July 20, 2018 Friday
Bedtime Story
Understanding Geometrical Symmetry
We shall continue to study the mathematical
ideas of symmetry tonight.
Last night I had told you that the terms “operations”
or “transformations” concerning symmetry are somewhat similar and are best
described in terms of a function or mapping.
Similarly, the terms “symmetry” and
“invariance” are used interchangeably, though “invariance” is often encountered
in relativistic physics.
By now, you would not be surprised if I
tell you that a geometric object can be symmetric in ways more than one.
For instance, take a simple geometric
object such as circle.
The circle if you were to rotate it by any
radians, such as pi, pi/2, pi/4 would remain unchanged and hence is symmetric under
the operation or transformation of rotation.
There is yet another operation under which
the circle will remain unchanged and that is reflection.
If a circle is reflected on a mirror, it
will still remain unchanged and hence it is also symmetric under the operation
of reflection.
Also, the possibility of symmetry will also
depend on what kind of geometric transformations or operations are available to
that particular object.
With this idea came the concept of
mathematical groups in symmetries.
All the set of transformations under which
an object maintains its symmetry form a group that is called the symmetry group
of the object.
The most common group of symmetries in the
Euclidean group of symmetries is the distance-preserving transformations in
space occurring in two-dimensions (plane geometry) and three-dimensions
(solid-geometry).
Such distance-preserving transformation
between metric-spaces is known as isometry.
These isometries are translations,
reflections, rotations or the combination of three.
Any symmetric object maintains its symmetry
under a subgroup of all isometries.
Such type of geometrical symmetries in
two-dimensions and three-dimensions are fairly easy to imagine for a mind of an
average ape.
Now let us see how symmetries are
applicable to abstract algebra where an average mind would hardly think
symmetry is applicable.
A symmetrical polynomial is such a
polynomial with n variables where interchanging of any of the variables results
in the same polynomial.
This sentence can be written mathematically
as follows.
Initially it sounds scary, but as you get
to understand how mathematicians talk in their language, it begins to feel
strangely very sensible.
We shall study the mathematical translation
of the symmetrical polynomial 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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