Category Archives: Mathematics

How to prove that space is curved…

Question: if you lived in flatland (a 2-d world), how could you tell if the land was curved in the third dimension?

Answer: geometry!

It turns out many of the mathematical rules we learned at school ‘fall apart’ if the working surface is curved. For example, can you draw a square on the surface of a sphere? No!

So can we use this insight to tell if our 3-d world is curved in a mysterious fourth dimension? Yes!

If we set off from earth, went in straight line for, say 1 light-year, then turned 90º, went 1 light-year, turned 90º again, and then did this yet again, you should have traced a perfect square, and be back exactly where you started. If you aren’t, something is amiss!

 

Now it turns out that it we do this, we will indeed discover an error; but why? And how do we know this?

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Newton told us that a massive object in motion will continue to travel in a straight line, unless acted upon by external forces. Some people think that Einstein overturned this insight, but he didn’t; indeed he extended it: he said that the force of gravity is not actually a force, and thus objects falling under gravity are actually going in straight lines! Indeed this makes sense, as anyone ‘falling’ does indeed not sense any acceleration, but rather feels ‘weightless’. Thus they are not actually accelerating, they are going straight – in curved space.

Now anyone who has thrown a ball can see this is absurd on the face of it, but Einstein was serious, and he is right, from a certain perspective. The ball is not going in a straight line through ‘regular’ space, but is going on a straight path in a 4-d construct called ‘space-time’. Likewise, he would argue that the planets are tracing straight lines around the sun; and indeed the ‘parabola’ of a baseball is actually not a parabola, but a very small part of the enormous ellipse that would be traced in the baseball could fall though the earth and go into ‘orbit’ §.

Anyway, Einstein’s model says that light travels in straight lines, but we have seen that light bends when it passes near to the sun (this can most easily be tested during an eclipse) – so… if one of the sides of your ‘perfect square’ were to pass near the sun, it would also be bent and if you followed the above rule to draw the square, you would not end up where you started.

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Physicists have grown used to Einstein’s model, and better tests for the flatness of space have been developed. For example, if you drew a circle on the surface of a sphere, the area would not equal Πr2, but would be less. Likewise, in 3-D space, we could plot a sphere and then measure the volume and if it did not equal 4/3Πr3, we would know something was amiss.

So physicists have looked at how light bends, and how the planets move, and found out, amazingly (but predicted by Einstein) that the error in this spherical volume calculation is directly proportional to the mass of matter within the sphere – proving that the warpage in space is proportional to (and thus caused by) ‘mass’.  Thus mass warps space.

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MC Escher: 'Grid'

But is space really warped in some ‘extra’ dimension?

Well, this is a good question. Maybe it is some extra ‘spacial type’ dimension, but you could also look at time as a fourth dimension, and argue that this space is not ‘curved’ at all, but rather that space and time simply vary in density in different locations. I personally like this way of looking at it, it eliminates the need for some vague ‘extra dimension’, and therefore swiftly removes the possibility that space could be ‘closed’ or fold back on itself in this extra spacial dimension. Occam’s razor thus prefers the ‘density’ model!

Footnotes:

§. In Wikipedia, they state that balls bounce in perfect parabolas, but note they also mention a ‘uniform’ gravitation field, and it is well to remember that the earth gravitational field is not uniform, but radial. Thus I stand by my assertion that missiles follow elliptical paths just like planets and comets. Of course, an ellipse is a close relative of both the parabola and the hyperbola, so this is not really that dramatic.

What exactly is ‘science’?

I used to think science was the practice of the scientific method; i.e. you propose a hypothesis, you develop a test of the hypothesis, execute it and prove the hypothesis.

That worked for me until the end of high school.

At university, I was a true nerd. I read all my textbooks cover to cover (mainly because as I was too shy for girls and too poor for booze). During this time, the definition above started to fail. So much of the science was maths, statistics, observation, pattern recognition, logic and quite a bit of rote learning. Not all of it fitted into my definition of science. I became a fan of a new definition: science is the study of the nature of reality .

But then I did post-grad, and I realised that not much in science is ‘proven’ (I guess this is the point of post grad study). Evolution, for example, is not proven. That the sun revolves around the earth is not ‘proven’. I discovered that the only things that could be proven were ‘ideas’ about ‘other ideas’. Bear with me on this one.

