giving a parabola its legs

I remember being taught in high school how to find the solutions to the equation: y(x) = ax² + bx + c = 0. The parabola in the figure clearly crosses the horizontal x-axis twice and gives us two solutions for y=0 that are in fact:

x = ( -b ± √(b² - 4ac) )/2a

But if you lift the parabola up so that its vertex is above the x-axis, there is no way this curve is ever going to cut through y=0. Or is there?

All those years ago, I was shown that the problem had to do with the square root in the expression above. A parabola that is above the x-axis corresponds to a negative value for b² - 4ac. Since we would have to take the square root of a negative number to get a solution, why not define the so-called imaginary number i=√-1 and just carry on! This is one way to motivate the invention of complex numbers, and I thought that was so cool or incomprehensible or both that I never went back to the geometrical picture of what's going on.

As many readers will know, the standard approach is to define a complex plane of numbers by adding an imaginary number axis perpendicular to the more familiar (real) number line. I have talked about the beauty of this before, here and here. A complex number is made up of a real part and an imaginary part, which is why you need a plane to describe these numbers geometrically. Now let's imagine what y=x² looks like when x is a complex number that can lie anywhere on the complex plane.

Our normal picture for y=x² is a parabola with a vertex at x=0 with arms rising majestically above the real x-axis. What about y as a function of a purely imaginary x=ai (where a is just a real distance along the imaginary axis)? Well, y = x² = a² i²= -a² (since i² = -1) so that y now takes negative values. In other words, we take a copy of our usual parabola, rotate it by 90 deg so that it is above the imaginary axis, and flip it so that it points down. This is nicely shown for a general parabola here.

For any x not on the real or imaginary axes, y is complex, which turns out not to be helpful for finding solutions to y(x)=0. But with two parabolae pointing in opposite directions, we've really got the y-axis covered. Now we can place the vertex above or below y=0 and we will have two arms ascending towards ever more positive y values, and two legs descending towards ever more negative y values. If our parabola has arms and its vertex is above y=0, all we need do is create the legs and follow them down to dig up the two complex roots to its equation.

When you think about it, the initial problem only occurs for polynomials of even degree, i.e. y=ax²+... or y=ax⁴+... because these are the ones that have arms or legs that end up pointing in the same direction. Obviously y=ax+b is going to cross the x-axis somewhere and so will y=ax³+... etc. The invention of complex numbers results in the even-degree polynomials acquiring both arms and legs so that they too are bound to cross y=0. Because we only have a single problem here - ensuring that all polynomials cover all negative and positive values of y - we don't need to invent anything more exotic than complex numbers. I love this geometrical picture, because it gives us an extremely informal demonstration of this so-called closure of the complex numbers, which is known in fact as the fundamental theorem of algebra.

brake cables and drip coffee machines

A while back, I had what might be called an epiphany of ignorance. In such a situation, what makes the sudden insight so mind-broadening is the realization that one has lived for so many years on this planet in complete ignorance of a simple aspect of everyday life. In this instance, a colleague at work said he was pretty sure that the plastic housing around bicycle brake cables have metal coiled around the inside. Being naturally argumentative, I started to disagree without having given the issue a moment's thought. Eventually I started to come around, and we eventually figured out that there has to be a push on the housing to balance the brake lever's pull on the inner wire, with confirmation from an informative website on bicycle minutiae. The author says it is all about Newton's third law ("every action has an equal and opposite reaction"). Being Newton's Ocean and all, I have given this rather too much thought, and I have come to the conclusion that it is more to do with the need to balance forces so that an object compresses rather than undergoes a bulk acceleration.

If we start by imagining a centre-pull style front brake, just pulling on a bare wire should work as long as you are sitting firmly on the seat, so that a force is also transmitted downwards to keep the front of the bicycle from lifting upwards. In principle, I suppose one could still end up lifting the front of the bike right off the road that way, so the use of a reinforced cable housing to transmit the necessary downward balancing force along the same path as the upward pull on the brakes via the inner wire makes more sense! Plus it allows the cable to bend and still transmit a differential force between the inner and outer components. It also allows for less symmetrically designed brakes such as side-pull and linear-pull systems, where the inner wire is connected to one side, and the balancing compression force from the housing is applied to the other side.

