Straightening the Space

The algebraic function y = 1/x is unique and had fascinated scientists and mathematician of the 19th century. The graph of the function is called a hyperbola which is a perfect example of an asymptote. It is shown below:

For those who are not familiar with mathematical jargon it can be explained in terms of numbers. The values of y for various values of x are as in the table below:

It is clear that as the value of x decreases the value of y increases, so much so that when the value of x becomes zero, the value of y becomes infinity. However it is interesting that if you proceed from x=1 toward x=0 the value of y increases to plus infinity, while if you approach from x=-1 towards x=0, the value of y decreases (or increases in the negative direction) until it becomes minus infinity. Thus at x=0, y exists simultaneously at plus and minus infinity. From the common concept of infinity it is mind boggling. How can something exist at both extremes of the universe at the same time? Such a thing could only happen if the space was a curved continuum. To put it simply, if two persons started walking in opposite directions an d continued to do so without deviating one day they would meet face to face on the other side of the earth. Hence came the idea of the curved space and the donut shaped universe.

But the scientists who must have been euphoric at the brilliant idea which must have been dubbed a great discovery seem to have missed out on the relationship between algebra and geometry. y=1/x can be written as x*y=1 which is the equation of a rectangle of unit area. As x decreases, y increases and the shape changes from square to oblong and gradually a thinner and longer rectangle until x approaches zero and y approaches infinity. But when x becomes equal to zero, the rectangle ceases to exist. It becomes a straight line. There is what one might call a geometrical phase change. Just like gas laws can not be applied after the gas has condensed into a liquid, the equation of a rectangle can not be applied to a straight line. X=0 is a straight line which extends from minus to plus infinity along the y axis. Is anything wrong with that?

So it seems that the notions of space being a curved continuum and the universe being donut shaped are mere illusions and any mathematical deductions based on these notions would produce more profound illusions.

If I live long enough or get an opportunity to devote full time to the subject, I would certainly like to remove as many illusions as possible.