Let us say we define the number system – this is an ‘idea’ or conceptual construction. Within this construction we can ‘prove’ that one and one is two. Because we ‘made’ the system, with rules, then we can make factual and true statements about it. We can’t do this about the real world – we cannot say anything with absolute certainly because we rely on flaky inputs like our own highly fallible perception.

It’s like that old chestnut: how can you be sure you are not living in a giant simulation? Of course you can argue that it is pretty unlikely and I would agree, and right there we have a clue to a better definition of science.

It turns out that much of modern science deals in ‘likelihood’ and ‘probability’ rather than proof and certainty. For example, we can say that the theory of evolution is very likely to be more-or-less right, as there is a lot of corroborating evidence. Science cannot be run like a law court – where the prosecution only need to reach a threshold of reasonable doubt to ‘prove’ someone guilty.

Aside for nerds: Science says you can use logic to prove things absolutely, but logic only works with ideas, and there is a breakdown between ideas and reality, so one can never prove things in reality. So it is thoroughly wrong for a court to say that someone has been proven guilty. The courts use this language as a convenience, to “draw a line under” a case as they have not found a moral way to dole out punishments based on probabilities. Imagine a world in which a murder suspect gets a 5 year sentence because the was a 20% chance he was guilty! Sports referees often operate in this decisive way, perhaps because it saves a lot of arguing!

Anyway, good science cannot just give up and say once there is consensus something passes from theory to fact. This is sloppy. We have to keep our options open – forever.

Think for example of Newton’s Laws of Motion. They are called ‘Laws’ because the scientific community had so much faith in them they passed from theory (or a proposed model) to accepted fact. But they were then found wrong. Strange that we persist in calling them laws!

It took Einstein’s courage (and open mindedness) to try out theories that dispensed with a key plank of the laws – that time was utterly inflexible and completely constant and reliable.

So it is that the canon of scientific knowledge has become a complex web of evidence and theories that attempt to ‘best fit’ the evidence.

Alas, there are still many propositions that many so-called scientists would claim are fact or at least ‘above reproach’. Evolution is attacked (rather pathetically), but the defenders would do well to take care before they call it ‘fact’. It is not fact, it is a superbly good explanation for the evidence, which has yet to fail a test of its predictions. So it is very very likely to be right, but it cannot be said to be fact.

This is not just a point of pedantry (though I am a bit of a pedant) – it is critical to keep this in mind as it is the key to improving our model.

Two great examples of models people forget are still in flux…

1) The big bang theory

2) Quantum theory

I will not go into global warming here though it is tempting. That is one where it doesn’t even matter if it is fact, because game theory tells you that either way, we better stop making CO2 urgently.

Back to the big bang.

I heard on the Skeptic’s Guide podcast today about an NSF questionnaire that quizzed people about whether they believed the universe was started with a massive explosion, and they tried to paint the picture that if you didn’t believe that, then you were ignorant of science. This annoyed me, because the big bang theory is now too often spoken of as if it were fact. Yes, the theory contributes viable explanations for red-shifted pulsars, background radiation, etc, etc, but people are quick to forget that it is an extrapolation relying on a fairly tall pile of suppositions.

I am not saying it is wrong, all I am saying is that it would be crazy to stop exploring other possibilities at this point.

You get a feeling for the sort of doubts you should have from the following thought experiment:

Imagine you are a photon born in the big bang. You have no mass, so you cannot help but travel at ‘light speed’. But being an obedient photon, you obey the contractions in the Lorentz equations to the letter, and time thus cannot pass for you. However, you are minding your own business one day when suddenly you zoom down toward planet earth and head straight into a big radiotelescope. Scientists analyse you and declare that you are background radiation dating from the big bang and that you have been travelling for over 13 billion years (they know this because they can backtrack the expansion of the universe). Only trouble is, that for you, no time has passed, so for you, the universe is still new. Who is right? What about a particle that was travelling at 0.999 x the speed of light since the big bang? For it, the universe is some other intermediate age. So how old is the universe, really?