So there you have it. The other day, this same colleague confided that he sometimes pours day-old coffee into the reservoir of our communal drip-style coffee machine, in order to reheat it. This led to a discussion about whether this would gunk up the machine, which led to a debate concerning how exactly such a machine works! The issue was whether the water would boil and deposit any residue in the reservoir. Now I kind of knew that the water didn't get totally turned into steam, but how does it move up the tube to get to where it drips down onto the coffee grinds? It turns out that pockets of steam and a one-way valve ensure that the heated water moves against gravity up the tube.

Perhaps the real epiphany is how much clever stuff there is in the "simple" things around us.

the trouble with physics by lee smolin

Lee Smolin works just down the road from me. Well that's a slight exaggeration - I live and work in Toronto, while Lee Smolin works out of the Perimeter Institute in Waterloo, about 100 km west of Toronto. The PI was founded in 1999 with help from the Canadian company Research in Motion (RIM), also based in Waterloo and best known for the BlackBerry.

Smolin is one of the originators of loop quantum theory, which is basically a rival to string theory. Reading The Trouble with Physics, published in 2006, I kept wishing for more about loop quantum theory, but I'll just have to get myself a copy of Three Roads to Quantum Gravity, and hope it's not too out of date (it was published in 2001).

Early in The Trouble with Physics, Smolin lays out the five big remaining problems in physics:

1. Quantum + gravity unification
2. Quantum foundations
3. Particle + force unification
4. Freely tunable parameters (in particle physics)
5. Cosmological discrepancies (dark matter and dark energy)

If I understand correctly, the reason Smolin favours more "foundational" methods - for example, to unify the quantum world with Einstein's gravity world - has to do with a preference for "background-independent" methods. Whereas Newton's laws play out on a fixed background of Cartesian (or Euclidian) space and an equably flowing arrow of time, Einstein's general relativity is well-known to define the very space and time that the events of the universe unfold across. So theorists who continue in the spirit of Einstein are like artists who do not depend on a predefined canvas but create everything from scratch. The ultimate appears to be a theory in which not only particles and forces can be seen as emergent properties of a vacuum, but space and time can themselves emerge - possibly in some quantized state. Needless to say, theories such as loop quantum gravity are background-independent whereas quantum mechanics, the standard model of particle physics, and (super)string theories are background-dependent. Now we all know that superstring theories require extra dimensions that are supposedly curled up so we can't see them. But I guess this is like an artist having a technique that requires a canvas with various quirky features, rather than a different artist whose canvas emerges as a natural part of his or her art.

A seemingly bigger issue is the fact that there are a huge number of string theories that can apparently be devised by changing any of a large number of free parameters. The "super" in superstring means proposing that fermions and bosons have supersymmetric partners, so a selectron is a boson partner to an electron, while a gluino is a fermion partner to a gluon. None of these particles has been observed, but with all those free parameters it is easy to just propose that they are way too massive to have been created in any accelerators. Abstruse mathematical models can be proved to be consistent (although even this is pretty hard at this level) but they do not necessarily correspond to reality.

Smolin talks a lot towards the end of his book about the sociology of today's physics departments, with hiring practices that favour those who will work on the now-popular string theory approach. But I think the more fundamental issue is the one he alludes to earlier concerning the adoption of the anthropic principle. Since it has become hard to test these theories experimentally, the anthropic argument implies that a physicist should pick a model universe in which he/she could exist, and which can be rigged to not look obviously different from our world. But this is hard to do, and perhaps these physicists have become just as intrigued with worlds that are mathematically possible rather than truly anthropic. In this case, theoretical physics departments may have turned into specialized mathematics departments, and string theory is now preferred precisely because it is such a rich source of mathematical worlds, even if none of them even remotely corresponds to our world!

the viruses and the worms

In the beginning was the hardware, although it wasn't much use without some software. And then, to spoil the Edenic harmony of Intel and Microsoft (I'm simplifying the history of computers a bit here!) there came the viruses and the worms, those evil packages of ones and zeros that every computer owner and network administrator fear.

The conficker worm has been in the news off and on since October of last year, and has of course become much talked about since it awoke from its dormant state on April 1 of this year. I was going to write about it anyway, and then I got to see a friend's PC a few days ago after it had been infected with something - although the symptoms didn't seem to fit those of any of the conficker variants. Believe it or not, this was the first time I'd actually seen a computer that was obviously infected. It was kind of spooky to see the internet browser jump to websites other than what you had selected. After installing Norton's Internet Security presumably way too late, it refused to run a full system scan and soon refused to fire up altogether.

