Question Mr Vishwas K Hajirnis , we are a math teachers from Jordan ,want to ask you aquestions which say :
1- prove that the parallel lines meet at infinity .
2- which is the best geometries ( Rieman , Euclidean , nuetral ) that is most credible and logical .
3- prove that the summation of the circle - square angels equal 360 degree
and we would be thankful if you could help us by answering our questiones
group names :
1- Mohamad Fakouri .
2- Jamal Abu Khader
3- Ahmad Attiah .
Answer Dear friends from Jordan
Let me clarify one thing at the outset: I am not a professional mathematician/teacher/researcher and desipte
the name of this site, I can hardly call myself an expert of this vast subject.
(I have mentioned the level of questions that I try to
solve.)
The questions you are raising are definitely interesting
and of fundamental nature but I am not competent enough to
handle all of them (especially question no. 2, about which
my knowledge is the least)
Any way, let me try the questions
"1- prove that the parallel lines meet at infinity."
I found 2 interesting pages on the web related to this.
Thus you will see that the answer to what you want to prove
is in the negative for Euclidean geometry. (And for
projective geometry the statement is more like a definition, and hence no question of proof ? )
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"2- which is the best geometries ( Rieman , Euclidean , nuetral ) that is most credible and logical ."
At the following web page there is a reference to logical
consistenecy of Non Exclidean Geometry
( http://mathworld.wolfram.com/Non-EuclideanGeometry.html )
The following statement appears here
"It was not until 1868 that Beltrami proved that
non-Euclidean geometries were as logically consistent as
Euclidean geometry. "
which seems to suggest that there is no question of
one geometry being better than other.
Following page also has good survey material on history of
Non-eucledean geometries
(Please note the above is a single link although it appears
as broken across two lines )
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"3- prove that the summation of the circle - square angels
equal 360 degree"
The word "degree" has many meanings in mathematics.
The most common meaning is the unit of angle measure
defined such that an entire rotation is 360°.
This unit harks back to the Babylonians, who used
a base 60 number system. 360° likely arises from the
Babylonian year, which was composed of 360 days
Thus asking a proof here is like a asking a proof for a
definition and you will agree there are no proofs for
definitions, axioms/postulates.
We only demand proofs for propsitions/theorems, lemmas etc.
You have also mentioned the word "square" here. I did not
understand this part. ( Surely you may not be asking me to prove that the sum of the angles of a square is 360
degrees.)
But for the sake of completeness here is the outline of the
proof,( if this indeed is being sought).
Join any one of the pairs of diagonally opposite vertices. (i.e. draw one of the diagonals.) Now this splits
the square into 2 triangles. The sum of the angles of these
2 triangles is 360 ( = 180 + 180 ).
Finally, please accept my apology in case you feel you have
wasted your time without getting any useful information here.
Perhaps "geometry-college" discussion forum could be the
appropriate place for questions like these.
There is also a dicussion group "geometry-research"
BTW There is a dicussion group especially for maths teachers
called Teacher2Teacher (T2T) ( All these available thru http://www.mathforum.org )
Thanks for the questions, I had a great time searching and
thinking about the answers.
NB (About me - I am from a town in the western part of India.)