This reminds us of the fundamental proposition of relatively – time is like a gooey compressible stretchable mess, and so is space, so the distance across the universe may be 13.5 billion light years, or it might be a micron (how it felt to the photon). It all depends on your perspective. It is much like the statement that the sun does not revolve around the earth and that it is the other way around. No! The sun does revolve a round the earth. You can see it clearly does. From our perspective at least.

Now, quantum theory.

Where do I start? String theory? Entanglement? Please.

The study of forces, particles, EM radiation and the like is the most exciting part of science. But being so complex, so mysterious, so weird and counter intuitive, it is super vulnerable to abuse.

Most people have no idea how to judge the merits of quantum theories. Physicists are so deep in there, they have little time (or desire or capability) to explain themselves. They also love the mystique.

I do not want to ingratiate myself with physicists, so I will add that the vast majority have complete integrity. They do want to understand and then share. However, I have been working in the field for long enough to know that there are weaknesses, holes and downright contradictions in the modern theory that are often underplayed. In fact these weaknesses are what make the field so attractive to people like me, but is also a dirty little secret.

The fact is that the three forces (weak nuclear, strong nuclear and magnetic) have not been explained anything like as well as gravity has (by relativity). And don’t get me started on quantum gravity.

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Anyway, thinking about all these issues, I concluded that science was (definition #3) the grand (platonic) model we are building of reality, ever evolving to best fit our observations.

My man, Plato

That works well for me. However, I recently came across a totally different definition for science:

# 4) “Science is a tool to help make the subjective objective.”

OK I paraphrased it to make it more snappy. It was really a discussion about how science was developed to overcome the fallibility of the human mind. Examples of weaknesses it needs to overcome are:

  1. The way our perception is filtered by preconceptions
  2. How we see pattern where there is none
  3. How we select evidence to match our opinion (confirmation bias)
  4. How we  read too much into anecdotal evidence
  5. etc etc.

I could go on. So ‘science’ is the collection of tricks we use to overcome our weaknesses.

I like this definition. We are all going about, and in our heads we are building our model of the world… and its time for an audit!

Is the Earth like a fractal?

Have you ever spent any time looking at the Mandelbrot set? I don’t mean a cursory glance, I mean really contemplated it?

Mandelbrot set

The Mandelbrot set. The horizontal axis is the number line,

And I don’t even mean the fancy coloured versions, just the straight black-and-white one (see image on right)?

It is really far more interesting than it looks…

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At first glance it just looks like a prickly pear gone wrong, so what is so interesting?

Well, remember this graph shows a set, that is to say it divides numbers into two groups, those inside the set, and those outside.

You could easily create such a set with circle – you can define the co-ordinates inside the circle is ‘in’ the set and that which is outside the circle outside the set. Such a set can be defined in a sentence: it is all points c less than the distance r from the origin.

The Mandelbrot set is just the same – a shape that divides space into two regions – in the image, the black area shows the numbers in the set, and the white area show the numbers out of the set. The only difference from a circle is that the Mandelbrot shape is more wiggly.

It equation is not much longer to define:

It is all numbers c, where if you square the number, then add the number, then square the result and add the number again, then repeat, it does not tend to infinity.

So zero is in the set, if you square it, then add it, it is still zero.

How about 1? No, it will run off: 1,2,3, 4, …

-1, on the other hand, squared, is 1, then added, goes to zero, where it will get stuck: -1, 0,0,0…

Figuring out the contents of the set is however complicated. Bloody complicated. Infinitely complicated in fact, and one of the marvels of the mathematical world.

To get a feeling exactly how complex this set is, take a look at some animations in which you zoom in on the perimeter; you can google “fractal dive” for more…

Mandelbrot Zoom Animation

You can choose what part of the set to zoom into yourself, if you want, here: http://www.h-schmidt.net/MandelApplet/mandelapplet.html

It seems that one simple sentence has been able to define an infinitely complex boundary, and begs some interesting questions: have we created a sort of universe? What is the information content of this set?

It makes my head hurt.

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The area of specific interest to me, is the relationship between the platonic ‘world of maths’ and the real world; so are there parallels between such complexity in the world of numbers and the real world?

