
some huge changes to geometry of True Geometry #35.7 Unitext 8th ed.: TRUE CALCULUS
Posted:
Nov 2, 2013 6:59 PM


Alright, in my prior posts, I mentioned how parallelism in True Geometry was no longer valid, since a line can be parallel to a given line and never meet at a intersection even if the lines go to infinity. We can imagine this in the 10 or 100 or 1000 Grid that a line of the xaxis can never intersect with a line that has a gentle slope of x= 0, y = 10 and x=10, y = 9.5 in 10 Grid.
Now we lose parallelism in True Geometry, unless, however, you want to include the idea that at some infinity point that the line y = 0 and the line of (0,10),(10,9.5) do meet and intersect at some distant infinity point in 10 Grid.
So, at this writing, I am not sure if parallelism falls to the wayside in True Geometry or whether we best preserve it as a far distant intersection at some infinity point.
So Parallelism is a major issue to contend with in True Geometry.
Another major issue is that in True Geometry there are no curved lines but that curved lines are all compilations of tiny straight line segments. Now that is true for Euclidean Geometry because we cannot have a Calculus with curved lines.
However, are "lines in elliptic and hyperbolic geometry" also compilations of straight line segments? Here I am not so sure. My first guess is that there are no curves in either Euclidean, Elliptic or Hyperbolic. I say that because I am going with the famous relationship that Euclidean geometry is the union of Elliptic with Hyperbolic. That relationship is a smooth one if no curves exist at all. But this needs further study. Perhaps we need curved lines in Elliptic and Hyperbolic by using the infinity numbers of empty space, just as I am not sure if Parallelism is abandoned in Euclidean Geometry.
So the above is a short summary of major changes for True Geometry in that we have points then empty space and then successor point, and we have no curves, and we lose the principle of parallelism (unless needed by something unforseen).
AP

