July 01, 2018 Sunday
Bedtime Story
Understanding Mathematical Manifold
Today I shall continue to grapple with the
concept of mathematical manifold and try to give you a visual intuitive grasp
using the metaphor of atlas charts and a general globe depicting Earth.
Now you would recall if you have flipped
through the colorful pages of Bartholomew atlas or any other atlas of similar
sort that very often several pages of atlas depict the exact same thing in a
two dimensional plane what a three-dimensional globe represents.
So the surface of the globe is being
described by several pages of maps or charts.
You can very well imagine that no single
chart of the atlas book can alone represent the entire surface of the globe,
yet any point or place on the globe will have at least one two-dimensional
chart that would accurately represent it.
In a way one can say that the surface of
the earth as represented by the globe is decomposed into several pages of
charts of the atlas.
This makes the surface of the 3-dimensional
globe a two-dimensional manifold.
Furthermore, any single point or location
on the surface of the three-dimensional globe can be perhaps represented with
not just one but several pages of the atlas where that spot find its place.
So how do you patch several of such flat
two-dimensional pages to give rise to a three-dimensional globe surface?
This exercise entails the use of
mathematical patching function which will map every point in the
two-dimensional flat charts to the surface of the globe.
For this explicit knowledge of functions of
two variables is necessary.
The atlas-globe metaphor given above is an
example of two-dimensional surface manifold.
Even simpler than this the surface manifold
is a one-dimensional manifold.
A circle forms another good demonstration
of this weird idea of manifold and topology.
Since a circle just a curved straight line,
it makes circle a one-dimensional manifold of a straight line.
Topology completely ignores bending and
hence a small piece of a circle is given the same treatment as a small piece of
line.
Both these examples of manifold were simple
and intuitively easy to grasp but as the dimensions increase, it becomes
impossible for our middle-world, three-dimensional minds to imagine the results
of topological manifestations.
Then the only thing that can help us lead
the way is mathematical charts (a collection of charts that represents a
manifold of some object is labeled as atlas very much analogous to the
geographical representations).
There are examples of manifolds in
topological space that has the capacity to blow away your mind or worse, our
minds will find it impossible to grapple with the ideas it presents except
proven geometrically on paper or blackboard.
We are nearly done with the manifolds and
whatever is left will be told 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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