Infinity

"Oh, what does it matter? It is like infinity + 1" , I said involuntarily.
"What do you mean- infinity + 1?" 
"Well, infinity +1 would still be infinity. It's an incremental change. Doesn't change a thing in the world", I went on.
And then my friend stumped me by quoting one of my own favorite lines- "But it made a difference to that one!"
"Inspiration story wise, I love that line. But mathematically speaking infinity+1 cannot be bigger than infinity. It just doesn't work that way."

Sparing you the rest of the conversation, let me get to my point. Infinity is really a fascinating idea. When somebody tells you- we are in an infinite cycle of birth and death (theologically speaking, or speaking of physics), I immediately think "But when was the first time you were born?" It's a chicken and egg problem- if I want to tell myself the universe is infinite I wonder how it all started. But if I try to reconcile that it is in fact, finite- then my mind goes "But what was there before it all started?" Well, it looks like I'd never get it. Maybe it is not just unknown but unknowable (ya, now I am an infinity agnostic :P)

Anyway, when I spoke of infinity+1, I spoke like it's any other number. Which, of course it isn't. When someone says God has infinite powers, they don't mean he has 9^(9^9) powers or something. They usually mean endless. There is just one infinity. And it's endless. All-encompassing. At least that's what I thought of infinity until yesterday... Yesterday, I happened to read a little about Modern Set Theory and cardinal numbers, and then I realized, infinity is not just a concept of endlessness- there are infinite infinities in this world. 

Wait, whaaa? 
(Click on Read more for more)

ECE

I was reading this math post. It is about puzzles and one simple method of solving most math puzzles- thinking of an extreme example. What he means is that you can assume some extreme values for the variables involved, and deduce the answer by logic. (I like the puzzles listed as well!)

Admittedly, this is a very useful method. Especially in your competitive objective type exam scenario (Though for this scenario, I personally think "Inky Pinky Ponky" is undoubtedly the method of preference!)

Few things I want to highlight though. One- You should think of the 'right' extreme example. Some extreme examples might not lead you to any conclusion at all! I remember how, in schools, teachers used to start explaining proofs by saying "draw perpendicular AD to BC" or "AM to NP" or some such thing. And immediately I would think how was I supposed to  know which perpendicular to draw? Later on I realized that if you understand geometry well enough, actually rephrase- if you know well enough to think from a geometrical angle (Pun unintended!) you will know exactly which perpendicular to draw. But point remains that you need to choose the "right" perpendicular (Oh! Again, pun unintended!) In the same way, one does not always know which extreme example to choose. And if you don't choose the right one, you might end up a) getting a long and roundabout proof or b) not proving anything at all!

Anyway, why I brought this up here is that this method is not only useful for mathematical problems. I find it a good tool to understand human behavior. For eg: When someone says I value money but I value family more, one easy way to verify it is to ask the question- "if you had to give up all your money so you could retain your family would you do that?" or something of that sort. Basically pick an extreme scenario and check your statement. In fact, this method helps even if you are simply trying to understand your own selves.

Anyway, I have used this method before, and though not mine, I thought someone should give a name for it. "Extreme Case Evaluation" sounds OK by me. Maybe I should publish a paper on it someday...
 

P.S: I hope people found out why the title is ECE... Ya ya, I know I am wicked .. :P

Lines. And points.

This is a math post.. Link rather... And folks who do not have a taste for it, don't blame me- you were warned!

I can't say I possessed a great liking for geometry at school, for whatever reason. Arithmetic was good (i am still in love with complex numbers), calculus was awesome. Algebra lost its charm later- maybe it got a little mundane, but for starters it was almost like solving some mystery ;) Trigonometry still leaves a bad taste as I think of the word. Computation is a word I learnt long after I had started liking it. Geometry on the other hand, and here I mean the Euclid kind of geometry, was one topic I developed a liking for after finishing school. As much as I hated the chapter on tangents and the 29 questions in the CBSE text, I loved geometry theorems after school... I still can't go past a new geometry result and resist the temptation to prove it. 

