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I am going to refer to one of those funny stories we all remember since the time we were at high-school. Remember the hen-house fence? Someone bought 8 meters of chicken wire to build a hen-house. He racked his brains to find out that, depending on the length of the sides of a rectangle, the floor surface of the hen-house could vary from zero to a maximum. After various projects he came to the conclusion that the maximum floor space his hen-house could have, is four square meters if he made it square. ......................................................................................... If you specify any area in the interval [0,4], it is possible to find which should be the dimensions of the hen-house. The solution requires the setting of a second-degree equation . When we learned how to tackle this problem we also were informed that 8 m of chicken wire would not fence any area. The maximum is 4 square meters. The man, a simple fellow, asks: Why not 5 square meters? ......................................................................................... Writing the second-degree equation, we have: ......................................................................................... he solution of this equation gives us a negative delta. Delta is -4. Roots are x = 2 + i and y = 2 – i. Naming the roots x and y, was done on purpose. You have probably been told that this problem has no solution! If you draw orthogonal axes X and
Y on the ground of your backyard, you will
have all you need to peg out your hen-house.Vertically, we shall put imaginary axis I.
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Ahaaa ! ......................................................................................... We can at least set the marks indicated by the algebraic solution of our problem. First mark: Origin (0,0,0). Second mark on axis X (2, 0, + i), that is two meters from
O and 1 meter above ground level on imaginary
axis I. It is easily seen that the other marks will
be (0, 2, - i), which is below ground level, and finally
(2, 2, 0), which is on ground level again.
We can draw a figure because we are not using our third
dimension Z; we put axis I in its place.
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Let us try to understand what is happening! ......................................................................................... As a matter of fact, our problem was transferred to the fourth dimension! There are no four dimensions in our three-dimensional space. And there is no solution to the problem! Simple, high-school level algebra took us to four dimensions! Let us imagine a space of four dimensions. The sum of the sides will be: 2 + 2 + I + 2 – I + 2 = 8 meters. The product of P = (2 + i) . (2 – i) = 5 square meters! ......................................................................................... Here, we are having a little difficulty because the floor of the hen-house tilts. Strictly speaking, the area can no more be calculated by the product of its sides; it will be slightly smaller, it is no longer a square. I am inviting you to calculate the exact area. Make the drawing and you will see that it is not difficult. But it was not this we had suggested to algebra ; we had come to the conclusion that the product of the sides ought to be 5, and that is what really happens. ......................................................................................... Let us use our brains; one should really bethink what is occurring, for it is too easy to say: there is no solution! ......................................................................................... All of us have been facing four-dimensional requirements since we were in high-school. We only did not understand the algebraic solution. But it is not too late to learn: when a problem cannot be solved in three dimensions, algebra provides us a complex solution requiring more dimensions. ......................................................................................... There is no way to exaggerate the importance of this fact! ......................................................................................... In the papers that follow, we shall see that the fourth dimension comes on in an absolutely natural way. As I said before, during our search for knowledge we have been facing the fourth dimension quite a few times, but, even so, we not quite accept the idea of Hyperspace. Man, after all, is the center of Universe! How could there be a place that is prohibited to Man? Universe was created for us! Nothing lies beyond the limits of our pride, nothing lies beyond our glory! ......................................................................................... |