Proving the Obviously Untrue

Today, two of my friends independently sent me a story about Tau Day which I had hitherto never heard of. One of them asked for me comment about whether this had any point to it. At first I thought the article was just mathematical trolling, thought about it a bit more, thought there might be a real point to it, thought some more and concluded it seemed rather silly.

The argument is about whether the mathematical constant pi, would be better being replaced throughout mathematics with an alternative tau, which is just twice pi (in other words, replacing pi everywhere with a half of this tau). It's suggested that formulae with tau will be more simple.

Basic Geometry

So this is all about the fact that pi was defined historically as the ratio of the circumference to the diameter of the circle, a very old classical reference stemming back to Greek geometry (incidentally pi is also known as Archimedes' constant since he attempted to calculate an approximation to it). Once upon a time, the formula used in schools would have been:



related the circumference C to the diameter D. But generally now, we use the radius r rather than the diameter. And so that gives us (for circumference and area):



The argument for tau begins by observing the extra 2 in the first formula, and wouldn't be nicer if we just defined tau to be twice pi so that these formula would be so much nicer. Would they?



Set aside for the moment the fact that pi is probably the most recognisable Greek letter in the world that speaks languages based on the Latin alphabet (aside from those that are, or appear to be the same). Set aside the fact that tau is used for other specific purposes in much of modern Mathematics, and in particular in the discipline of Topology. The first formula might be nicer, but the second one is probably worse, and by enough to make the improvement of the first rather parlous. OK. But the article talks about this being the problem behind radians, so maybe that's where we get the big gain. Let's explore that.

Radians instead of Degrees

There's nothing particularly clever about using degrees. It's an arbitrary choice (360 degrees in a circle) that probably owes a lot to do with historical factors in one civilisation. It is true that when you start to do some significant mathematics with degrees, it starts to look quite unwieldy. The classic two formulae to consider are the length of an arc and area of a sector.

Suppose we have a circle of radius r and we want to work out the length of an arc (a part of the circumference) where the angle subtending this arc is theta degrees (don't panic, no more Greek to come). Then in degrees the formula will be:



The reason why is that the fraction on the left is the fraction of the relevant angle out of all the angle available, multiplied by the total arc length available (the whole circumference). The formula is not beautiful, and the similar formula for sector area is also a big ugly.



You will note that in both cases there is a 360 on the bottom of the fraction and a 2 pi on the top. This looks like nature's way of trying to tell us something. What would happen if we used an unit of angle so that, instead of having 360 of them in a circle, we had 2 pi of them in a circle (proponents of tau will just say tau of them in a circle)? The formula, derived using the same logic, become much nicer.




So we get



Now these are beautiful, elegant formulae, and the underpinning of why radians (the unit of angle we are talking about here) are used instead of degrees in much of higher mathematics, the formula are much simpler (particularly true when using calculus). Also, look at that first formula, it has all the resonance of F = ma. 1 unit of arc length is found in a circle of radius 1 unit with an angle of 1 radian. So beautiful is this that it used as the definition of the radian in many books. So far, so good. Did we really need tau to produce these? Does it matter that it it tau and not pi that cancels out? I can't see why.

Fourier Series

Another example owes to the work of Fourier, who showed that repeating patterns can be broken into sums of the most basic repeating functions, the ones that are most simple are the sine and cosine functions. These are used to model waves of any sort which are of course ubiquitous in nature. It turns out you can build up more odd shapes like triangular and square waves out of these sinusoidal ones. Being able to do this is important in many aspects of Science and Engineering, and eventually this theory leads to all sorts of cool stuff like the way data is compressed in photos and more.



This graph is labelled in degrees, so you can imagine if we change it to radians as we should then the 360 would become 2 pi. This is where again, proponents of tau will argue that replacing the 360 with a simple tau makes things easier. And it does, if all you want to do is to label that diagram. But the foundation of Fourier theory is building functions up in combinations of these:



The formulae you need to be able to deal with to do this are (among others):



which I grant, strike fear into the hearts of many. But they don't look nicer with tau (and are a little more awkward):



Remember, the people arguing for tau are claiming it simplifies formulae, not making them look worse.

Euler's Identity

Finally, I cannot leave this without talking about Euler's identity considered by most mathematicians (including myself) to be one of the most beautiful results in Mathematics.



This result can be written in a few ways, but this way is very commonly used. This is because in this form you can see how this identity connects the five most important numbers of Mathematics: 0, 1, pi, i and e. With tau, it just doesn't have the same beauty:



so I will stick with pi. Thanks all the same.

Posted by Colin Turner

Today I have found myself reading about three authors. These three men had various levels of fame and in each case, the interest today was more about them than it was about their work.

