My personal silence is broken; I must apologise for the low level of activity on my part I have been very busy finishing up the first year of my second degree – this is now behind me and I am on to the second year of my second degree! That is not easy to read.
You may have noticed that I now have a co-author Mekhi; who I reblogged some posts from previously. I am very excited to now be part of a team (if not a little intimidated by the quality of the team-member!); hopefully in the coming months you will be getting more regularity from Rationalising the Universe. After all, if something isn’t worth doing habitually it isn’t worth doing at all.
Today I write a short post about something close to my own heart; and to so many others in the field of mathematics which is the notion of mathematical beauty. That is, deriving aesthetic satisfaction from mathematical work (which in my opinion is on a level far above any painting, sculpture or photograph can ever achieve). There is no one defined notion of mathematical beauty; as with the more regular notion of of human beauty there often is a general consensus but it is down to personal taste. Here I summarise some areas in which mathematics in my eyes is at its most beautiful.
Surprisingly simple results
In this area I take a commonly cited example of Euler’s formula, which was showered with praise by the late great Richard Feynman, who used to refer to the formula as “our jewel”. Let me first present the general version of the formula:
This is actually very simple; it is saying for any real number, we can represent this as the real element, the cosine of x and the imaginary number, the sine of x. This is in itself what is often referred to as a deep relationship – it links trigonometry with the exponential form. However this formula is still waiting to blossom into its full beauty. There is a special form of the Euler formula, which is the form when x is equal to pi. When we have this wonderful things happen – the cosine drops out to -1, the sine disappears and we are left with the Euler identity below.
That is incredibly powerful mathematics summarised in a short, elegant looking formula which revels so much. To me this is incredibly beautiful.
Mathematical proof is the weapon of the pure mathematician. In fact mathematics is essential built of proofs – the proofs themselves are just not often presented. It is interesting how many people can for example apply the logic of Pythagorus’ theorem, but how many can actually prove, in a general sense that works for all cases, that the theorem is valid. The answer is not many!
One of my favourite proofs I already blogged my own version of; as a young university fresher (and yes, I now look back at myself as being young then) I will always remember the first time my head was turned to something truly beautiful – and it was not covered in fake tan in Tiger Tiger on a Thursday night, it was the proof of infinite primes. That is the proof that there are infinity many prime numbers – something that feels like it must be true, but it requires a proof. Rather than present my own wording here, because this is about mathematical beauty I will not butcher it. I will leave it to the grand master Euclid, 300 years before the birth of Christ:
For a reductio, suppose the theorem is false. Suppose there are finitely many primes, say p1, …, pn, where pn is the largest prime. Consider the number N = p1×p2×…×pn + 1. Either N is prime or not. If N is a prime, then N is a prime larger than pn. Contradiction. If N is not a prime, then it has a prime factor, say q. But q cannot be p1, or p2, … or pn. Hence, q must be a prime larger than pn. Contradiction. So, there must be infinitely many primes. QED
This is of course a proof by contradiction – a method of beauty in itself. Start of by assuming a universe where the converse is true. Stroll around in that universe for a little while and reduce it to the absurd. Then you know that you have to go back to the other universe; the true universe. This logic is, might I say, delicious. If we were to personify the two examples, Euler’s formula is visually stunning and the proof of the infinite primes exudes elegance, and dare I use this word, swagger.
I like the next result because it seems like a gross mathematical abuse but it is actually quite accurate. I promise you I wouldn’t commit it to the digital realm if it wasn’t, I have more respect for our mutual digital playground.
1 = 0.999999999…
That is to use an equals sign, and write two numbers that appear to be slightly different but are actually correct. It’s playful, it’s fun and I like it.
There is also beauty in being surprised; when you have an idea of what something will be in your mind… and then it just isn’t. Well that is exactly what happens when you turn to the harmonic series. What we are saying here is take one. Now add a half to it, then a third, then a quarter etc. You get the idea, keep adding and adding and what to you get? You get infinity. To me anyway this seems counterintuitive at first (although you don’t need to think too deep to get through the logic). It is just a very beautiful result; take a series where the next element is getting smaller and smaller, sum it and you have a result that tends to the infinite. Nice.
There are many series like this, which have a sort of beauty about them; sine and cosine for example are just Taylor polynomials which are centred around 0 but I just find them a little clunky. This is where I get a bit illogical; but with mathematical beauty, dare I say it, you just get a feeling.
Finally we have the equations that are of course why I get up in the morning; the equations that describe a huge amount of the physical world around us. Now these equations are not in themselves all that simple; they do not fall into the category above. They are more complex, you probably would first meet them at undergraduate level however they are beautiful when you consider their power and what they describe.
The first is the wave equation;
a neat hyperbolic partial differential. Yes it is difficult, however the equation applies to all waves. It describes the propagation of the waves which travel across the oceans of earth as well as the cosmic microwave background, the noise of the universe. So when you consider what is actually is describing; credit where credit is due; this is a beautiful result.
Finally, let us present the Dirac equation; raw mathematical beauty:
In this equation there are all manner of wonderful constants, elegant symbols are pure hardcore mathematics. In words, what this equation did was provide some insight into unity between the quantum and the relativistic; it reveals the wavelike nature of electrons moving close to the speed of light. Not only does this equation look beautiful and describe something amazing, through it quantum field theory was born and the existence of antimatter was predicted. All i’ve done today is a few hours of mathematics in a coffee shop….