April 04, 2017 Tuesday
Bedtime Story
Constructible Number
Trisecting an angle means constructing an angle that is one-third
of a given angle.
Bisecting an angle had been solved and I am sure you all must have
done it in your classrooms or at home at a very young age.
I remember my father teaching me how to use the geometric set box
that he had got one for me when I was in my junior school.
It was fun then.
Yet it was only in 1837 that the French mathematician Pierre
Wantzel proved the impossibility of this feat.
Similarly, the construction of a second cube whose volume would be
double that of the first cube using only unmarked straightedge and compass was
proved an impossibility by Pierre Wantzel.
By the way, if the side of the first unit cube is 1 and volume 1
(13 = 1), then the side of the cube doubled its volume would be x3
= 2, so x =
.
This is exactly where the problem lies.
What on earth is a constructible number?
Is the root of 2 a constructible number?
What about root of 3 or root of any prime number?
Well, you can approach this question geometrically instead of
algebraically and ask the question:
What is a constructible point?
Well…a constructible point in an Euclidean plane is constructible
if given a fixed line segment of unit length, the point can be constructed with
unruled straightedge and compass.
With an unruled straightedge and a compass in your hand, you can
do the following things:
1. Draw a line between two points
2. Draw a circle through a point with center being at another
point
3. Create a point which will lie at the intersection of two
non-parallel lines
4. Create one or two points by intersecting a line and a circle
5. Create one or two points by intersecting two circles
Did you notice something about this geometry?
It has a faint resemblance to logic.
The points and lines are equivalent to initial set of symbols.
What you are allowed to do with them is equivalent to algorithm or
rules of the game.
Geometry seen this way closely resembles axiomatic algebra and
Gauss was perhaps the first to realize this.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
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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