathematical curiosity? After all, given a line and a point outside that line, we really cannot draw 2 distinct lines that are in the same plane and are parallel to the original line. So why care about non-Euclidean geometry?

The answer, as we know now, came via Einstein. His work on the General Theory of Relativity proved that space is not Euclidean. You can really construct a triangle in space whose angles total to more than 180 degrees. This is one of the many seemingly bizarre, counter-intuitive ideas that have been brought to us by the 20th century. Here, the author focuses more on how our understanding of space changed with the theory of relativity, and less on other paradoxical aspects of the theory and justifiably so.

The last segment in the book is about how String Theory is changing that understanding even more radically, with its 10+ spatial dimensions. That’s another challenging task, and again I think the author did a great job.

I have to add a note on the style. Leonard Mlodinow has a peculiar sense of humor. I enjoyed his witty remarks throughout the book. He also tries to illustrate some points by creating scenes involving two brothers, most likely his sons, if I remember correctly. I was mildly annoyed by these examples. These two aspects of the book might irritate some readers.With these caveats I definitely recommend this book. This is not a mathematical textbook, it’s more a history. The reader is not expected to be a math graduate, but just a curious individual. So go for it.

1 comment:

  1. Thanks Abhay for sharing this book review, I would love to read the book, some day. It is hard for we humans with limited capabilities of our 5 basic senses to comprehend or imagine what can happen when the small quantities we are able to easily perceive with our basic senses are extended to extremely large values or infinity.
    I don't know if I fully grasp what it means to have a triangle in space sum of whose angles is greater than 180 degrees or if the parallel lines in a plane can ever intersect. I recently saw a video presentation where the author showed how 1+2+3+4.....+infinity = -1/12 (this is supposed to be an important calculation in strings theory) . I had hard time believing it and still think it was a "mathematical trick"!
    I guess we are still a species whose evolution has been constrained by or rather evolved in harmony with our surroundings and as we contemplate things like space travel, teleporters and what not, so that we can screw up larger part of the universe, some day our senses will catch-up to our new reality.


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