Clearly fractals sometimes have similarity to things in the real world – such as crystals, feathers and broccoli florets. We see many reminders of complex structure in the real world, and it brings me to think of the earth as a sort of giant 3-d fractal – where the solid matter is ‘in the set’ and the gas of the atmosphere and the vacuum of space is “out of the set”.

We can see that the interface, the earth’s surface, also has complex structure, including such things as crystal caves or the lining of your lungs – and like with the Mandelbrot, we also seem able to zoom in to many levels.

However, just as there are no perfect spheres in the real world, there are no perfect fractals, and it seems that the structure falls apart once we get to subatomic levels of zoom (or does it?)

Some part’s of the earth’s interface are fractal-like due to the iterative nature of their construction (tree growth, crystal growth, sea-shells, etc.), and you can see that a simple rule of “branching” in a plant can make a complex shape. However, some of the complexity, specifically organic and bio-chemistry, still seem different (to me anyway).

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Krzysztof Marczak Mandelbulb

Krzysztof Marczak's Mandelbulb rendering

Now it turns out that my idea of the earth as a fractal with its skin as a complex interface can be more closely matched to the newly invented (discovered?) “Mandelbulb”, the amazing 3-d set. The interesting story of their development is told by one of their key developers, Daniel White, here.

The Mandelbulb is amazing because, as with many other fractals, you can zoom in and see ever more fascinating detail, and with incredible variation, and if you zoom into it continuously it would not be unlike zooming in on planet earth using Google Earth.

This excellent video below shows the idea of zooming into the Earth off well  (it seems to be derived from a book I happen to have called “Powers of Ten“).

So of course, you can do a similar trick on the Mandelbulb:

Enough said!

The Statistics of Fear

By all rights we should be rather scared of cars. We should regard them in much the same way we would regard a tarantula on the bedsheets or a short stroll on a tightrope. Yet we aren’t. Curious. Cars kill far more people than lava flows, piranhas or anthrax.

Perhaps they are too new for evolution to equip us?

Now if we look into the stats you realise we should be even more terrified of crap food, and the word “sedentary” should be a glare-inducing swear-word. If our bodies knew what was good for them, we would get grouchy after a day without a vigorous walk, and start shaking weakly after two. Sugar, so long sweet, should soon evolve to be utterly disgusting. Not soon enough for some.

Family Tree Nonsense

For many, learning about their family tree can be a real joy and pleasure.

Realising that a long distant grandfather was in the guild of barber-surgeons, or that you have a criminal or judge or, heaven forfend, someone famous in your lineage, can be a real thrill, and provide you with significant ego-boost, or perhaps a nice feeling of belonging.

The reason why it’s so often a positive experience, is due to an interesting mathematical oversight.

What do I mean?

This fractal can teach us something about ancestry: we have more ancestors than we tend to suppose...(image credit: Northwest Liberty School http://nwls.us/)

This fractal can teach us something about ancestry: we have more ancestors than we tend to suppose...(image credit: Northwest Liberty School http://nwls.us/)

I mean that people can read anything they want from a family tree, and this is made easy by the exponential nature of ancestry.

Would you find it remarkable for someone to say they were directly descended from Isaac Newton, Henry VIII or even Jesus? Would you think more of them?

What about people who say they are ‘from’ somewhere? – “my family originally come from Brittany…”, or “my family fought in the  revolutionary war…”

Analysis…

Now anyone who knows about the maths of ancestry, knows that there is actually very little remarkable about relations with famous people, especially from a long time back. If you have two parents and four grandparents, eight great-grand parents and so on, you can guess the numbers get big quite quickly.

The American revolutionary war was around 230 years ago, so perhaps eight or nine generations[1].  Nine generations back we had perhaps 2^9 (or 512) ancestors, and their folks were probably still around you can add them to the mix (another 1024)  giving over 1500 relatives, all swanning about somewhere in the world around the time of the war.

OK, you might want to reduce the number a bit due to some folks appearing  multiple times in your tree; (yes, in-breeding happens to all of us), but the number is probably still well in excess of  a thousand.

So, with over a thousand ancestors around at the time, the chance of having at least one involved in the war is pretty darn good (especially if you are were born to US citizens). I would argue that for anyone who can trace back three generations (to your eight great-grandparents) in the US, it would be far more remarkable if they didn’t have ancestors that fought in the war.

