Another type of number which is difficult to understand and has a complicated relationship to reality is the imaginary number denoted by i which is equal to square root of -1. Now, any negative or positive number multiplied by itself yields a positive and not a negative number, so the square root of -1 cannot exist in the field of real numbers. (square root of -3 can be written as 3i and so on..) But this kind of number has been defined in order to solve certain algebraic equations where roots of negative numbers appear. (They do so because the field of real numbers is not algebraically closed i.e. the solutions lie in a different field to the coefficients) A complex number consists of a real and imaginary part and can be written in the form of a+bi, where a and b are real numbers and i is the standard imaginary unit. So now a real number can be thought of as a special case of complex number where b=0. A complex number can be plotted on a graph with the real part on the x-axis and imaginary part on the y-axis. Using complex numbers means that we need never get stuck at solving tricky equations (the field of complex numbers is algebraically closed). They are very useful in many fields but in the field of Quantum physics, they are not just useful they are essential.
With real numbers we can describe the geometry of solid, geometric shapes like squares, cylinders, spirals etc. Benoit Mandelbrot discovered (accidently) that complex numbers could be used to describe complex shapes. Consider the equation zn+1 = zn2 + c ; where z and c are imaginary numbers and n increases by 1 each time. This means that the output squared and added to c is fed back to the input for calculation of another output. Mandelbrot wanted to know for which values of c, the magnitude of zn would stop growing when the equation was applied for an infinite number of times. He discovered that if the magnitude went above 2, then it would grow forever but for the right values of c, sometimes the result would simply oscillate between different magnitudes less than 2. He plotted these values with the help of a computer and was amazed to see a complex pattern which when magnified revealed a similar hidden pattern and this pattern went on infinitely. He named this pattern a fractal. A fractal has the self-similarity property of having the same (irregular) shape at all levels of magnification. Mandelbrot soon realised that fractal shapes appear everywhere in nature. A mountain range for e.g or a coastline, lightning or systems of blood vessels. These shapes cannot be predicted exactly in their details but the general shape can be approximated. Since then a variety of fractals have been discovered and some of them lie at the heart of a new branch of mathematics called chaos theory. Fractals are also used as the basis for digital art and animation created with the help of a fractal-generating software.
Fractals have an amazing complexity that surprisingly comes from a very simple equation. These images look like nothing you`ve seen before yet the individual patterns are familiar. Here is an example of a mandelbrot set:
The Mandelbrot set zoomed in:
Here is one that is created with a fractal generating software written in C called Sterling:
Another interesting number (this time,real!) is a constant denoted by c which is the speed of light in vacuum (approximately 186,282 miles per second which is essentially the speed limit of the universe). That the speed of light is constant regardless of the frame of reference of the observer is a very strange phenomenon that we are not normally aware of because it is so incredibly fast. If you are travelling on the highway in a car, for e.g, at 60 mph, your speed relative to the stationary objects like the trees that you pass by is 60 mph, but your speed relative to a football in the next car seat is 0 mph, likewise your speed relative to your friend driving by your side at the same 60 mph is 0 mph. If he is travelling at 60 mph in the opposite direction, then you will see him as driving away from you at 120 mph. If you shot a bullet from your car(!), then the total speed or velocity of the bullet would be the speed at which you are travelling + the speed of the bullet. Then, you would naturally expect that if you switched on the headlights of the car, the total speed of the light would be the speed of the car + the speed of the light but this is not so. This is actually very wierd and easier to visualise (that it is wierd!) if you imagine yourself travelling in a rocket at a speed close to that of light. You would always see light travel at c regardless of the speed you yourself are travelling at and is the same even if you were travelling in the opposite direction! How can this be?
Albert Einstein predicted time dilation as an explanation of this special property of constant speed of light in his theory of special relativity. Since v = d/t where v is velocity, d is distance and t is time , if the velocity remains constant something else has to give and that something is time. Let us imagine that we have a clock made of two mirrors opposite to each other in a train. A photon (a particle of light) bounces between the two mirrors and each tick of a clock is made by a photon bouncing off the mirror. Since the speed of light is a constant and the distance between the two mirrors is constant, this results in a very accurate clock. Suppose that now the train starts to move. Since the train is moving, the photon bouncing off a mirror will have to move a longer distance in a diagonal path to get to the second mirror, and back again in a diagonal path to the first mirror. Its path is now on a zig zag line since it is not `carried` by the train as would a sound wave or a ball. It needs to move a greater distance even though the two mirrors are the same distance apart so each tick of the clock is now after a longer duration! The faster the train moves, the slower the clock ticks and when the train reaches the velocity of light, time stands still! Time dilation has been tested many times by high precision clocks onboard jets flying around the world. After their flight it has been found that they run slower than the ones on ground, though this change is only in the order of a fraction of a second. Here is a cute video demonstrating this effect:
http://video.google.com/videoplay?docid=-6173480890398763094#docid=3095948330513072764
What does time standing still mean? Imagine that you were massless and hitchhiking a ride on a photon in a light ray travelling from the sun to the earth. For you, the instant that you leave the surface of the sun would be the instant that you would land on a beach on the earth, it does not take any time at all. (To an observer on the earth, however, the photon has taken 8 mins and 19 seconds to reach the earth.) This means that all distances are reduced to zero for the photon.This is really very strange for us to imagine…
Special theory of relativity also predicts length contraction (decrease in length of objects travelling). The general theory of relativity predicts (among other things) gravitational time dilation – gravity influences the passage of time. The more massive an object is, the slower time runs; the further away you are from the object the faster time runs. This notion of space, time and velocity being interdependent (where before Einstein, time was thought to be constant) has forced us to think of the universe as a space-time continuum in four dimensions, 3 spatial and one of time.
The nature of the photon or any subatomic particle for that matter is another of nature`s mysteries. Quantum physics is all about thinking very,very small and it first started taking shape during Einstein`s time. Just as classical physics does not work at relatavistic speeds, so too at the atomic and sub atomic level, these laws are pretty much useless. Things happen at that scale which if they were to happen in our world, would seem very bizzarre indeed! Einstein spent the last years of his life searching for a unified theory that would unite the general theory of relativity with electromagnetism. A developing theory of today is string theory (which reconciles general theory of relativity with quantum physics) and M-theory which talks of not 5, not 6 but a total of 11 dimensions! Because I have moved further and further away from the subject of art, I will sign off for now leaving you with a link of some videos about string theory which I am sure you will thoroughly enjoy:
Watercolor Journal 