Back

...........................................................
Come on, my friend, give me your hand. I want to take you on a fantastic trip, a trip that would amaze Dante and Alice in Wonderland. Dante, because you will know where the dwelling places are, and Alice in Wonderland because it´s so fantastic. I want to take you to Cosmos. ......................................................................................... Mathematical models are like words. Never are they absolutely appropriate, but we cannot do without them when communicating. Now, pedants are not able to make themselves understood with words because understanding is the intimate cognition that takes place deep inside of us. Complicated formulas, Greek and Latin words are currently useful to those who have commensurate knowledge, not to beginners. ......................................................................................... Cosmos encompasses the entire Universe and much more. Cosmos is what Ancient Greeks used to refer to when they meant the entirety of everything that exists or may exist. To me, the term is quite correct, and perhaps you agree. Relax, raise you head, don´t be afraid, come on, and we shall try to find out what it is all about. I shall show you something that no one has ever seen. ......................................................................................... So, what is the Point? ......................................................................................... A point is space whose dimension is zero. A point is a place, the dimension of which is zero. Some say that dimensions are related to the degree of freedom. Consequently, Point is a place with zero freedom, that is with no freedom at all. Point is a cell. Point is cloister. In the real world there is nothing that exactly equals mathematical beings. And, in our world, mathematical beings are contaminated with the realities of the world we live in. Some may argue that a cell is a three-dimensional space. Yes, strictly speaking, it is. Now get out of the cell and walk freely, and you will find out that cloister is a Point. ......................................................................................... Let us go for a trip now, that is a lot more interesting. We are in Sangatte, near Calais, France, and we go to Folkstone, England. Take your seat and let us have a chat during this short, but pleasant journey. ......................................................................................... The train enters the Eurotunnel. And I am going to show you what one-dimensional spaces are. These are so important to the comprehension of all the other ones that we shall not loose any time on this trip. In one-dimensional spaces we have one single degree of freedom: forward, or backward. All straights, all curves, in their mathematical significance, are one-dimensional spaces. All lines, wires, pipes and the Eurotunnel are one-dimensional spaces, too. ......................................................................................... To locate any point on a line, we need no more than a point of reference, the origin, and say at which distance we are. We are at three kilometers from Sangette. There is no doubt about this. ......................................................................................... Now we have learned that one single number (the distance to the origin) defines any position on a line, straight etc. When the line is a straight, we use to call it X,
and define any point on the straight by one single
real number, negative before the origin, and positive after
the origin.
.........................................................................................
First conclusion: one single number defines any point in one-dimensional space. ......................................................................................... Now, let us imagine an other train, inadvertently entering our line in Folkstone. And even worse, a third train leaving Sangette, heading to Folkstone. ......................................................................................... What is going to happen? ......................................................................................... We are stuck. ......................................................................................... It should be noted that, in a one-dimensional space, the border, anything that closes it off, that limits it, is a space with one dimension less, in this case, a point. Two points close off an inter-space of a straight. ......................................................................................... This is an absolutely general principle, but we should go slowly. Let us go deeper into our situation. We are stuck, unless one of the trains backs off so we can get out of the Eurotunnel. ......................................................................................... Second conclusion: two points limit, close off, a section of a straight. ......................................................................................... If the train leaving Folkstone notices its error and backs off, we can continue. Now look what a highly interesting situation: the train that has left Segatte after us will not ever reach the end of the Eurotunnel before we do. There is no way to overtake in a one-dimensional space. Whatever may happen in the tunnel, what got in first, shall get out first. I would suggest you to put small colored balls into a transparent plastic hose. Balls should be allowed a certain clearance. If a red ball is put in first and then a blue ball, there is no way to make the blue one come out at the other end, unless the red ball gets out first. ......................................................................................... Third conclusion: in a one-dimensional space, there is order. ......................................................................................... Because these very severe limitations exist in one-dimensional spaces, the Eurotunnel has three tracks, one hither, one thither, and one for maintenance. Yet, one-dimensional limitations exist in each of these three lines. ......................................................................................... We are arriving in Folkstone. When we are in a space of n dimensions and want an additional dimension, we need no more than to get a single point that does not belong to our Space. To exemplify: we are in a straight; in a one-dimensional space. We want to go to a space with two dimensions, a two-dimensional space. Nothing more is needed than to find one single point that does not belong to the straight under consideration. As a matter of fact, geometry teaches us that a straight and a point that does not belong to the straight, define a plane. A plane is a two-dimensional surface. ......................................................................................... In a two-dimensional Space, we get another degree