As you get further back in time, the numbers get more serious. A thousand years back , or forty generations, the straight maths gives 1 099 511 627 776 ancestors. Of course, this is impossible, as there were not enough people in the gene pool; the real number is clearly much lower and this is due to our old friend, in-breeding – where the family tree morphs into more of a family ‘web’ and involves the majority of the (breeding) population of your “gene pool”, the group that share enough in-breeding to behave somewhat like a super-organism. Where cross flow of genes between parts of the pool becomes retarded, (most usually by geographic barriers) the pool may divide and racial difference may develop.

Of course, we live in a time of great ‘connectivity’, and the US is a great example or a melting pot, with a very ‘open’ gene pool. This means that statistically, the chance that all 1000+ of one’s ancestors were around in the revolutionary war is hopelessly optimistic (unless there were special circumstances, like a closed community with a high degree of in-breeding, as may be the case with some religious groups).

So basically, anyone who says their family is all-American, “since the revolution” is being highly selective in their analysis.

Of course, western society does tend to invest much importance in the male line – which is far more specific – and would only give a couple of  chaps alive in ~1780, and if you can indeed prove this then the claim may be considered more interesting.

However, the argument that the male line is more important in some way (such as in the forming of character, or of any particular heritable trait) is pretty unconvincing. So even if you can trace a direct male line to Isaac Newton, this is no guarantee that you will pass your physics tests! Any advantages he had, will have been diluted by the 16,000 or so other folks who contributed just as many genes.

The male and  female lines can actually be traced (using mitochondrial DNA for the female line and Y-chromosomal DNA for the male line), but though this makes it easier to trace these ancestors,  it is perhaps still unwise to assume this line is more important than the thousands of other ancestors.

In the case of the USA, there is another factor, the large family sizes, and the resulting high population growth rate. The population present during the revolution have, by all accounts, been very fruitful. That means that even if you could trace your male line right back to, say, Thomas Jefferson, the chances are, you are not unique.

The Opinion Bit…

Hard-earned privilege...
Hard-earned privilege…

I am constantly annoyed by selective analysis of ancestry. I hope that the above simple illustrations alert the reader to this trickery, or at least confirm the reader’s suspicions (or convictions) that much of this is wishful thinking. What is most important to our own ‘value’ in the world is surely what we ourselves decide to do, not what our remote ancestors may have done.

However, I cannot deny that family research is still hugely interesting, even if what it really confirms is that we are all brothers and sisters, and none of us is superior due to our ancestry.

Don’t even get me started on so-called “royalty”!

[1] How long is a generation? http://www.ancestry.com/learn/library/article.aspx?article=11152

Open question about relativity

A quick open question for physicists:

If you accelerate off in one direction, and keep accelerating until you are travelling fast (a relativistic speed), special relativity supposedly says the universe contracts in the direction of your travel. Fine, I can see how that makes some sense.

Now consider a massive body, such as the sun – it warps space time in its vicinity, presumably roughly equally in all directions, creating a symmetrical ‘dent’ in the fabric of space-time (if you like the trampoline analogy).

But if you fly past at a relativistic speed, and space is contracted in the direction of your travel, will the sun’s sphere of influence also be contracted, turning it from a “sphere of influence” into an ‘oblate spheroid of influence’?

Or will its shape be maintained for some beautiful reason (which is what I suspect)?

Thx.

The interesting implications of our theory of gravity…

The evidence is now pretty strong that Gravity is just a symptom of ‘curved’ space time.

While it’s cool to have gravity all figured out, like so many matters in science, the answer raises even more interesting questions.

Like what is the nature of the curvature? Well, people (including me) are still trying to figure this out. In the meantime it is a good pastime to pontificate about the implications of curved space time. Here are two of my most recent theories/perspectives…

Perspective 1: Trees and apples switch places…

Each mass has a ‘destined path’, a path it will follow if left to its own devices. Just as Newton suggested in his First Law of Motion, things only change velocity when experiencing a net force.

However, he thought that gravity was a ‘force’ that made apples drop, however, the new theory of gravity suggests the apple was stationary – it was the tree and the meadow that were accelerating (upwards), a result of being pushed by the ground.

