July 02, 2018 Monday
Bedtime Story
The Creator of the Mathematics of Manifold
It was Gauss who first studied abstract
spaces and came to the conclusion that they can be treated as mathematical
objects in their own right.
While Gauss’s work dealt more with the
curvature of surfaces, it was Bernhard Riemann who extended the idea of Gauss
to higher dimensions that allows both angles and distances to be measured that
was intrinsic to manifold.
That is why such higher dimensional
manifolds go by the name of Riemann manifolds.
Riemann took up the theory of complex
functions for his thesis or dissertation.
You now know what are mathematical
functions, but complex functions are even more impressive; these are functions
whose arguments are complex numbers, meaning numbers linked to
(solution to x2 = -1).
The study of functions of complex variables
is known as complex analysis and is the gateway to higher dimensions (without
really intending to be).
The study of complex analysis by Riemann
yielded to him surfaces which I can safely say is beyond the imagination of
most average apes, even accomplished artists.
These surfaces today, in his honor, are
called Riemann surfaces and it is here that the word manifold arose.
Every Riemann surface is a two-dimensional
real analytic manifold – even this sentence makes little sense verbally unless
it is backed by mathematics which only very few apes on this planet are
equipped with to understand.
Riemann was fortunate that he had Gauss to
present his dissertation thesis (1854 in Göttingen titled ‘On the hypotheses
which underlie geometry’) as it is unlikely that any other mathematician would
have been able to perceive the sharp and piercing intellect behind this work.
I myself have downloaded the paper that has
been translated into English and was quite surprised to see that the paper
essentially talks about space and what kind of geometry would best describe it.
The paper actually begins with extreme
criticism of the past great geometers, explicitly naming Euclid and Legendre (not
to be confused with Lagrange of Lagrangian mechanics who showed little interest
in gemoetry) and their failure in the description of the space.
Please allow me to quote some first few
lines of this paper:
“It is known that geometry assumes, as
things given, both the notion of space and the first principles of constructions
in space.
She gives definitions of them which are
merely nominal, while the true determinations appear in the form of axioms.
The relation of these assumptions remains
consequently in darkness; we neither perceive whether and how far their
connection is necessary, nor a priori, whether it is possible.
From Euclid to Legendre (to name the most
famous of modern reforming geometers) this darkness was cleared up neither by
mathematicians nor by such philosophers as concerned themselves with it.”
We shall continue with the 1854 paper of
Riemann ‘On the hypotheses which underlie geometry’ in the nights to come.
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.
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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:
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