December 28, 2016 Wednesday

Bedtime Story 

Banach Tarski Theorem in Other Forms

Reassembly of all the infinite points of the sphere after its decomposition is again mathematical that involves rotation.

It is at this crucial point of reassembly that the axiom of choice is invoked to allow reconstruction of the sphere from an infinite and uncountable number of choices.

Let me list out all the other forms in which Banach Tarski Theorem can be stated:

2^nd Statement

A sphere in a Euclidean space can be doubled using only the operations of partitioning into subsets, replacing a set with a congruent one, and reassembly.

3^rd Statement

A 3-D Euclidean sphere is equidecomposable with 2 copies of itself.

4^th Statement

Any two bounded subsets of a 3-D Euclidean space with non-empty interiors are equidecomposable.

5^th Statement

S² is SO(3)-paradoxical as is any sphere centered at the origin.

Moreover, any solid ball in R³ is G₃-paradoxical and R³ itself is paradoxical.

G₃ is the group of isometries in R³.

The group of isometries is the group of bijections from R³ to R³ that preserves distance.

The word paradox is also mathematical and needs to be precisely defined.

Consider a group G acting on a set X. 

So suppose there is a subset of set X that can be broken up into 2 disjoint sets each of which can be split up into finitely into many disjoint joints and then reassembled into the copy of the original.

This is paradoxical.

Enough of this mathematical jargon.

Let us now try to understand the chain of reasoning of the Banach Tarski Paradox in unpretentious simple English.

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