It lets us think of falling objects as ‘free from force’, and obeying Newton’s First Law.

Now, switch gears. Think what would happen if you could walk through solid things like walls. You may think it useful, but it would certainly cause some inconvenience, as you would presumable fall through the floor and plunge into the Earth’s molten core. You would fall past the centre and then start slowing; you would then briefly surface on the other side of the Earth, only to fall again. You would thus oscillate on some sort of sine wave. This is your ‘destined path’, the straight line through space time that your mass and location intend for you, where you to follow Newton #1. It is simply all the floor tiles and rocks preventing you from going straight in space-time. You are thus constantly being pushed, and thus curving off that path, thanks to the force of the floor. Lucky thing really.

Perspective 2: Slow time really is a drag…

A gravitational field can also be thought of as a gradient in the speed of time. It is possible (to me at least) that rather than supposing space-time is curved, it may well be that it simply varies in ‘density’. How? Well if time passes at different speeds in different places, that can be thought of as a density difference.

Now, we know that even when standing still, we are still plunging ahead – through space-time – in the direction of time. However,  thanks to Earth’s gravity, time is going slower down at your feet, they are sluggish, stuck in the mud. Now if you have a pair of wheels on a fixed axle, what happens if your right wheel gets stuck in the mud? It slows and you turn to the right… and in the just the same way, your body is trying to ‘turn’ downwards toward your feet – the gravity you feel!

When I first thought of this model, I was smug and pleased with myself. Until I found someone else[1] had already used it to accurately model planetary orbits. Read about it here – they have shown that waves (and therefore particles) will curve for the above reason combined with Fermat’s Principle. Bastards! 😉

 

Refs:

[1] Landau, LD; Lifshitz, EM (1975). The Classical Theory of Fields (Course of Theoretical Physics, Vol. 2) (revised 4th English ed.). New York: Pergamon Press. pp. pp. 299–309. ISBN 978-0-08-018176-9

Winning the toss

WARNING: Please do not even try to read this unless you are cricket fan. If you are not it will only irritate you 😉 thx.

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Test Cricket: Australia vs. South Africa: 6 tosses to ZERO, nada, nought. Time for a change in the rules.

by Jarrod Hart

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Winning the toss in cricket can be a very significant advantage, and lead to unfairness. Let me explain…

Firstly, I am going to admit that I am an avid cricket fan. And being a nerd too, this makes me doubly susceptible to a love affair with sports statistics.

I probably learnt most of my maths skills working out how many runs Graeme Pollock required to improve his season’s average, or calculating the run rate required by Clive Rice’s Transvaal team to win the Benson & Hedges night series final at the Wanderers.

So what is my problem with the toss?

In many sports there are environmental factors that skew the game – the sun shining in your eye when you serve, the dew on the putting green, the wind behind Jonny Wilkinson, and so on.

A cricket pitch is no different. Being composed of sand and grass, it is not wholly predictable – it is also prone to evolve over time.

It can therefore be a big advantage to bat first, or perhaps to bat second. The skilled captain can often tell what the pitch will do from looking at it. To make things fair, a coin toss is used to see who gets to choose who goes first. Fine. Over the course of years, all teams will win some and lose some. However, with test matches taking five days each, the teams only play each other once every few years, so the ‘fairness’ may take a generation to arrive. 

The stats are clear. Winning the toss helps. Get the latest stats from wikipedia; at the time of writing 34.7% of toss winners had won the match. 30.8% of toss losers had won. The remainder draw/tie.

Some people will still say, oh, that’s not too big an effect, less than, say, the home advantage in football. Yes the numbers aren’t too dramatic, but what no-one seems to be pointing out is that that 3.9% difference is actually coming from a somewhat bigger difference in a smaller subset of the matches.

For example, many cricket tests are one-sided. That is to say, the favourite wins. In cricket upsets are fairly rare (rarer than in football for example). Draws yes, but complete reversals are not that common. This means that there have been lots of outclassed teams winning the toss (statistically, even the most outclassed team win the toss half the time). They have of course, mostly lost.

Secondly, there have been matches where the pitch really was constant. That is to say the toss didn’t help.

So we are left only with the matches between evenly matched teams on pitches that change. The 3.9% positive toss effect must be a stronger effect coming from this subset of the results.

