Interesting Post of the Week - 3

Firstly, apologies!! Last week I was traveling and so couldnt plug-in to my blog to post the last week's interesting post.


This week's most interesting post comes from Food for Thought. A few days ago I mentioned about a good book by Dr. Douglas Hofstadter on Formal Systems and Strange Loops. The moment I saw this post on Food for Thought, I knew that this must be one of the things Dr. Douglas Hofstadter has written about in GEB. It is infact mentioned about in one of the chapters. Food for Thought is a quiz blog maintained by enthusiastic people. Food for thought gives a daily dose of one question and many questions lead me to hitherto unheard of stuff, which upon further digging lead to some new book or some article on internet enhancing the knowledge horizons into unknown knowledge co-ordinates.

They posted the answer in two days' time and the answer to the question is Klein's Bottle. Alas, if I had read GEB before, I would have answered this question. This post gave a good reason to go a little deeper into Klein's Bottle and then into Mobius Strip. Well you dont find this bottle intersting, do you? Well try this:

Klein's Bottle has basically got four dimensions. Yes, you cant see the fourth one. But you have to read this to understand more.

Klein's Bottle, when dissected gives Mobius Strip. So what is so special? - you ask. Let me tell you what is so special.

Assume a ribbon, yeah a cicular cloth ribbon i.e., a ribbon connected at the ends on the same side. If an ant starts walking at some random point along the ribbon, it would reach the point where it started after some time, but it has only walked one side of the ribbon. It has to flip over to walk the other side. If the same ant walks on the Mobius Strip (simply put, Mobius Strip is the same ribbon with both ends pasted to each other but with one side twisted by 180 degress and pasted to other side of the ribbon), it will reach the same point where it started only after walking both sides of the ribbon i.e., inside and outside of Mobius Strip are not disconnected. Mobius Strip might thus be looked at as a blurred demarcation of inside and outside. Check it out for yourself.

The same explanation holds for Klein's Bottle. There is no inside an outside of these objects. An ant which starts walking at a point inside the Klein's Bottle will have to eventually walk the outside as well as inside to reach the same point again.

Heres an animation describing how Klein's Bottle can be formed and from it, how to arrive at Mobius Strip. You can infact make a Mobius Strip yourself. Go ahead try it!

It is certainly a challeging task to describe such structures mathematically. The philosophical statament these algebraic topologies make is unmistakable. No wonder Douglas Hosftadter was amazed at strange loopiness many objects exhibit!!

PS: I am certainly going to write a lot about this Strange Loopines, especially from a Hindu Context!!