(Ahem.. And if you are still thinking about the 29 questions- no I don't remember the exact number. I just conjured up a convincing one ;))

So when I came across this post on Pascal lines I thought I would post it for fellow math lovers to read... The beauty of geometry lies in its essential simplicity- who would have thought that one circle and some lines across it could result in Steiner points, Kirkman points and so many other points named after people*. AND Cayley lines. AND theorems with Cayley lines and Steiner points.. It is really all beautiful. Especially when I think about whether my circular pen stand and all the refills it holds form pascal lines and hexagons and blah de blah... (Curse my imagination, huh!)

Anyway, check out this link... :) And for today,

 Image Courtesy: http://talent.randomoriginal.com/2010/09/13/thats-all-folks-the-open-call-is-officially-closed

*Someday I should catch hold of some new points and name them after myself to leave an indelible mark in the math world :P

Knotty....

(Statutory warning: This page may contain material enuf to send anyone nuts.. fulla theory, so ppl who hate math, pls save urselves!!!)

(Disclaimer: The author is in no way responsible for any damage to anyone's brain as a result of reading this post!!!)

Its a vague memory now.. of one saturday morning wen "The Hindu Young World" taught me the English custom of making knots in handkerchiefs.... (Readin my posts shud ve told u by now tat im one ardent Hindu reader. Tho, personally, I find the Express a better paper, The Hindu is one I ve grown wid. N the Young World used to b a sorta addiction a long time ago) I don think anybody in Indian homes does that, but apparently it ws d commonest custom abroad, I mean that of keeping knot reminders...

Neway, it led me to ponder on how a person could remembr what they made the knot for.. ( Bein a very forgetful person myself, the fact that the kerchief knottin cant help me much, struck me badly, i guess!! ) Anyway, that was my first tryst with knotting. N after that, I completely forgot the significance or existence of knots, (except for occasional reminders by the Indian cinema of the "importance"(??!) of "the three knots", n ill refrain from commentin upon that!), til a few days ago wen i stumbled upon dis awesum theory of knots...

"THEORY of knots????", I went flabbergasted. Why shud anyone write a theory on knots???? I mean, on somethin as simple as knots?!!!

But, then, a little browsing opened my eyes to the romance of the world of knots....

Wikipedia says, even simple activities such as running a load from the hardware store to home can result in disaster if a clumsy twist in a cord passes for a knot. (lolz....) They are also "crucial" accessories(?) for truckers, sailors, netting, fabric-makers etc.. (I ll leave that part for u to read in wiki.. since im not a gr8 believer of ctrl+c ctrl+v s...). Wel, jokes apart, apparently, knots re so useful tat they are used even in genetics. N wat follows is my interpretation n comments on the knot theories i leafed thru..

Mathematical knots are defined as embeddings of circles in 3- dimensional euclidean space... meaning that, it is as if different circles wer superposed in 3 dim... (a very fundoo way of explainin)... N there are difrnt types of mathematical knots.. the diferent knots are described by a parameter that they cal the knot invariant...

There are also knot equivalents (meanin different knots tat can be reduced to d same type of knots) n knot diagrams (to show knots projected onto a single dimension) n some sorta moves (t s actually some mathematician's name, bt too unpronouncabl, so i leave it to overinterested ppl to find out!) that can be usd to find if 2 knots are equivalent or not.... n d list goes on.. der are knot polynomials (as u mite ve guessed, polynomials to describe knots) , n knot sums(!!!) n wat knot!!!! ;)

The weird thing is that math ppl ve done so much research on knots, n.........

O, i forgot.. there is also the "UNKNOT", which is thread wid ends joined together at both ends, that doesn form a knot!!!!

Neway, the point is that our mathematicians ve actually done so much of work on knots, n propounded such interestin theories on em.... N wat started wid d gordion knot, has now come a looooooooong way...

Kudos to mathematicians!!! (as long as thes theories don enter our textbooks! ;) )

N ya, hope they find sumthin for us knotheads to keep reminders in, better than handkerchiefs!!! (now, don reply cellphones!!!)