I also have anecdotal evidence (all scientists, wince now). I have often watched a match, where the first team has scored 600 of a flat pitch, and then seen the second team face a ball that suddenly stays low or cuts around. We have recently seen some almighty thrashings, some by an innings plus. These by teams that were, just before the match, considered fairly equal.

Now we come to the present time. The #1 and #2 teams in the world have been locked in battle for several months. I am of course referring to the ongoing test series between South Africa and Australia. And Australia have just won their sixth toss in a row. They have won the last three matches, on ‘evolving’ pitches, and of course I am bitter, of course I am looking for excuses. But I am rational enough to see unfairness when it strikes.

I was not going to write about it. I knew it would come over as whining. Until a friend pointed out a simple solution: they could have one toss at the start of the series and then alternate the choices for the remaining tests, thereby preventing one team heaping the unfairness too high.

Today’s sixth win in a row for Australia was too much. We need a change in the rules. The South Africans have lost their #1 ranking to the toss, and it stinks.

The last 6 tests:

Test no. 1899 Toss won by Australia, match by SA (In Australia)

Test no. 1902  Toss won by Australia, match by SA (In Australia)

Test no. 1904  Toss won by Australia, match by Australia  (In Australia)

Test no. 1910 Toss won by Australia,  match by Australia (In SA)

Test no. 1913  Toss won by Australia,  match by Australia (In SA)

Test no. 1916 Yes, Toss won by Australia,  match ongoing (In SA)

Foot-note: Being a cricket stats nerd, it was of course sad news to hear that Bill Frindall, a peerless cricket analyst had died. He acted as the eye of a the nerd-storm that has been raging for years, about which most of the world was blissfully unaware. His death has left us all without bearing, little squalls in the night.  CricInfo is not not quite the same. Yes, it’s rammed full of passionate staff, many of whom are nerds and scholars of the game, but it seems to lack the Frindall touch. I ask you this: who will care about this problem with the toss in this new world order? Bill would have.

Gravity explained in 761 words

People seem to be harbouring the impression that there is no good theory of Gravity yet. I asked a few friends – most thought Newton had explained it, but couldn’t explain it themselves. This is rather sad, 80-odd years after a darn good theory was proposed.

Of course there is still some controvery and the odd contradiction with other beloved theories, but the heart of the General Theory of Relativity really does a great job of explaining gravity and it is really wonderfully beautiful, and can be roughly explained without recourse to jargon and equations.

This is a theory that’s just so darn elegant, it looks, smells and tastes right – once you get it. Of course, the ‘taste’ of a theory doesn’t hold much water; for a theory to survive it needs to make testable predictions (this one does) and needs to survive all manner of logical challenges (so-far-so-good for this one too).

This is not a theory that needs to remain the exclusive domain of physicists, so for my own personal development as a scientist and writer, I thought I might try an exercise in explaining what gravity is – according to the general theory of relativity.

For some reason, my wife thinks this is strange behaviour!

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The story really got started when Einstien realised that someone in an accelerating  spaceship would experience forces indistinguishable from the gravity felt back on Earth. 

He or she could drop things and they would fall to the floor (assuming the spaceship is accellerating upwards)  just as they would fall on earth.

So perhaps that’s all gravity is… some sort of accelleration? Let’s see.

In the spaceship, it’s clear to us that the objects would appear to fall to the floor, but in reality, it is the floor of the spaceship that is rushing up towards the objects – this explains why things fall at the same speed whether heavy or light, matching Galileo’s own test results when he dropped various things, supposedly from the leaning tower of Pisa. It further implies that things will ‘fall’ even if they have no mass at all… such as light beams.

The thought experiment goes thus: Consider if you had a laser-beam shining across the spaceship control room; it would curve slightly downwards, because the light hitting the opposite wall would have been emitted a little time ago, when the spaceship was a little way back, and going a bit slower (remember, its accellerating).

We know the light is not bending, it is just that the source is accellerating, resulting in a curved beam. Imagine a machine-gun spraying bullets across a field – as you swing the gun back and forth the bullets may form curved streams of bullets, but each individual bullet still goes straight.