How much is a computer virus like its biological cousin anyway? Since a biological virus is basically a computer program written in the four base code of its DNA or RNA, and it replicates itself using the cellular machinery of a host organism, the computer version is obviously quite aptly named. A big difference, of course, is that computer viruses have not been configured to mutate as they spread. (Earlier versions of conficker can apparently update to the latest version, but that is a different can of worms altogether.)

Meditating on both forms of virus makes one think about the stuff we call "information." Viewed from one angle, a biological virus seems like such a harmless assortment of letters from the genetic code of life. But of course those letters are inextricably tied to the powerful workings of life, and the information packed into a virus has the power to spread horrible diseases among humans and many animal species. Biological viruses are such curious entities because it is debated whether one can even define them as life forms or not. It is therefore hard to view them as intrinsically evil. (Then again, a single-celled bacterium does not sit around and ponder whether to wreak havoc on a population of humans either!) Computer viruses are also just pieces of information, but - being the product of intelligent design - carry the explicit malevolence of the designer. It would be a cheap shot at the ID community to make the obvious logical step concerning an assignment of good or evil to a designer of biological viruses. But it is surely an interesting starting-point in a discussion of whether the "stuff" of good and evil appears to be a basic constituent of our universe, or whether it in some sense simply emerges within the context and structuring of life, where information and knowledge can unleash the power to create and destroy.

laser-induced fusion

Well I've been busy trying to make a start on some basement renovations but I decided to come up for air and to make contact with my poor neglected blog. I just happened to notice the curious image (left) on the main page of wikipedia which led me to today's post. A few days ago, the US national lab in Livermore announced that their National Ignition Facility is now completed, tested and ready to start doing real experiments in inertial confinement fusion.

Many years ago, I remember being aware of magnetic confinement experiments using donut-shaped prototype fusion reactors. These "tokamak" reactors with electromagnetic coils wrapped around their donut shape are rather like small versions of particle accelerators, with the fusion fuel forced to circulate around the middle of the donut. Fusion holds the promise of clean energy (just like the Sun's!) since the idea is to use the same nuclear fusion reactions involving isotopes of hydrogen (deuterium and tritium) that occur in the Sun. This, of course, doesn't obviously indicate that the energy production would be clean. Deuterium and tritium combine to form helium innocuously enough. But all the unreacted tritium is radioactive, which has to be taken into account during decommissioning, but its half-life is only 12 years so it doesn't represent a long-term headache. However, the sides of the reactors themselves become radioactive due to exposure to energetic neutrons that are produced along with the helium. Nevertheless, if fusion energy became feasible via magnetic confinement, these radiation concerns are very different from today's nuclear fission reactors that create long-lived radioactive waste in direct proportion to the amount of energy created. The real issue is that no one has managed to get more energy out than they have to put in to keep the deuterium and tritium confined and hot enough to react.

But I am ashamed to admit that I was totally ignorant of the laser approach to the problem of attaining temperatures and pressures equivalent to the interior of the Sun. I feel so out of touch here it's as though I've been hiding in my basement for years not days! This laser inertial confinement method is reminiscent of the H-bomb design, whereby a fission explosion is used to implode the fusion material to create the higher temperature and pressure for the fusion reaction to proceed. Indeed, the number one motivation behind the National Ignition Facility is in fact to continue H-bomb research without doing full-scale tests. They will use 192 extremely powerful lasers which converge on a tiny pellet of fusion fuel inside the target chamber shown in the picture. The final amplified laser energy will give a total power of 500 trillion watts. This is about 1000 times the electrical power that is being used across the whole of the US at any one time, but it is only maintained for a split second. After this the optics get distorted from the heat involved in the light amplification and the system has to cool down for several hours before the lasers can fire again. Ignition is said to occur if more energy comes out than goes in.

Whereas an isolated laser firing performs a valid simulation of an H-bomb detonation, a power station would require a constant stream of fusion fuel pellets to be ignited, so that a continual stream of fusion energy could be made available to convert water into steam to run turbines for electricity generation. France has a similar project called Laser Megajoule expected to be completed next year. A large European High Power laser Energy Research facility (HiPER) aims to focus on more efficient ignition methods by using one laser pulse to provide adequate compression when combined with another laser pulse to heat the compressed fuel to reach ignition conditions.

These big science projects are all very impressive displays of technological prowess, but for many years, the promise of power stations running on nuclear fusion has remained a far-off dream. There's something comical about playing around for years with extremely expensive facilities where you always end up pumping in way more energy than you get out. No wonder we all got excited by the possibility of cold fusion!

pyramid schemes and falling dominos

So Bernie Madoff has pleaded guilty to a pretty massive investor defrauding scheme. Like Charles Ponzi, he started small, but as former chairman of NASDAQ, his fall in reputation has been rather greater. But was Madoff's truly a pyramid scheme?

