Subject: [HM] When is Playfair's postulate important?
From: Daina Taimina (dt34@cornell.edu)
Date: Mon Jan 10 2000 - 10:28:29 EST
Dear list memebers,
I am asking you the question which comes from Prof. David Henderson
(Cornell University):
"I am rearranging the order of the material in the new edition of
"Experiencing Geometry", and struggling with how to treat parallel lines
on the plane. I came up with the following questions: Where do we ever
care that parallel (non-intersecting) lines are unique? Is there any
constuction that needs this result?
In all applications I have thought of what is actually used is uniqueness
of parallel transport (that is, on the plane, parallel transport is
independent of the path along which one transports) -- this is an immediate
consequence of "holonomy = 0 on the plane" or "angles of a triangle sum to
180deg". To go further and prove that that parallel (non-intersecting)
lines are unique is more complicated and involves (as far as I know)
non-contructive and/or infinite arguments.
The key question is "If you have a pair of non-intersecting lines, are they
parallel transports along some transversal?" Why do we care? Note that on
a hyperbolic plane there ARE pairs of non-intersecting lines which are not
parallel transports.
Is uniqueness of non-intersecting lines (by itself) useful except as an
historically important distinction between the plane and the non-Euclidean
geometries?"
We would really appreciate any comments.
Daina Taimina
This archive was generated by hypermail 2b28 : Mon Jan 10 2000 - 11:41:13 EST