So Einstein suggested that perhaps light beams will bend in this same way here on earth under a gravitational field. Now Newton’s theory of gravity says light beams may also bend if they have ‘mass’, but the mass of light is a dodgy concept at best (it has inertia but no rest mass, but that’s a whole different blog posting). Anyway, even it it does have mass, it would bend differently from what Einstien predicted. So the race was on to see how much gravity could bend light…

This bending of light prediction was proven by a fellow called Eddington who showed that during a solar eclipse, light from distant stars was indeed bent as it passed near the sun, and by exactly the predicted angle.

Einstein went further though, suggested that light beams on Earth are, just like on the spaceship, really travelling straight, and only appear to bend, and that this can be so if space-time itself is curved. They are going straight, but in curved space.

We know that the shortest distance between two points is a straight line, but if that line is on a curved surface, supposedly straight lines can do strange things – like looping back on themselves. Think of the equator. This model therefore allows things like planets to travel in straight lines around the sun (yes, you read right).

The model has been tested and shown to work, and gives good predictions for planetary motion.

So what can we take home from all this?

Well mainly, if this model is right, we need to let it sink in that gravity may not be a force at all, but an illusion, like the centrifugal ‘force’ you experience when you drive around a corner.

Secondly, it is an open invitation to think about curved space and its marvellous implications!

Extrapolating your way

There is a very powerful scientific reasoning tool that I use, that, it occurs to me, I wasn’t actually taught… the simple art of extrapolation.

Most people have a pretty good idea of what extrapolating is – its where you look at a trend and predict what will happen if that trend persists. 

For example, if I said it took me 6 months to save £500, I can use extrapolation to predict how long it will take me to save £2000; its something we do all the time – yesterday I was driving down from Bristol, I could count off the the miles, and knowing the distance, I could predict if I would make it for dinner (I didn’t).

Scientists use this too. A good example is the way we can calculate the temperature of “absolute zero” by looking at the volume of a balloon as you heat it up. If you had a balloon at 25C, and you heat it to about 55C its volume would increase by about 10%. What does that tell us? It tells if we cooled it, it would eventually have no volume – and that this would happen at around -275C (-273.15C actually) – absolute zero.

Of course, the method relies upon assumptions – usually the assumption that the trend will continue in the same way (people often use the term “linear” to represent relationships that form straight lines when plotted on a graph).

What if the relationship is non-linear? For example, if little James is 5 feet tall when he is 10, how tall will he be when he is 20? Clearly he won’t be 10ft tall – that is because the relationship between height and age is “non-linear”.

Most of us are smart enough to extrapolate without knowing the jargon, but when the relationships get complicated a bit of maths and jargon can help.

For example, if we want to examine the population of bacteria in a petri dish, or the spread of a virus (or a rumour) through a population, our mental arithmetic is not always up to it. Luckily, some scientists have realised even these complex affairs have some predictability and although “non-linear”, they can still be modelled – graphs can be plotted and extrapolations made.

If this interests you, I refer you to books on epidemiology; I will move onto another sort of extrapolation – one used to check people’s theories by identifying ‘impossible’ extrapolations.

Let’s say, for example, that the want to predict  how the obesity epidemic will progress in the coming decades. If the media says obesity in a certain group increased from 14-24% between 1994 and 2004, and then goes on to predict that obesity will therefore reach 34% by 2014, does this withstand scrutiny?

Never mind that the definition of obesity may be faulty (BMI), never mind that they are extrapoliting from 2 data points – let’s rather ask if the linear trend is justifiable. This can be done by extrapolating the prediction to try to break it. 

If the model is right, obesity will go on increasing and soon enough 100% (or more!) of the population will be obese. This is clearly wrong – obesity is not likely to get everyone – vast swaths of the population are likely to be immunised to some extent against obesity due to active lifestyles and good dietary educations, or perhaps its in their genes, the lucky things. 

The truth will of course be more complex – the first group to become obese will be the most vulnerable, so an increase from 14-24% may incorporate that group, but each successively 10% will be harder fought.  All this is enough to suggest the predictions made for 2014 are doubtful, and those that go further are downright shameless. But it doesn’t stop them

I am sure you can think of other suspicious trend-based predictions, like those for peak-oil or global warming. They could do with some improvements, so get to it!