Samuel Clements aka Mark Twain, the great American writer was coming to my attention because his century long delay in publishing his memoirs is coming to an end. I must confess that I've never been the biggest fan of Twain's works, though many I have enjoyed. His life is very interesting however, filled as it was at times with great pathos and controversy as well as success. There are many theories as to why his memoirs have been so delayed, but one of the front runners is that some of his comments about (at least orthodox, in the broad sense) religion attracted attention. I'm sure it will be broader than that. Nevertheless, one of his quotes is already in the random section in my blog.

Also it has another name - The Word of God. For the Christian thinks every word of it was dictated by God. It is full of interest. It has noble poetry in it; and some clever fables; and some blood-drenched history; and some good morals; and a wealth of obscenity; and upwards of a thousand lies… But you notice that when the Lord God of Heaven and Earth, adored Father of Man, goes to war, there is no limit. He is totally without mercy - he, who is called the Fountain of Mercy. He slays, slays, slays! All the men, all the beasts, all the boys, all the babies; also all the women and all the girls, except those that have not been deflowered. He makes no distinction between innocent and guilty… What the insane Father required was blood and misery; he was indifferent as to who furnished it.

I have great sympathy for this quote, and it puzzles me when I meet people who cannot remotely consider the possibility that, even if there is a perfect God, perfect in His grace, that it might be the case that the humans who wrote the bible might not have listened so well. Because the only other possibility is that such acts are, in fact, OK to be committed by people when protected by divine wrath.

You may recognise this quote, it is possible that it is illegal to publish it in the Republic or Ireland which is another story altogether. Anyway, it will be interesting to see what else Mark Twain will offer the world in his memoirs.

Martin Gardner is my second author under discussion. I read of his death today, albeit at a ripe old age. His collection of mathematical puzzle books adorned my shelves (they currently languish in a box for now) for many years and with collections of his own puzzles and those of many others, they gave me insight into the nature of mathematics. I still use some of the puzzles I read about in my classes.

I was quite surprised to read that he had written extensively about Lewis Carroll and his work, although as a mathematician I was long aware of the mathematical implications of Carroll's works. So I'll perhaps have to acquire that book for summer reading. However I have long suspected that the more controversial aspects of Carroll have prevented him being enthusiastically claimed by mathematics in the public imagination.

Gardner's book introduced me the Fibonacci Sequence, and so it is fitting that I saw this the day before. I think he would have loved it.

Nature by Numbers from Cristóbal Vila on Vimeo.

Douglas Adams was the third author. I couldn't write about authors on Towel Day and not mention him. I have been marking much of the day, but when I've been out and about running a few errands, I have always known where my towel was. I reflected how Douglas would have marvelled at the fact that already, many of us already have the closest thing to Hitch Hiker's Guide in our back pocket, a smart phone connected to Wikipedia, and the rest of the total of human knowledge on the Internet (and the sum total of gibberish too).

Hmm. And having just added more to that pile with more rambling even than usual, I'll say so long and thanks for all the fish.

Posted by Colin Turner

Last week I wrote a little about the size of black holes, and incidentally discussed very primitive calculations I did on the lifespan of any black hole created by the LHC.

A few days later, this interesting article showed the results of professional physicists on just how little such little black holes could grow, in some cases even if their lifetime was not restricted. Enjoy.

Posted by Colin Turner

I allowed myself to become rather distracted by my second years last week as the class was finishing. They were talking about an episode of Horizon that discussed General Relativity and theories of Quantum Gravity. What followed was a free ranging discussion on the nature of infinity, mentioned briefly in the program. But we also talked about the nature of a black hole and its size.

It's surprisingly easy to calculate this with reasonably elementary maths and physics. I first did this when I was about 17 (how very sad) using classical physics equations, and was astounded to discover that even so, the answer was correct (I checked it in the Encyclopedia Britannica in the library at the time).

Here is Newton's universal law of gravitation, between two bodies. It describes the force F between two bodies that are r metres apart. Let's take the one with mass M to be the black hole. G is a small (though mysterious) constant.



You can work out the energy needed to escape the black hole using the old stand by equation that work done is the force times distance traveled against that force, but that only works with a constant force, this force will change as we move, so we need to use the big daddy of multiplication, integration.



Specifically, we will work out the energy needed to escape from the event horizon, the surface at which the escape velocity is the speed of light, which is c (299,792,458 m/s). So the energy will be given by moving my little mass m from the radius of the event horizon, let's call is R to infinity, to show we have broken away.



Now, this should just balance the kinetic energy possessed by my little mass m traveling at the speed of light.



and if we rearrange for R we get that



In other words, the radius of the event horizon, the bit we think of as the "hole" is dependent entirely upon the mass of the object. Please note this is based on a very simple model of a non rotating black hole. Nevertheless we can do some nice calculations from this.

The Sun, would have to be compressed from its diameter of about 700 million kilometres into a radius of just under 3 kilometres. The Earth's mass would need to be compressed so much to form a black hole you would need to squeeze its radius of over 6 thousand kilometres into a radius of around 9 millimetres. That's how dense we're talking here.

