Monday, 24 October 2011

The long shadow of Euclid's ghost

In my earlier blog entry I spoke a bit about non-Euclidean geometries, specifically the spheroid geometry of Riemman's and Lobachevsky's hyperbolic geometry. I've been thinking about this, and I've come to the conclusion that Euclid's fifth postulate on parallel lines is inescapable even in the spheroid and hyperbolic geometries or the elliptical for that matter: in the spheroid, the center point creates the first chord and the tangent creates the second chord 90 degrees to the center and the lines never meet though the surface is finite but unbound; in the hyperbolic, the center point is outside the curve but the tangent line is also inescapable there.

It is when one considers only the surface geodesics thereof the lines begin to converge or diverge depending and screw up the Pythagorean theorem on the triangle summing to 180 degrees. But the center point in three dimensional spheres and hyperbolic geometry can still create the right conditions for Euclid's fifth to apply. In fact, any curve in any n dimensions would still obey it because the fifth is inescapable as an iron-clad geometric fact.

my apologies to my readers for this little episode... but I've been kind of under stress and such "epiphanies" tend to ooze out when messy reality bears down on me. But of course I'm not a mathematician and my above "proof" makes no pretensions of rigour: I just love contemplating something so beautiful as Euclid's axioms. Silly boy.

How's that for talking eskimo, Mr Anderson? Can we do such things in Nunavut, if not Saskatchewan?

Jay

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