December 26, 2016 Monday

Bedtime Story 

Demystifying Banach Tarski Theorem

Based on all the above material, Banach and Tarski wrote that famous paper of theirs in 1924.

The paper itself does not label the work as paradox but merely as a theorem.

It is the writers of popular mathematics (is that an oxymoron, popular mathematics?) literature that love to baffle the gentry with the word paradox.

The Banach Tarski theorem or paradox can be stated in several ways and let me list a few of them out:

1^st statement

Given a solid ball in 3-D space, there exists a decomposition of the sphere into a finite number of disjoint subsets, which can then be reassembled back in a disparate way to furnish 2 identical copies of the authentic sphere.

I have to make a few points clear about the nature of the sphere, the act of decomposition and then the act of reassembly.

First of all, the sphere is more of a mathematical idea (rather than a physical atomic ball) which consist of arrangement of infinite points, the key word being infinite.

That means we are dealing with an infinite set which takes us back to the ideas of Cantor and Vitali set.

A sphere of radius 1 is mathematically defined as:

S = {(x, y, z)|x² + y² + z² <= 1}

The act of decomposition of the ball is yet again a mathematical one that was defined by Hausdorff which involves mathematical rotation and translation.

It is denoted by SO(3) which is a group of rotation about the origin of 3-D Euclidean space R3 under the operation of composition.

Hausdorff had proved that a surface of a unit sphere (radius 1) in space is a disjoint union of 3 sets B, C and D and a countable set E such that, on the one hand, B, C and D are pairwise congruent, and, on the other hand, B is congruent with the union of C and D.

This proof of Hausdorff is extremely fascinating and should itself be seen as a paradox by itself.

By the way, Hausdorff was the son of a Jewish merchant, and like so many others he was subjected to bitter humiliation after that infamous Kristallnacht of November 9, 1938.

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:

http://skmclasses.weebly.com/

All his books can be downloaded for free through this link:

https://zookeepersblog.wordpress.com/science-tuition-chemistry-physics-mathematics-for-iit-jee-aieee-std-11-12-pu-isc-cbse/

For edutainment and English education of your children, I recommend this large collection of Halloween Songs for Kids:

https://www.youtube.com/channel/UCd14DRdYKj454znayUIfcAg