of freedom. We can go forward or backwards, but not to the right or to the left. Cars and boats are vehicles that move on surfaces, that is in two-dimensional spaces. We, humans, normally move the two-dimensional way: we walk on surfaces. Monkeys, pulling from one treetop to another, move in a three-dimensional space. But let us proceed slowly. ......................................................................................... Now we have to decide whether we continue by boat, or by car. In spite of those inconvenient British traffic habits we shall go by car. At first sight, a car on a road may look like a vehicle moving on in a one-dimensional way. We should not forget that cars can also be driven in deserts, which normally are near to plane. Even on roads, we can choose a lane. We are free to slow down so someone in a hurry can overtake us. As we have seen, overtaking is not possible in tubes and railroad tunnels. Viaducts and roundabouts (traffic circles in Britain) are means to overcome two-dimensional limitations. ......................................................................................... ‘Geometry’ means measurement of Earth. (Ge = earth; metron = measure). Plane geometry according to Euclides has originated on the ground of our yards. For many centuries, it was the only geometry we had. Any point of a plane can be identified through two indexes. Look at a town map. To find a street, we have to go to square F3, for example, that is column F, third line. In mathematics, we use orthogonal axis X and Y,
and the position of any point is given by way of two real
numbers. Axis X is not new to us, and we may take
coordinate
y as the distance between the point under consideration and
axis X. In other words, and keeping in mind how we
define two-dimensional space, we have coordinate x, and new
coordinate y represents to which extent the Point keeps us
off our one-dimensional system X.
.........................................................................................
Now we shall go to three dimensions. Let us go to the sea. The surface of the sea is obviously a surface, and like all other surfaces, it is two-dimensional. To define a three-dimensional space, the one and only thing we need is a single point that is not part of the surface. I shall now choose the hook the seaman is using to fish. As the seaman´s hook is below the surface of the ocean, I can define any point in the ocean. ......................................................................................... This also applies to aircraft. When the captain of an airliner informs his position to the control tower, he gives his geographical coordinates (two numbers) and his height (the third number). If he does not say how high he is flying, it should be clear to anybody that the information is dangerously incomplete. ......................................................................................... The captain of a submarine always informs his steersman in which direction and at what depth he must go. ......................................................................................... Carefully watch this third value, depth or altitude. It indicated how much the vehicle is away from the two-dimensional surface. ......................................................................................... Let us now make a huge leap forward. We are in our three-dimensional space, and are going to search for one single point that does not belong to it. The study of particle physics suggests insistently on at least a fourth dimension. Brian Greene´s super-strings theory suggests many dimensions. In the book ‘Hyperphysics’ I am currently working on, I demonstrate mathematically the existence of at least one more dimension. ......................................................................................... This is a fact; it is not illation. ......................................................................................... With one single point in four-dimensional space, we can define the entire four-dimensional space. Keep in mind that this new value, like always, gives us the distance between the point and our dearly loved three-dimensional space. ......................................................................................... When reaching this point of my story, I like to tell the tale of the fly and the ant, just to relax a bit. ......................................................................................... We just finished dinner, and I would like to do and experiment. Put a piece of candy on the tablecloth. Draw a circle around the piece of candy, using a piece of poisoned chalk. ......................................................................................... This is merely an exercise. The circle is a two-dimensional surface. Its limits are space less one dimension. Two minus one is one. The poisoned line is one-dimensional, the limit of the circle. ......................................................................................... This is merely an exercise. The circle is a two-dimensional surface. Its limits are space less one dimension. Two minus one is one. The poisoned line is one-dimensional, the limit of the circle. In no time an ant, an essentially and thus by definition two-dimensional animal, smells the savor of the candy and comes. If it steps onto the poisoned chalk mark it shall die; the ant is quite aware of this. It scouts the neighborhood, but finds no way to get through the poison barrier. A fly, an animal capable of three-dimensional motion, is also attracted by the smell of the candy. The fly flies, lands on the candy, feeds up and takes off. Poor little ant is watching. Not quite stupid, poor little ant sadly understands the fly´s superior ability. ......................................................................................... We are exactly the same situation as the ant is. We are completely confined in a three-dimensional space. However, we can, and shall, study space with a greater number of dimensions. ......................................................................................... Fish is a further example of an animal living a three-dimensional existence. Poor little fish is subject to an additional restriction: it shall not get out of its volume of water. It is not able to. We can make a very interesting analogy with fish. The fourth number indicates the size of the distance that separates us from three-dimensional space. Let us imagine ourselves four-dimensionally, whereas the fish is three-dimensional. This may help us get an idea of how a being feels that is living beyond our dimensions, of how it looks upon us. We may put a hand in the water, so we become reality to the fish. We may enter the fish´s world at any time we want. Our reality, however, is everlasting and the fish cannot, normally, see us. ......................................................................................... Analogy is not very good, as