Euclid's Window : The Story of Geometry from Parallel Lines to Hyperspace

Leonard Mlodinow is a science writer with a sense of humor. In Euclid’s Window, he tells the story of the development (or should I say, evolution?) of geometry from the pre-Socratic Greeks to modern day.

The Greeks made some remarkable discoveries, among them the fact that the length of the diagonal of a square could not be expressed as a ratio of the lengths of its sides—or, as we would say today, the square root of two is irrational.

Not much is known about the person known as Euclid, but he seems to have systematized all that Greek civilization knew about geometry. His work, The Elements, shows how the Greeks had demanded rigor and logic in their approach to mathematics. To them, it was not sufficient to be able to calculate, one show more had to prove that the method of calculation was demonstrably valid. The entire work consists of many theorems derived from just a few definitions and (he thought) self-evident postulates.

But one of his postulates seemed just a little less self-evident than the others: the so-called parallel postulate, which asserts that parallel lines never intersect one another. For 2,000 years, geometers attempted unsuccessfully to show that the parallel postulate could be derived from the other postulates. Euclid himself may have been aware that there was something fishy about the postulate since he refrained from using it in his proofs of his first 28 theorems. It turns out that the parallel postulate is “true” only in a special kind of space, now called Euclidian space, which is basically a “flat” plane, that may be infinite in extent.

It was not until the 19th century when Carl Friedrich Gauss (and others, independently) figured out that logically consistent geometries could be created in which the parallel postulate was not true. For example, the postulate (and thus, much of Euclidian geometry that depends on it) is not valid on the surface of a sphere, like our planet earth. Nevertheless, Euclidian geometry is accurate and valid for practical purposes unless the objects being studied are large enough to be affected by the curvature of the earth.

Other exotic, but logically consistent, geometries were developed in the late 19th century. They found a practical use when Albert Einstein was wrestling with what came to be his general theory of relativity. He found he could make sense of what we call gravity if space itself was “curved.” He was delighted to discover that curved space geometry that fit his theory already had been worked out.

Geometry has come to play a role in modern efforts to combine quantum mechanics with general relativity. The Uncertainty Principle of quantum theory decrees that certain physical traits form complementary pairs that possess a certain limitation: the more precisely you measure one trait, the less precisely you can measure the other. The value of these complementary quantities beyond their limiting precision is fundamentally undetermined, not merely beyond the scope of our current instruments. And when you apply the uncertainty principle to gravity, you are driven to some rather bizarre conclusions about the geometry of space.

Efforts to make quantum theory consistent with general relativity have led to the development of string theory or M-theory, which are driven by insights of mathematics, not physical principles as Einstein’s theories were. Mlodinow writes:

“M-theory appears to have the property that what we perceive as position and time, that is, the coordinates of a string…are really mathematical arrays known as matrices. Only in an approximate sense, when strings are far apart (but still close on the scale of everyday life) do the matrices resemble coordinates—because all the diagonal elements of the array become identical and the off-diagonal elements tend toward zero. It’s the most profound change in the concept of space since Euclid.”

This can be pretty heady and heavy stuff, but Mlodinow makes it pretty enjoyable. He peppers his discourse with wry asides, for example, he observes:

“In the case of the Crusades, ‘contact’ with the Europeans was about as desirable as contact with the Martians in War of the Worlds.”

Whenever he needs two real life examples to explain a concept, he uses his impish young sons Nicolai and Alexei. When the reader is likely to want a simple answer to a complex issue, he admonishes, “Dream on!”

This book might have been subtitled A History of the Concept of Space. It shows how mathematics as well as science develops by building on pre-existing ideas. It is a well-told tale, well worth reading.

(JAB) show less

I was very plesantly surprised when I picked this book up at a small independent shop of Drury Circle in DC about 6 years ago. I thought it would be interesting but I didn't expect it to also be so entertaining. Mlodinow does a remarkable job considering the subject matter is the history of Geometry yet he brings the story to life and he even managed to get me to chuckle a few times.

But then again, maybe I'm just a math geek?

Overall, an excellent book that covers geometry all the way from Pythagoras (so not just Euclid) to modern day String Theory. It's probably not a good choice for a pop science book, but for someone interested in the history of mathematics/science, it's very well done.

There's a lot in this book. He covers early geometry by Pythagoras and Euclid, on to others like Descartes (and his predecessors), then on to people like Gauss and Riemann. He then discusses the impact of their work on Einstein which leads into String Theory and M-Theory where he ends up.

There's some interesting revelations early. Most notably, that Pythagoras effectively had a cult around him and many of the things that Jesus was said to do in the Bible (like walking on show more water), were attributed to Pythagoras by his cult members (and this all happened years before Jesus). So, did Jesus' followers borrow from Pythagoras when writing the New Testament? It's certainly a point that many Christian fundamentalists wouldn't like, but I find it interesting that Pythagoras whom most people only know because of formula was a cult figure back in Ancient Greece. show less

Mlodinow tackles what some people would think would be a dry topic and manages to infuse some wit into it. You can tell that he really loves his topic and wants the reader to as well. He explains the math and gives you examples to help you understand. And they are very helpful (although I must say that his examples using his sons start to get a little annoying after a while.) He explains the beginnings of geometry and how it progressed and reasons why it was, at times, held back due to politics and religion and other things (which puts a lot of history in the book that you normally wouldn’t think of as having anything to do with math.) It starts out with things I learned in school, like the Pythagorean Theorem and coordinates on a x/y show more graph and other things I recognized and then moved on to more complex things like string theory which I had no prior knowledge of. I started out fine and could follow well enough but as the book went on and the theories got more complex I had a harder and harder time keeping up and often had to reread a passage to understand it (and sometimes never totally did.) It is obviously a book for a particular audience and is not for everyone but if you are interested enough to pick up the book in the first place I don’t think you will be disappointed. It is well written and Mlodinow knows his stuff and his love of the topic comes through and infects the reader. show less

Euclid's Window, by Leonard Mlodinow, is a tale of the development of geometry and its application since its codification by Euclid. I was first introduced to Mlodinow's work when I received The Drunkard's Walk, his 2008 bestseller, for my birthday this year; shortly thereafter I also received Euclid's Window (my grandfather loves for me to read books about mathematics, so he is my supplier around birthdays and Christmastime), which was published in 2001. The Drunkard's Walk is considerably better and Mlodinow's development shows in it, but his humor misses the mark in Euclid's Window and he is less effective as a conduit of ideas as well.

In his defense, through much of Euclid's Window Mlodinow has to deal with what is called "science" show more but by any reasonable application of the philosophy of science must be more truly categorized as nonsense. Though Einstein's theories of Relativity fail not as nonsense but as nonutilitarian--if you can't use it, what is the purpose of science?--much of the work in Einstein's aftermath is purely ludicrous. It is the work of experts gone mad, insistent upon displaying how stupid they really are by forming overcomplicated theories and justifying them by noting correlations and confusing them with causality, hoping that everyone else will be confused as well. But in choosing his topic, Mlodinow knew he would come across this kind of thing, so I can't totally absolve him of responsibility.

Still excellent for an early book in a writer's career. show less

Prety good history of geometry through the lives of Euclid, Descartes, Gauss, Einstein and Witten. But yes, Alexei and Nikolai WERE rather annoying...

Very entertaining and informative: plus being (mostly) comprehensible to a mathematical thicky like me! The least likeable aspect is the continuous (and boring) use of his sons names for examples.