The beauty and mystery of Nature (part 2)

 

                 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  zwould 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:

             http://www.youtube.com/watch?v=E7FV9aaiwKQ

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The eternal beauty and mystery of Nature (part 1)

 

                     During the pythagorean time, a mysterious new type of  number, now called the irrational number was  concluded to exist from the famous Pythagorean theorem which states that

       The sum of the squares of the sides of a right-angle triangle is equal to the square of the hypotenuse (the side opposite the right angle).

                     Now, if we take the two smaller sides in a right-angled triangle to be 3 and 4 units in length, then the remaining side would be 5 units in length because 5*5 = 4*4 + 3*3. So far, so good; but what if the shorter sides were 1 unit each in length? The larger side would have to be x where x can be written as x*x = 1+1 = 2. But what can be the value of x which when multiplied by itself would give 2? It must lie somewhere between 1 and 2, so that means it should be a fraction. But it turns out that there is no fraction that when multiplied by itself gives 2. It was finally calculated to be a value running forever without any regular pattern: 1.41421356…….. and written as squareroot of 2 and is an example of an irrational number. A rational number can also run forever but it has a repeating pattern to it (for e.g 1/3 = 0.3333…. ) That these type of numbers could exist which are not fractions of whole numbers was so shocking that the truth was suppressed at the beginning. Now we know that irrational numbers are the norm rather than the exception. Between every two consecutive integers and two fractions there can exist an infinite number of irrational numbers.

                   The root of 2 may not be a very special irrational number but there is another irrational number that is, and which is considered almost magical – phi (1.616033988749894…………) and has become the most popular number to be used and studied  in biology, art, architecture and music since its discovery. The golden ratio or golden or divine proportion as it is called is formed when a point B is chosen on a line AC such that the distances AC:AB = AB:AC  (fig a). Phi can be calculated from the arithmetic mean of 1 and the square root of 5. It can also be obtained from the Fibonacci series (the integer obtained from adding the two previous integers before it with the first two integers being 1 and 1) The Fibonacci sequence produces numbers that, when divided, forever keep getting closer and closer to the value of phi.  

                  A golden rectangle is a rectangle which has a length over width ratio equal to the golden ratio. A golden spiral can be drawn inside this retangle by dividing it into squares and drawing a quarter circle with radius equal to the side of each square. (An approximate golden spiral can also be made from the numbers in the Fibonacci series) This spiral is a special type of what is called a logarithmic or equiangular spiral.                   

composition

 

                   The logarithmic spiral has a property of self-similarity meaning that the shape of the spiral remains the same for each successive curve. It keeps getting bigger and bigger but its shape remains unaltered. Logarithmic spirals are found very often in nature – in snail and sea shells, in the arms of galaxies, pinecones, fruits and vegetables like broccoli and pineapple, in the way a hawk approaches its prey etc. The cross-section of a  nautilus sea shell forms one of the finest, most beautiful natural example of a logarithmic spiral. The inside of its shell has an iridescent mother-of-pearl coating. The animal living inside the shell needs the shell to grow along with it in order to keep accomodating it. As it grows it moves into a larger area and seals off the vacated chamber. As it always places new shell material at the open edge and faster at one side than the other, it makes the shell grow in a spiral manner. (If the shell was circular in shape, the poor creature would not be able to grow) Each type of animal has a different rate of deposition and just slight changes in these rates result in a stunning array of beautiful shell shapes. Horns of animals make a two-dimensional version of the same concept. They would grow straight if the rate of material which is added at the base is equal on both sides (like in nails) but if one side grows faster than the other, the horn starts to coil to the side which has less material added. Below are some varieties of sea shells:

 

