December 24, 2016 Saturday
Bedtime Story
Giuseppe Vitali, Felix Hausdorff and Henri Lebesgue had Laid the Groundwork
This paper “On the Decomposition of Point Sets into Respectively
Congruent Parts” in turn was based on two papers published earlier, one by
Giuseppe Vitali and the other by Felix Hausdorff.
Giuseppe Vitali in 1905 had described a set of real numbers that
is not Lebesgue measurable, meaning even a unit set of real numbers, that is
[0, 1] on a number line which is a subset of real numbers R
has the same cardinality as
that of the continuum R.
This Vitali Set is based on the Axiom of Choice which allows
selection of non-measurable infinite sets.
Felix Hausdorff went a step further and wrote a paper on the
paradoxical decomposition of a sphere S2 (a 2-dimensional sphere in
R3) somewhere around that time.
It states that if a certain countable subset is removed from S2,
then the remainder can be divided into 3 disjoint subsets A, B and C such that
each is congruent to the others and all are congruent to the set B
C.
Essentially what Hausdorff proved was that for a sphere S2
there is no finitely additive measure that is defined an all subsets.
This implies that the measure of subset A is both 1/3 and ½ of the
non-zero measure of the whole sphere S2.
This concept, as we will see, becomes central to the paper
published by Banach and Tarski.
Here, the mathematical idea of 3D rotation group plays an
important role.
Mathematically, a rotation about an origin is defined a
transformation that preserves the origin, Euclidean distance and orientation
(handedness of space).
A rotation group (group of all rotations about the origin of 3-D
Euclidean space R3) is designated by SO(3).
French mathematician Henri Lebesgue made an important contribution
to the set theory by introducing a novel concept to measure subsets of a set.
It goes by the name of Lebesgue measure and it assigns a measure
to the subsets of n-dimensional Euclidean space.
For n = 1, 2 and 3 the Lebesgue measure happens to correspond with
the standard measure of length, area and volume.
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.
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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