IKEA has the answer! Who Knew?
Looking for an image to go with the post i could not go past that one but my post was meant to explain WHY things are really much more complex than we ever realised.
Simple —-> Complicated, in 1 ‘easy’ step.
Ever wonder how something so ‘simple’ can go so wrong? Or go so wrong so FAST???
Me too but here’s a little example that may offer – if not help on how to keep things simple – some reason for why we never saw what was coming, or came the way we never expected!
Lets start off VERY simple. (Warning this will involve some numbers but DON’T WORRY! There will only be two of them – Zero and One – that’s all OK? I said I’d start off simple).
So, two numbers, just two ‘things’, they can be anything you wish. The idea is that all there is is two different ‘countings’, either there is one of a thing – or there is no thing, nothing. All (one) or Nothing (zero). Simple – yes? Good.
As simply useful as this can be – to tell if we actually have something or if we have nothing – it doesn’t quite cover ‘all’ cases. In life ‘nothing’ is ever this simple, sadly. There is often a ‘complication’.
Sometimes things have ‘opposites’. By reason of it’s position/location (or sometimes it’s image) we can have the ‘opposite of some(one)thing’ In number terms this is simply written as negative, or minus, one (-1). So if we add this to what we started with we now have one thing (1), nothing (o) and the opposite of one thing (-1), often seen written down simply as: -1,0,1.
In case you are a little confused already think of ‘one thing’ as being your Right Hand – you have one of those (one thing: 1). Think of the ‘opposite’ of your right hand (by reason of it’s location and it being a ‘mirror image’ of your right one), your left hand, as being minus one (–1) thing and when you hold them both straight out in front of you in between them you have no hands – no thing (0). See? – Easy as!
Ok – we started out with two things – one (or All) thing and zero (or No thing) and then added an ‘opposite’ for practicality. (How would life be if you only ever had one hand or one foot?)
Now in life we often find in our day to day world there is another complication. So far the things we have considered are only really being ‘described’ mathematically in one ‘dimension’ – a straight line starting from the left with –1 (opp.of thing) moving through the middle no-thing-ness of zero and ending at the point opposite to where we started +1 (thing). Or you could do it the other way round if you like reading ‘backwards’ 😉 Maybe the Hebrews, Asians and Muslims have it ‘right’ and we should all be reading right to left?
Life is seldom something we live in just one ‘dimension’ however. Most mathematicians and physicists will tell you there are at least 3 distinctly separate dimensions needed to truly define the ‘space’ we all live in. All solid objects need 3 dimensions to define every point in them.
To simply visualise this – imagine a cross: + (2 dimensions) and then rotate it by holding any ‘arm’ and twisting it through 90 degrees, keeping your hand still. The arms have then pointed in all three ‘dimensions’.
In simple terms we might call these dimensions, length, width and depth; or left, up and forwards, with their respective ‘opposites’ of right, down and backwards.
OK – so now, from our beginning with simple two ‘things’ (all or nothing), we added one mirror ‘opposite’ and put them into each of the three dimensions that can completely represent the physical world we live in – almost!.
Still simple – very simple; right? Two numbers (1,0); the number (thing) or it’s opposite (1,-1); and these can be oriented or aligned in any of three dimensions in space. Lets call the 3 dimensions x, y and z for ease of distinguishing them: x is left, right; y is up, down; and z is forward, backward.
OK then – what’s the problem? Well, given that there are only two numbers, two forms (thing or it’s opposite thing) and 3 dimensions they can be arranged in, how many ‘possibilities’ are there?
2? Three (-1,0,1)? Nine? (three times the 3 dimensions?)
In actual fact, there are 27 different possible ‘arrangements’ of these two (or three) ‘things’!
How can that be? Twenty-Seven?? From just 2 numbers? Yup!
Imagine a simple 3D cube like common dice. Every ‘edge’ is a representation of the two numbers and the opposite. –1,0,1; each ‘corner’ is a single ‘thing’ (number:1) away from ‘nothing’ (0) (the edge’s centre point).
The cube has eight corners: Two of them we could show as the ‘points’ (1,1,1) (one in the dimension or direction ‘x’; one in the dimension or direction ‘y’; and one in the dimension or direction ‘z’ from the centre) and (–1,-1,-1). These two corners are on the exact opposite points of the cube (as the numbers clearly show, ones and their ‘opposites’) through the centre of the cube, which is at point (0,0,0).
The other six corners or points can be written as: 1,1,-1; 1,-1,1; -1,1,1; (and their opposites) –1,-1,1; -1,1,-1; 1,-1,-1.
Besides the 8 corners we also have the middle ‘point’ (0,0,0).
There are also the centre points of each six‘faces’ of the cube (imagine the dice used in a game of craps and the point on one representing the ‘number’ one – a single point in the middle of the face. If we can only have the number one (or it’s opposite, minus one) on our dice then each face will have a point which is defined as being 1,0,0 or 0,1,0 or 0,0,1 and the ‘opposite side face points’ –1,0,0: 0,-1,0; 0,0,-1.
With the middle and the corners we now have a total of 15 separately defined points from just ‘one’ number away from zero and zero itself (in 3 dimensions).
So I said there are actually 27 different possible arrangements; where are the other 12?
Well, our cube has a centre (0,0,0), 8 corners and the middle of each of 6 faces so far – but we have not mentioned the ‘sides’. Because the height, length and width of our ‘one number’ cube extends from –1 to +1 they are actually two ‘units’ long; and each side therefore has a ‘middle’ that our one number defines.
There are 12 ‘sides’ to every cube; 12 middle points. So 12, plus our first 15, equals 27!
(if you want the exact 3D co-ordinates for each point simply write down all the combinations you can where there is exactly one zero in the 3 coordinate numbers and you’ll have them – here’s 2 to be going on with: (1,1,0) and (1,-1,0) ).
So from just two numbers (things) 1 and 0, allowing for the ‘opposite’ of 1 and considering in our daily world there are 3 separate dimensions we could ‘place’ them in, we find we have 27 possible alternatives to choose between!
Sadly – the ‘complication’ has one final ‘twist’.
In our everyday existence I have so far said we need 3 dimensions to fully account for our place and space in the Universe. While in some conditions this is true – it does not actually fully define the world all of us live in. There is one ‘extra’ dimension we have not considered. That of ‘Time’.
Those 27 alternatives all assume there is no change in time, only a change in place/position. Effectively this is just when nothing ‘moves’ in time – like in a photograph. That is not how we all live.
We have to consider the possibility of a movement of one ‘unit’ (so as to keep it at the simplest possible level) in time, as well as in space.
How badly does that complicate things?
It effectively means that EVERY alternative we defined before has THREE possible states in time: Now (‘zero’ time), a second later (time +1) and a second ‘before’ (time –1).
That means – from just two numbers – in the ‘real’ world. We have Eighty-One (27 x 3) possible ‘alternative’ states we can define!
Staggering isn’t it?
See why sometimes – even when things might appear simple, they can in fact be very, very complicated?
For the more mathematically advanced of you, you may, if you have enough time to spare, consider the possible number of ‘complications’ we would have if we limited what existed in our world to just 10 different things… give them the numbers 1,2,3,4,5,6,7,8,9 and 0 (and their opposites)… in 3 dimensional space – and time?
©Bob Thursfield 2009