                      In the heads of flowers, the most beautiful and well known example being that of the sunflower, phi is used in seed organisation. The heads consist of small florets which later mature into seeds. The seeds are produced at the centre and migrate outward to eventually fill out all the space. Each seed is produced at an angle to the previous one. Let us suppose that the seeds are produced for every half turn, that is 180 degrees. (fig a) This will cause the pattern of seeds to be in a straight line which is clearly not a very efficient way of filling the space since there are a lot of gaps. The same is the case for one quarter of a turn (fig b) or for any decimal fraction(fig c) for that matter. Fig d shows the seeds being produced at an angle of pi (3.14..) which is an irrational number. Now there appears a spiral pattern but it is still not an optimal filling. For that to happen, the rotation has to be the most irrational number possible and that is phi. The corresponding angle, that is the golden angle is the smaller of the two angles created by sectioning the circumference of a circle according to the golden section. This is approximately equal to 137.51 degrees. You can see the resulting pattern corresponding to this golden angle in the daisy head photo. It appears as two sets of crisscrossing spirals one going left and the other going right and the number of these spirals in each direction will be two consecutive numbers in the Fibonocci series.  i.e. they can be 5 and 8 or 8 and 13 or 34 and 55 etc depending on the size of the flower head. Also, since the petals are formed at the extremities of the spirals the number of petals in a large number of flowers are also Fibonocci numbers. What is simply amazing is that a flower can make such a precise measurement of angle, even a very slight change of 1/10th of an angle can disrupt the optimal filling! In the same way, a plant uses this angle in leaf growth to ensure that all leaves receive the maximum amount of sunlight and dew (fig e). Leonardo da Vinci was the first to describe this type of leaf arrangement in plants.

 

                     Phi is also found in the human body, like in the ratio of the forearm to the hand, in the spiral of the ear, in the DNA molecule… In fact, the occurence of phi is so common in nature that any form that is constructed based on these proportions is generally believed to be aesthetically pleasing too. For e.g in art, a canvas in the proportions of the golden ratio, can be sectioned again in this ratio lengthwise to get the horizon line. I`ve used them in this drawing of the human face; the length of the face:width of the face, the distance from the eyes to the chin:distance of the eyes to mouth, distance of eyes to mouth: distance of eyes to nose. 

golden proportions

 

                      The Vitruvian man, a drawing made by Leonardo da Vinci to illustrate a book entitled `De divinia proportione` (The divine proportion) by the mathematical innovator Luca Pacioli has become a universal symbol of human perfection and the integration of mind,body and spirit. The name Vitruvian comes from the Roman architect Marcus Vitruvius(ca. 70-25 BC) who was the source of many of Pacioli`s ideas. Leonardo di Ser Piero da Vinci was a multitalented genius from Italy who lived in the 15th century. He was a painter, sculptor, architect, musician, scientist, mathematician, engineer, inventor, anatomist, geologist, cartographer, botanist and writer and had an unquenchable curiosity to understand the universe. He described his anatomical works as a cosmografica del minor mondo, a cosmography of the microcosm and believed that an anology could be drawn from the workings of the universe to the workings of the human body. The Vitruvian man is the first of its kind to bring together artistic and scientific objectives in the proportions of the (male) human body. Each part is a simple fraction of the whole. The body is inscribed in both a circle (symbolising the spiritual existence) and a square (symbolising the material existence) with the centres of the two being different. The combination of the two different arm positions and two different leg positions creates a very dynamic composition of sixteen different poses.

         The proportions as written by him in the accompanying text in mirror writing are as follows:

a palm is the width of four fingers
a foot is the width of four palms (i.e., 12 inches)
a cubit is the width of six palms
a pace is four cubits
a man’s height is four cubits (and thus 24 palms)
the length of a man’s outspread arms (arm span) is equal to his height
the distance from the hairline to the bottom of the chin is one-tenth of a man’s height
the distance from the top of the head to the bottom of the chin is one-eighth of a man’s height
the distance from the bottom of the neck to the hairline is one-sixth of a man’s height
the maximum width of the shoulders is a quarter of a man’s height
the distance from the middle of the chest to the top of the head is a quarter of a man’s height
the distance from the elbow to the tip of the hand is a quarter of a man’s height
the distance from the elbow to the armpit is one-eighth of a man’s height
the length of the hand is one-tenth of a man’s height
the distance from the bottom of the chin to the nose is one-third of the length of the head
the distance from the hairline to the eyebrows is one-third of the length of the face
the length of the ear is one-third of the length of the face
the length of a man’s foot is one-sixth of his height

vitruvian man

 

             The original drawing of da Vinci`s can be seen here:

            http://en.wikipedia.org/wiki/Vitruvian_Man