Symmetry and dance

If the dancers are allowed only 3 positions of the arms and two positions of the legs, there would be 35 different possible positions. Five of these are symmetrical so if the dancers wear masks on the backs of their heads, rotating through 180 degrees produces a mirror image of the same position. In other words, the dancers have not changed shape.

Each of the 15 asymmetrical positions produces a new one when rotated through 180 degrees. So the dancers would appear to have changed shape.
Moving from one position to any other requires a real change of shape (i.e. a move through at least one other dimension).

Multi-dimensional space - and dance

In Flatterland* the 'Spacehopper' explains that a bicycle has seven dimensions:


The Spahehopper nudged the bicycle handlebars through a small angle. "There - a fourth dimension!"
"But you just moved it in three dimensions."
"Yes, but it has changed shape - if it had stayed the same shape, you could make it fit its current position by rigid motion."
"So if it has moved through a new dimension, in which direction does that dimension point?"
"The turn-the-handlebars direction." said the Spacehopper "Different from the turn-the-front-wheel direction and the turn-the-back-wheel direction and the turn-the-pedals direction. That makes seven dimensions in all."
So every time we change shape, we enter a new dimension. Dance is a truly multi-dimensional art-form. 
* Flatterland (Like Flatland only more so) by Ian Stewart

Duration and size

This graph plots Duration (how long something exists) against Volume (size). The axes are both log scales with quite a few gaps to accommodate the biggest things. 
The arrows show things which change in size during their lifetimes. I could not think of many things which decrease in size, but added the polar ice caps as an example.

Self-consciousness, movies, halls of mirrors and immortality

Self-consciousness could be defined as 'awareness that experience is happening'. A cat wakes up feeling hungry and 'decides' to hunt, catches a mouse and eats it. I wake up feeling hungry, make breakfast and eat it. 
One difference between me and the cat is the concept of self.
I can think 'I am eating breakfast' - the cat has no language and can't formulate such a thought. Essentially self-consciousness involves removing myself one step from reality so I can invent an 'I' which can be referred to. This is a uniquely human ability which can get us into all sorts of trouble:

• We can pretend we are in a movie in which we are the main protagonist.
• We can create endless chains of self-consciousness: "I am aware that I am thinking 'I am eating breakfast'" .... and so on to infinity.
• We can delude ourselves into thinking that the 'I' we have invented actually exists - and is immortal.

The Anthropic Observation

Lee Smolin* poses the question “Given there seems to be many sets of consistent ‘Laws of Nature’ which could reign in many possible universes, how come our universe has such finely-tuned parameters which make life possible?”

Answer 1) There are very many universes, so the chances of one with life in it is greater.

Answer 2) We can imagine a universe without life but only from our fortunate position of being alive! If a universe existed without life there would be no observer to perceive it – so would it exist? In other words, maybe a universe without life is impossible.

Answer 3) The universe passed through many phases of evolution to its present state. (this is a bit like Answer 1 – but time replaces space.

Maybe the universe is driven by a mechanism which favours consciousness.

Our role as observers or perceivers of the universe may not be a passive one. We have to ask the question about relativity – ‘Who is this observer?’ - What happens when she goes away?

Hume decided there was no way to show that the world exists when we stop perceiving it.

Bishop Berkley said ‘That proves the existence of God who is perceiving everything all the time – thus thinking it into existence.’

But who invents the idea of God? – We do of course (maybe not individually but collectively).

So perhaps the universe is brought about by our collective conciousness.

* In The Life of the Cosmos

Pi = 3

In the Old Testament, King Solomon’s pool had a diameter of 10 cubits and a circumference of 30 cubits. This gives a value of pi = 3. 

Suppose our instruments were not accurate enough to detect that pi is actually a bit more than 3. Or suppose we had a reigious conviction that pi must be a whole number. When measuring a circle with string, we would stretch the string a bit to accomodate the error. 

One day a pedantic mathematician with a length of un-stretchy string would be sure to challenge this belief. To avoid being burnt at the stake, he (or she) suggests that space-time must be curved just enough to make pi ‘seem’ to be a bit more than 3. In other words the circle is 'really' shaped like a dutch cap and the diameter is being measured across the arc of a circle (see diagram below).

From God’s point of view it is exactly 3 of course! 

This is dicussed in this Physics Forums link:

http://www.physicsforums.com/showthread.php?t=9869

The unknowable

Since the speed of light is constant, there is a limit to how much of the universe we can ever see or explore. Beyond this limit, everything is unknowable. Does it make any sense to ask what could be in this unknowable region? Is what we could know of the universe simply a definition of the universe? Does God know the unknowable bit?