We can also consider the radius as described by the contained energy of the black hole, since we know that



and so, replacing our M in our above equation we get



Wow. Remember c is a big number, taking it to the power of four is a lot. So why do this? There's been a lot of speculation about the possibility the Large Hadron Collider (LHC) could create a black hole. This has caused a fair degree of panic, and at least one suicide. It's a physicist's dream that a black hole might be created. I just looked up the "high" energies used by the LHC, and high is a relative term. It plans to bash protons together with 7 TeV (Tera electron volts) of energy each, or lead nucleii with 574 TeV each, let's take the latter. Just how much energy is that in a collision? Well, doubling and converting to good old Joules gives 184 micro Joules. That's really not a lot, 184 millionth's of a Joule. A 100W light bulb uses 100 Joules each and every second. How big would the radius of such a black hole that might form be, from that energy? Check the maths, because so far I haven't but I get.

metres

which is 0.000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,003 metres wide (I hope, I didn't double check the zeros either), which is probably not the planet swallower of people's imagination (but this is fun). But the problem is people think this tiddler will grow very rapidly, but that's because they don't know about Hawking Radiation. This is an interesting quantum effect that means black holes aren't really black, they do emit a little radiation. Large holes would gather surrounding matter faster than their low radiation rate, but small holes have the opposite situation, they radiate more rapidly. The maths for all that is pretty complex, and you need to make lots of assumptions, but the time taken for our little black hole to "evaporate" is (hurriedly calculated by me)



a tiny, tiny fraction of a second. Even allowing for the ambient temperature and some fall in of matter, this little baby is not in equilibrium, it's not getting mass fast enough to accumulate more. It's safe*.


* All disclaimers apply. No liability is assumed for foolish unvalidated experiments done by you or other members of your species. Do not attempt to create black holes in your garage. Any subsequent destruction of your civilization, planet or solar system is at your own risk, and any "EPIC FAIL" signs placed by aliens on the remains is not due to me or my calculation. No calculations have been done on the matter of strange matter either. If you break the planet / system / galaxy or universe you own all the parts.

Posted by Colin Turner

Derren Brown has been back on our screens recently with a series of big events. It's brought him a lot of publicity, and a lot of the public comment has been amusing to say the least. I've been watching everyone comment on the TV and Radio that he's a trickster, and his explanations of how he does things are often bogus.

Yes.... Yes... That's the point. He's a magician, a showman, he tells you right up front at the start of the show. Misdirection, magic, showmanship and more. It's entertainment. He is the first to make it clear that what he does is a trick. He doesn't claim otherwise. If you can't bring yourself to enjoy that, I humbly suggest you watch something else. And while we are on it, I'm relieved people see it's a trick. I think David Blaine's closeup magic is among the most impressive I have ever seen, and I shudder when those around him don't even seem to consider it's just a trick.

Personally, I love Derren, I love his cheeky smug as hell smile as he gets away with it. I enjoy his faked discomfiture. I enjoy it when he fakes near success in tricks to make them more convincing. I enjoy calling him names as I laugh as he lies through his teeth to the audience. I have casually studied a bit of magic in my time, and he is a fine magician. Many big stage magicians rely on assistants, who actually do all the hard work, but he is clearly the architect of most of his own tricks. I'm reading his book and it is informative and very amusing to me, and I very much respect him as a fellow in fight against (rampant) irrationality (a little bit can be a fine thing).

Sadly I've missed a few of the "events", since it clashes with my Iaido class and I wasn't organised enough to record it. I'm slowly catching up on 4OD. But I did see the program on how he stole the lottery. And I'm a mathematician, and I laughed and laughed at it. I didn't believe a word of it, but even the deceit was cleverly convincing and it's not trivial to say exactly what he really did.

He claims he averaged the results of a crowd of people picking lottery numbers. There's a consequence to this. Consider this: how many ways can 50 people pick a number that averages to 1? How many ways can they pick numbers that average to 30? If you understand this point you'll see the possible "answers" from the crowd have an odd distribution. But the cover story was very amusing in its own right. You don't want to know how he really did it, it'll be shockingly dull.

So yes. We know he didn't show the numbers in advance of the draw. Yes, we know his explanation is nonsense. It's a trick. That's the point. Enjoy it and smile along with his insufferable smugness .

Posted by Colin Turner

The website for Number 10 Downing Street has a new petition calling for an apology for the treatment which led Alan Turing to commit suicide at the age of 41, rather than submit to having his homosexuality "cured" by chemical castration.

Turing was, not only a founding father of Computer Science, but a leading member of the dedicated team at Bletchley Park, who decoded the Nazi Enigma code. This work saved thousands of lives.

As reported on the BBC, the originator of the petition, John Graham-Cumming, is not only suggesting an apology, but a posthumous knighthood for Turing. It would seem the least his country could do for him, after all he did for his country, mathematics and computing.

If you're a UK resident, please consider signing the petition.

Posted by Colin Turner