A pure pyramid scheme collapses when the ever growing numbers of new recruits that are required to fuel it hits a limit imposed by finite resources. Ponzi's scheme relied on the rather ineffectual-sounding business of buying international postal reply coupons in Italy and selling them in the USA. In his case, the finite resource was the reply coupons themselves. So when a newspaper pointed out that his business would require about 6000 times the available supply of coupons, it became obvious that he must have turned to forging them.

But was Madoff's scheme - fraudulent as it evidently was - bound to collapse eventually? I have no idea, but he had been going strong for quite a while and his weakness only became exposed when the domino effect of the sub-prime mortgage fallout hit. Now don't get me wrong - fraud is fraud and I'm not trying to find excuses here. While a reasonably informed investor knows that his or her money may get eaten up by the market, any losses should all be above board and not due to a broker using your investment capital to feed someone else's fake returns, even temporarily. The funny thing is that if Madoff's scheme was primarily intended to demonstrate reliable but not unrealistic returns by temporarily borrowing from investor Peter to pay Paul during dips in the market, then he might have got away with it in perpetuity as long as there wasn't any panic-driven run on his "bank."

Again, I stress that I am not condoning his activities in any way, especially as I am too ignorant of precisely what was involved anyway. But I think it is correct to say that even conservative banks would have a hard time if everyone decided to pull out their cash at once. Since regular bank accounts are insured by the federal government, there is no reason why there would be a run on these accounts. Needless to say, this is not true of the stock market, which still has inadequate safety systems to prevent the sorts of extreme volatility that occur when people start to panic simply because they know others are panicking. The only thing we have to fear is fear itself etc.

In terms of physical systems, the stock market is like an elevator on a bungee cord. Instead of dropping down smoothly to a lower floor in response to some newly emerging reality that "the economy" needs to cool off, the market overreacts and drops down way further than required for a realistic correction. The system is mostly designed to enable microeconomic transactions to occur very quickly, with potential investors able to get their hands on up-to-date information. It is precisely these design criteria that cause problems when the mass media start to broadcast panic signals not about one particular company but about the economy as a whole.

So yes, Madoff should be made to pay for his recklessness and what was clearly totally fraudulent practice. But what would be so much more valuable than throwing all the Madoff's in jail is if we could figure out a way to keep the market as responsive as it currently is when a regular number of investors jump in or out, but add some sort of natural brake when the herd instinct starts to occur. Communist countries have traditionally enjoyed control over the mass media. How ironic to muse on the fact that the free market can get damaged so badly by the pronouncements of doom that are fed to us by the free press!

happy belated pi day

Last year, I wrote my sixth post on March 14, in order to help celebrate Pi Day. This business of celebrating a bit of math on 3.14, just because these are the first three digits of the irrational number known as pi, is a definite case of math trumping physics, since we should really have been celebrating the anniversary of Einstein's birth on the 14th.

Einstein was a pretty straightforward and rational kind of guy. The number pi is also a straightforward enough character, being the simple ratio of a circle's circumference to its diameter, but everyone knows it is irrational. Well I thought this was well-known until I browsed through the historian Fernandez-Armesto's Ideas that Changed the World recently, where he confidently tells us that pi = 25/8 or some such rational approximation to the real pi. I don't recall what he thought the number pi had done to change the world, but it wasn't anything to do with its irrationality.

A safer bet for revealing the wonders of mathematics is a book like Bridges to Infinity by Michael Guillen. I have a feeling it is out of print - my copy dates from 1983, when it was initially published. I confess I've only ever dipped into it - I don't know why I haven't read it cover to cover because it contains some nice accessible mathematical insights. For example, Guillen demonstrates how Georg Cantor could be sure that there were more irrational than whole numbers, even though - of course - there are an infinite number of the latter. Here goes:

Imagine you can list all the irrational numbers in order and you label them using whole numbers:

1 .17643...
2 .23482...
3 .62346...
etc...

Obviously these aren't the first three but you get the idea. Cantor then showed that you can make a new irrational number that is different from every single one on this list, even though it's supposed to be the complete list! You take the first digit of the first irrational number on the list and pick a different digit to start the new number, so you know it will be different from the first number. You change the second digit from the second number and so on. Clever huh?

Happy belated irrational Pi Day!