Is the Earth like a fractal?

Mathematics, Physics, Science, Science communication, , , , , , , ,

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…

[youtube=http://www.youtube.com/watch?v=fLrSapMYUlI]

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“).

[youtube=http://www.youtube.com/watch?v=9BjHvwSvpOw]

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

[youtube=http://www.youtube.com/watch?v=LxsniUIVfEM]

Enough said!

The Statistics of Fear

Evolution, Mathematics, Science, , , , , , , ,

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

Evolution, History, Mathematics, Science, Science communication, , , , , , , , , , , , , , , ,

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

Climate change solutions: mass behaviour simulation

Economics, Environment, In the media, , , , , , , , , , ,

Saving our planet from catastrophic climate change might require an unprecedented mass co-ordination of all the people on our lonely little planet.

However, it requires co-ordinated sacrifice, the west are living unsustainably, the east have not had their fair share yet, Africa is unmanageable… how will we pull it off?

Pondering this issue, I am sure none of the experts will have any good prediction of how people will behave – when will the zeitgeist be strong enough to allow governments take the massive steps required? When will china be satisfied that they have pulled up their living standards enough such that it would be fair for them to sacrifice too?

With such imponderables, it seems to me, we might gain some insight if we can create on on-line simulation, a ‘game’ if you like, with a large number of participants, each with their own minds, their own priorities, their own feeling of what constitutes justice.

In this game, some would be wealthy, those that had benefited from the industrial revolution, the slave trade, etc, etc, and fiercely protective of their way of life, many more would represent the 3rd world, the developing nations, the disenfranchised, the war-torn…

These people would thus all live in a 2nd-life style world in which carbon emissions are sure to cause catastrophe (no-one knows when!) but carbon emissions are associated with the luxuries used by the people. Will people be able to co-ordinate themselves to reduce overall emissions? Or will they each take the ‘every man for himself’ route, ensuring that the fit survive, but perhaps with a lower total survival rate?

Could such a game be set up?

It would require a committed community of computer experts (which exist) and a committed community of environmentalists (which also exist) – but do they overlap?

How could we go about trying to make this happen?

Unproven medicine : an alternative name for alternative medicine

In the media, Medicine, Science, Science communication, Skepticism, The scientific method, , , , , , , , , , , , ,

“Alternative and complimentary therapies”. They sound so nice. So warm and fuzzy. Surely they augment the cold clinical scientific approach to regular medicine, and have a more holistic approach catering to the soul and spirit as well as the flesh?

I argue not. Hear out my logic…

Any treatment that has proven to provide reliable benefit, is automatically added to the canon of ‘western’ medicine. Therefore the only treatments left available for ‘alternative’ to claim, are those that are unproven, or worse, treatments known to be actively harmful.

Promoters of alternative medicine will argue that western medicine is still woefully weak, and not tuned into holistic and spiritual matters and that such things defy proof. This is clearly claptrap. If you do a well designed double-blind, placebo-controlled test of an ‘alternative therapy’ and the outcomes are no better than for the placebo, then the participants who got the treatment are no better off, spirit or no spirit.

I personally prefer the sort of benefits that can be detected!

How did this situation come to pass, where unproven medications have such a grip?

I think there are three main ingredients:

  1. People make money from other people’s fear (in both western and alternative medicine) and that causes folks on both sides to hide or twist the facts – and also erodes the public’s trust.
  2. The fact that complimentary medicines do actually offer benefits – the well-known benefit of care and attention and also the benefit of the placebo effect – muddies the waters.
  3. It is however the human weakness of putting far too much value on anecdotal evidence that assures the future of unproven medicine.

I think that people who understand this do a disservice to our communities by giving this bad medicine the label ‘alternative’ or ‘complimentary’, so I would like to propose the term ‘unproven medicine’. I would however welcome some more lyrical suggestions!

