The Road to Reality II

The next two chapters of The Road to Reality (i.e., chapters 3 and 4) covered number systems: integers, real, and complex (aka imaginary) numbers. I'm comfortable with all of these though I haven't dealt with complex numbers since college. If these were new to me I'm not sure how well I would have followed his discriptions, which makes me nervous.

Penrose asks something about each of these number systems that I don't remember from my math classes. That is, do they have some real meaning in the real world? While -3 has some obvious meaning when balancing your checkbook, what does it mean to have -3 cows in a field? It turns out that electron charges are a physical thing that require integers to represent them.

For real numbers he lists a few things that we use real numbers for. While there's a limit to distances we can measure, when you consider volumes you cube them and the numbers get large. On the small scale, irrationals are useful, but when dealing with quanta there might be a limit to how small we have to measure. He promises that real numbers will be necessary for calculus when we get to it.

For complex numbers I remember manipulating them but never knowing why they were important (aside from being how to deal with the square root of -1). First off it seems arithmetic and arbitrary roots in complex numbers always yields solutions in complex numbers. This wasn't the case for say the rational numbers where the square root of 2 gets you into trouble quickly. Also all polynomial equations have one (or more) complex solutions.

I also don't remember graphing complex numbers with the x axis for the real part and the y axis for the imaginary part. And I certainly didn't remember using such graphs to know if infinite series converge or diverge. And I didn't realize that Mandelbrot sets were on a complex plane (I never studied them in school, I just saw pretty pictures). Apparently complex numbers come up in quantum mechanics. Next up, logarithms.

Physics Books, Crochet, and the Universe

Since the start of this sabbatical I've been reading some books about astronomy and the nature of the universe. If you want a really fun gentle introduction I highly recommend A Short History of Nearly Everything by Bill Bryson. As I read more I realized I wanted not a chronological history of what mankind has discovered but something organized differently starting with simple concepts and working up to the more advanced stuff, like a text book. Brian Greene's The Fabric of the Cosmos did that but after a while all the layman's descriptions of these things get a little stale and I wondered if I could handle some of the math. I was always good in math and my BS is actually in Applied Math/CS so maybe I had a shot.

I found out that Roger Penrose recently wrote a 1000 page tome called The Road to Reality: A Complete Guide to the Laws of the Universe. His approach included math because he felt that's the only way to really understand it. So the first 400 pages are a math primer. This is just what I was looking for. I waited till a new edition came out this January and picked it up last week.

Penrose begins with the pythagorean theorem, something I'm quite comfortable with. To prove it he shows a lattice of squares (that is squares next to each other like a tile floor) overlaid on another lattice of squares at an angle. Ok. And then he started talking about Euclid's 5 basic postulates which are all pretty simple including the 5th which says if you have an infinite straight line and a point not on the line, then there is exactly one line through that point that does not intersect the line. Basically it's the parallel line.

But then he starts imagining a geometry where this postulate is not true, what if there were more than one line through the point yet still parallel to the original line? To demostrate this geometry he shows me drawings by Escher of a lattice of black and white fish in a circle, and he wants to know if the Pythagorean Theorem is still true.

It turns out the drawing is a representation of hyperbolic geometry in 2 dimensional euclidian space (a fancy way of saying on a sheet of paper). Then he talks about other other forms of representing this space but after going through a few more sections, I'm realizing this chapter isn't about the Pythagorean Theorem but is about hyperbolic space and I'm lost. Well that didn't take long. All I know about hyperbolic geometry is that many scientific calculators have extra trig buttons called sinh, cosh, and tanh and I remember an older cousin telling me no one ever uses them. Ok, time for some help, off to the web.

A google search for hyperbolic geometry yielded the usual suspects of wikipedia and Wolfram's Math World which I found defined it in math terms I didn't understand. But there were many other pages found too. I found this page helped a lot because of the Java applets that let me move points around and see how the lines are drawn. But then I found The Institute for Figuring had these Hyperbolic Space Crochet Models created in 1997



The bit that helped me was he description how she created these. "Taimina intuited that the essence of this construction could be implemented with knitting or crochet simply by increasing the number of stitches in each row. As you increase, the surface naturally begins to ruffle and crenellate." And crenellate my OSX dictionary tells me means "provide (a wall of a building) with battlements" so isn't that clear. :)

So I'm working my way through this. I think it becomes important because the universe might not be what we think of as flat (just as the Earth isn't flat even though it looks that way to us). Here are some other drawings suggesting this and in crochet form it turns out the shape of the universe may be a pair of pants.



I gotta keep reading.