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 Subject: RE: What is a triangle? Author: Mathman Date: Jan 2 2005
On Jan  2 2005, rabeldin wrote:
> I have no quibble with the definition. My concern is that the
> unconscious substitution of a definition in terms of sides for one
> in terms of angles may lead to errors when we generalize to non-
> Euclidean spaces.

We say "triangle" and think "trigon". In
> Euclidean space they mean the same thing but in all non-Euclidean
> spaces?

In any case, I am more concerned with the general class
> of error involved in using informal definitions. This is but a
> single example of such an error.

You say "triangle" and I can think "trigon" as do you, but I can as often and as
readily think "three points in space".  I'm not sure why the fuss.  If teaching,
then a formal explicit definition is required.    Defining one way under one
circumstane does not invalidate another under other circumstances.  The rombus
can be defined as a parallelogram whose diagonals bisect each other.  That is
sufficient.  Or, it can be defined as a quadrilateral with all four sides qual.
That is sufficient.  Or it can be defined as a parallelogram with any two
adjacent sides equal and that is sufficient.  So long as the definition contains
sufficient information I see no problem.  Defining the triangle as three
distinct points in space will be suitable in Euclidean or non-Euclidean space;
the only difference being that "linear' in Euclid would be an arc on, say, the
sphere.  the "trigon" in Euclid" is bounded by three lines joining the points,
but by arcs on the sphere.  It's symmetrical definition, not entirely different
in significance or meaning.  If "most people" wish to misinterpret, that's their
problem, but that has not been my experience in God knows how many years.  I've
just once seen a student get entirely mixed up and draw a "triangle" which had
the U-shape you describe, but she was a "special" student and had enormous
recognised learning problems.

David.