the fourth dimension is as close to us as we want, while the fish, in the aquarium, may remain at distance. ......................................................................................... We are diving into the fourth dimension. Our universe is just a tiny part of the four-dimensional space that harbors unlimited three-dimensional spaces like ours. As the plane contains unlimited straights, three-dimensional space contains unlimited planes. ......................................................................................... Imagine a being able to move four-dimensionally. To us, it will be what our fly is to the ant. ......................................................................................... Possibilities of new dimensions do not end here. As a matter of fact, possibilities have no limit. Uncountable dimensions really exist, but this is not cosmos yet. These spaces which, in one way or another, allow to be measured with a ruler, metron, we use to call metric spaces. ......................................................................................... Sharpen your attention. Watch. We shall now measure time, observing the time a piece of bread takes to get moldy. We put a piece of bread in the refrigerator. And now we begin to count time. A few days later, we take another piece of bread and put it in the cupboard. And we start another time count. After a few days, the second piece of bread will be moldy. The second bread sample overtook the first one. This was the last conversation about physics I had with my father in February, 1985, one month before he died. ......................................................................................... You did not like this – too simple – experiment. That´s okay. ......................................................................................... Let us talk about the paradox of the twins resulting from Einstein´s theory of relativity. Marc and Mary are twins. Adventurer Marc leaves on an inter-galaxy journey, at a maximum speed equal to the speed of light. Mary stays on Earth, gets married and gives birth to a son, John. John grows up. He hears about his uncle Marc´s adventure. He dies before uncle Marc returns. One day, Marc will come back; still young. He will not find his nephew John, because he will be dead already. His sister will have lived many years, but will have died as well. ......................................................................................... This story is absolutely possible, although quite difficult to take place. The period of John´s lifetime will have begun to count when he was born, a long time after his uncle Marc was born, but will have ended before his uncle´s lifetime. ......................................................................................... We have seen earlier that overtaking does not exist in one-dimensional space. So time must necessarily be, at least, a two-dimensional space. ......................................................................................... This idea is absolutely new. It is shown here for the first time. ......................................................................................... These spaces, measurable by means of a clock, are called chronic spaces. ......................................................................................... We live in, or better say, we are absolutely confined in, a tree-dimensional metric space and a one-dimensional chronic s pace in mandatory progress. ......................................................................................... Why does our time go on in a mandatory manner? ......................................................................................... The main responsible for all this is the second law of thermodynamics that defines the direction of processes, most of which are irreversible. Let me explain. What happens when you put two bottles in contact, one containing hot water, the other one cold water? The hot one cools and the cold one warms. Nothing impedes that the cold bottle becomes colder and the hot bottle becomes warmer. What makes this happen is the Second Law. As the process h as a direction, we may say that time is passing. Time is passing because we are heading toward equilibrium. ......................................................................................... Drop a ball halfway to the top of a slope. The ball goes down the slope, and its energy turns into heat. Nothing impedes the ball to cool and go uphill. Once again, the Second Law compels what must happen. An iron bar exposed to atmosphere rusts. Nothing but the Second Law impedes the bar to cool, release oxygen and become new again. Time passes as rust progresses. ......................................................................................... In our bodies and environment, physical and chemical processes following the Second Law of Thermodynamics cause us to feel that time passes in a given direction, in a mandatory way. And now a question comes up: is there any phenomenon that is not submitted to the Second Law, and that can make time flow backwards? Possibly, the answer is yes. And this question is of extreme importance to particle physics. ......................................................................................... Time, as we perceive it, does simply not exist. Time, as we know it, is simply a limitation of our matter. Confined in three dimensions. ......................................................................................... Cosmos is the conjunction of all infinite chronic spaces and of all infinite metric spaces. ......................................................................................... In cosmos there is no beginning and there is no end. There is no past, and there is no future. Looked upon from cosmos, Marc´s life can be seen in any sense. The beings who, possibly, live in spaces superior to ours, may canvass our lives in any sense. The feeling of proceeding toward our death is absolutely illusive; a fruit of our limitations. ......................................................................................... This is strictly mathematics-based, physics-based and science-based. ......................................................................................... The Supreme Being, who may live , or who may be Cosmos proper, can, possibly, be eternally present in all points of our humble Universe. For the first time in human history has Science opened a door to the Supreme Possibility of showing the infinite, and unimaginable, greatness of Cosmos. ......................................................................................... The present article was completed on April 27, 2003, and registered at the Cartório de Registro de Títulos e Documentos, in São José do Rio Preto, Brazil. The article is available for publication. ......................................................................................... |