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Fantastic Geometry

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Posted by Giota on November 6, 2000 23:40:03 UTC

Space is tightly connected with the total of its physical properties, that we can check up with experiments, so they (the physical properties) seem to us logical. Describing the space through its properties is a matter of the geometry. The well known to us geometry is the Euclid's geometry.
Euclid's geometry is built on the known Euclid's axioms, that are not mathematically proved, but we accept them like "axiomatic logical"!
Say! who can prove the fifth axiom that says "From one point out of a straight line we can draw only one straight line which is parallel at the first"?
None! All the 'solutions' are driven to a thought which we have to accept as an axiom (without be proven).
Before one and half century the russian mathematician Nikolai Ivanovich Lobatschewsky was one of the first that tried to prove the fifth axiom using the "reductio ad absurdum" method. He supposed that the fifth axiom is wrong and so there are at least two straight lines passing through the same point and are parallel to a straight line which is not passing through that point. But he concluded at no contradiction! What is that meaning???
Of course that we can start our thoughts based on an idea which is opposite to the fifth axiom and make a chain of theorems. In this way, the new theorems could 'built' a new non-Euclid's geometry, which Lobatschewsky on 1826 named "fantastic" as that period he couldn't find a field that this geometry was fitted and besides the results of this geometry was not according to the 'picture' of the space that is around us.
But as A. D. Aleksandrov says: "If we are interesting in geometry as a logical theory, we must look for the logical accuracy of the thoughts and not if the thoughts are fitted on the common and known shapes."
Now, after all the above are written, I think we can talk about Fantastic Geometry!

In the two dimensional space of the Euclid's geometry a theorem says that if a straight line is parallel to another, every point of the one forbears from the other constand distance.
This property isn't valid in the two dimensional space of Lobatshewsky's geometry. In such a space, two parallel straight lines don't forbear constand distance, but they converge without coinciding at one of their directions, while at the other direction their distance extend to infinity.

->So let's say that a pair of non parallel lines diverge from one side or both sides.

Another axiom of Lobatshewsky's geometry is: "Through a point which doesn't lie on a given straight line innumerable straight lines can pass, which lines don't intersect the given line."

Do you want some more? Or are you tired reading? ok...

->The sum of the angles of a triangle is always less than 180 degrees. If the triangle is 'growing' so its heights are getting bigger and bigger, then both three angles are tending to zero!

->The perimeter, l, of a circle isn't propotional to its radius r, but it is accordant to l=p.k.(e^(r/k)-e^(-r/k)), where p=3.14 and k is a constand which depends on the measuring units.
Let me note that r/k is much smaller than l=2.p.r. This means that when r is too small Lobatshewsky's perimeter is almost equal to Euclid's perimeter!

But I think I've tired you... If you think this subject is interesting (it is for me), I'd love to have your replies.
Hey! Don't forget the 'straight lines' of the universe!

see you around

P.S.: Sorry, in advance, for the syntax and orthographic mistakes. I did my best. It is a foreign language for me.

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