Thursday, December 05, 2002

Experience Mathematics # 21 -Euclid's fifth axiom


Euclid’s fifth axiom says that given a line $l$ and a point $P$ not on the line, there is exactly one line parallel to $l$ passing through the point $P$. For centuries people thought that Euclid’s fifth axiom was “obvious”. But some mathematicians did not find it obvious. 

Finally, Reimann and Lobachevsky, both modified the axiom and tried to derive a new geometry. 

Reimann began with the axiom: Given a line $l$ and a point $P$ not on the line, there is no line parallel to $l$ passing through the point $P$. Reimann derived many geometrical theorems that are applicable on the surface of a sphere. For example, he showed that the sum of angles of a triangle is always greater than 180 degrees. Try drawing a triangle on a sphere and see why this has to be true.

Similarly, if we take a hyperbola ($y=1/x$) and rotate it around the y-axis, then we obtain a surface where Lobachevsky’s geometry holds. Lobachevsky’s geometry contains the axiom: Given a line $l$ and a point $P$ not on the line, there is more than one line parallel to $l$ passing through the point $P$. In this geometry, the sum of angles of a triangle is always less than $180$ degrees.

There is a property of the surface (known as curvature) that determines the geometry. Only surfaces with curvature zero follow the Euclidean geometry. Another example of a surface is that of a saddle (of a horse). Can you tell which geometry is applicable on this surface?

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