Evil: a baseless construct

Atheism, In the media, Psychology, Science, Science communication, The scientific method, , , , , , , ,

This morning on BBC Radio 4’s “Though for the day”, the Right Reverend James Jones claimed “Evil triumphs when the imagination is inebriated with evil”.

So as a logician I would like to know what exactly “evil” is. Can it be measured (like energy)? Or detected by our (5) senses? Does it conform to the known laws (models) of physics?

For something so darn vague it is amazing how much we use it day to day. We blame so much on it, and justify so much in its name.

But in a strange dichotomy, if you pay close attention the the professions (medicine, law, engineering, etc) you will find scant mention of this concept – it does not help in the treatment of criminals or the mentally ill it does not explain earthquakes or building collapses – it seems has no use in the real world, but is used by politicians and preachers like a moral blank-cheque.

I therefore suggest that the concept of evil is a relic from a mystical past in which gods were invoked to explain thunder and demons to explain crop failure.

Surely all talk of someone being ‘evil’ or an act being ‘evil’ has no place in our secular world?

God vs. logic

Atheism, Science, Skepticism, The scientific method, , , , ,

It occurs to me, as it may well have occurred to you at some point, that the very idea of anything “supernatural” existing is self-defeating because anything that blatantly defies our methods of analysis and understanding would totally undermine the entire show. Thus our system of understanding which is so incredibly consistent that it’s able to convince electrons to run such complex mazes that the whole internet results, would be fatally flawed if even the smallest miracle were possible.

The application of logic has yet to be shown to fail.  The world has never thrown up any scientifically verifiable evidence of anything outside of our model. Thus there has never been any evidence for any god, any miracle, any ghost, any anything supernatural.

Yet its not as if God (as rep for all things supernatural) is deliberately covering his tracks. If He was, no one would know about Him. Clearly his followers have detected some “evidence” – but amazingly no science or logical analysis has ever trapped any of it.

So if you believe in God, how can you have any faith in such a poor system, that can’t detect this God you see so evidently? The system so good it cures diseases, determines the compositions of distant stars and genetically engineers mice so they glow in the dark can’t see something so big and important three quarters of the planet believe in it.

There is indeed an elephant in the room.

Post-theism

Atheism, Evolution, Science, Skepticism, The scientific method, , , , , , ,

I inhabit a post-theistic world. God or the idea of God has long-since lost its influence on my thinking – my morals, my emotions or my research. I am thus not so much an atheist as a “post-theist”. To me the debate is over, so the categorisation into the do’s and do-nots is now only an interesting study of human gullibility rather than a deep & heart-rending dilemma. 

Anyway, I thought it would be good to put into words my position on the whole God-vs-Science debate, which is really rather important and most certainly cannot be ignored.

I have frequently been tempted to write about this topic, but have opted to keep my powder dry, because I was not sure which approach would actually increase the net happiness in the world (groan away, but what other approach is better?). Options included:

  • Explaining how I came to be an atheist: my idiot christian school taught me about other religions, and I quickly added two-and-two and realised if they were mutually incompatible they were probably all false
  • I could go on about the scientific implausibility of the whole show. The total lack of evidence, not to mention the blatant disregard for logic.
  • I could explain how there is evidence human brains are designed, by evolution, to believe in concepts like God. 
  • I could go into the details of infectious memes, and how religion is a darn good example of a viral idea that has all the properties you would try to put into a computer virus if you made one (which I did once, for a HP48s, a once popular calculator).
  • Lastly, I could come out like a raging fundamentalist pointing out how religion is the ‘root of all evil’ (to borrow a phrase) – how religion justified the great wars and the subsequent suppression of nations /women / freethinkers / witches / etc.

However, I realise that Hecht, Dawkins, Harris and several others have walked this path (rather well) for me. Perhaps I should focus my efforts on subjects not yet so well covered? Some would argue that atheists need to rally together and ride the current wave of interest and publicity – we need to gain the critical mass.

This is the real question. What would help?  

I don’t know! Please help…