August 17, 2018 Friday
Bedtime Story
Felix Klein Reconciles Euclidean with Non-Euclidean Geometry
Last night we had been introduced to the Prussian
mathematician Felix Klein in our bedtime story that is now trying to integrate
group theory to the mathematics of symmetry.
He is one of the few great mathematicians
who is also famous for being a mathematics educator and for giving Göttingen
that formidable reputation of mathematics research center.
Future formidable research centers of
mathematics in the United States such as IAS, Princeton would later go on to
copy its model with dedicated mathematics reading room and library.
It would be Klein who would select David
Hilbert in 1895 who would continue to enhance the reputation of Göttingen along
the lines of his predecessor.
Klein had this very unique view of
geometry; He saw geometry as the study of properties of space that is invariant
under a given group of transformations.
According to him, there need not be two or more
geometries (as was the great debate then between Euclidean and non-Euclidean)
but the essential properties of a given geometry could be represented by a
group of transformations that preserve those properties.
Transformation in mathematics is
distinctively a subject that belongs to group theory and relates to action on
its elements.
So now group theory was becoming
incorporated into geometry which initially arose in the problem of insolvability
of polynomials.
One can see Klein’s Erlangen program as an
attempt to reconcile the Euclidian with the non-Euclidean geometry using the
concept of group theory and transformations or invariance.
Let me explain to you how he proposed all
this which was not easily accepted those days when Euclid was still considered
to be flawless, impeccable and unshakable from his lofty pedestal.
At first it seemed almost impossible to
reconcile two geometries with one fundamentally different axiom – namely the
parallel postulate.
Whereas in the Euclidean geometry it was
considered ‘a self evident truth’ that parallel lines never meet, its opposite
counterpart could easily prove that parallel lines always meet (and in some
cases never meet).
How can then such fundamentally different
geometries ever come to peaceful reconciliation with each other.
The answer, Klein proposed, lay in
something which had already been discovered by mathematicians who applied
mathematics to the study of art centuries earlier.
It was the mathematics of perspective.
You might know vaguely what we all mean by
perspective, but if I were to ask you to define what it means, you might be at
a loss.
In the nights to come we will devote
ourselves to the story of perspective and how both mathematics and arts combined
together in perfect harmony to create a parallel development of stunning art
and architecture and whole new mathematics.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night Mon Ami and my fellow cousin ape.
Advertisements
Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
No comments:
Post a Comment