Subject: Re: [HM] When is Playfair's postulate important?
From: Daina Taimina (dt34@cornell.edu)
Date: Thu Jan 13 2000 - 17:42:49 EST
Dear list members,
I have received several comments on my question about uniqueness of
parallels, some not answering my question because I was not formulating it
well. The most helpful answer came from a high school teacher and his
geometry class. I will now try to clarify the SETTING, give the
REFORMULATED QUESTION, give the PARTIAL ANSWERS, and then state a FURTHER
QUESTION.
SETTING: {two lines (vectors) are parallel transports (PT) along a line
(geodesic) l, if the angles (in the same direction) they make with l are
congruent.}
Consider:
PPP: Playfair's about existence and uniqueness of parallel lines (the
"usual" PP in high school)
EFP: Euclides Fifth Postulate (a transversal with interior angles summing
to less than 180 implies intersection)
H=0: Holonomy of triangle = 0. [Holonomy is the angle between a vector
and its image after being parallel transported in turn along each side of
the triangle.]
<s=180: sum of the angles of triangle = 180
PT!: If two lines are parallel transports (PT) along one line then they
are PT along ALL transversals.
NI=CP: Non-intersecting lines have a common perpendicular.
In the context of absolute geometry (Euclidean w/o PP) all of these are
equivalent (in other words, all are true on the plane but false on a
hyperbolic plane). BUT they are not equivalent in broader geometric
contexts (for example, EFP is true on a sphere but PPP is not). They, in
general, have different meanings that have diverging sequences in different
geometric settings.
The standard uses of parallel lines in Euclidean geometry are properties of
parallelogram, angles of triangles, similar triangles. These applications
directly use <s=180, PT!, or H=0 and any of these can easily be proved by a
simple argument using any one of them .
REFORMULATED QUESTION: Are there applications that can be proved using
PPP, EFP, or NI=CP more directly than using <s=180, PT!, or H=0?
PARTIAL ANSWER:
(a) A high school teacher, Steve Weissburg, and his geometry class came up
with the Theorem: (3 non-collinear points on a plane determine a unique
circle). In the standard proof where we construct the perpendicular
bisectors of two cords, EFP is needed to show that the bisectors actually
intersect, there does not seem to be a direct way to use <s=180, PT!, H=0,
or PPP.
(b) Theorem: (the composition of two reflections in the plane is either a
rotation or a translation). The proof involves looking at the two lines of
reflection; and if they intersect then the composition is a rotation about
that intersection point; and if they do not intersect then NI=CP implies
that there is a common perpendicular and the composition is a translation
along this perpendicular. Again, there does not seem to be a direct way to
use <s=180, PT!, H=0, or PPP.
Note that EFP and NI=CP are essentially contrapositives of each other
since if there is PT along some line then there is always a common
perpendicular. Note also that these applications do not on their surface
seem to have anything to do with uniqueness of parallel lines.
FURTHER QUESTION: Is there an application that can be proved using PPP
more directly than by using <s=180, PT!, H=0, EFP, or NI=CP?
If there is no such application, then why should we use PPP except as part
of a historical discussion?
Thanks,
Daina Taimina and David Henderson
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