Mountain (de)Range(R/D)s...!! :)

Making Mountains out of a mole-hill...! ;-) :)

It was Christmas, & I was lazing around with my friends in the utter cold weather...

And my dear friend Prashant "woke me" up with some of his suggestions on making 'nice-looking' mountain-ranges when he happened to see my following creation, which is essentially a Mathematica(l) plot of Sin(2x)+ Cos(2y) where both x & y are in the closed-interval [0, 3 Pi][click the pic below to see better! ]

He wondered if we can have better approximations to real mountain ranges...

And that got me thinking...(& literally, rummaging the cup-boards in my memory-p(a)lace (a dingy hut rather..!! :P))...

And soon, i found something interesting, an utterly simple way of making mountains...based on pairs of Parentheses which can be nested together as well as shown below...[click the pic below to see better! ]

I was working on an alternative way of making mountains out of "strings' while my dear friend Prashant came up with this beauty (http://on.fb.me/vOOgx0)... :)

Jusst made my own, "verbose" (like me moron..!:P) version...which is more like instructing a robotic arm(or a child) to draw the mountains, with a simple rule as below:

• Traverse the string of brackets...given as input
• If you encounter a "(", ASCEND i.e., draw a line moving towards the UPPER-RIGHT hand corner.
• Else (if it is a ")"), DESCEND i.e., draw a line moving towards the LOWER-RIGHT hand corner.

Here's the output of my baby Mathematica(l) program: [click the pic below to see better! ]

As I came running DOWN the Stairs & shared with Prashant [it's always the Eureka! phenomenon, albeit with my clothes on..! ;) ], the thing of JOY (& BEAUTY!) that I discovered, & want to point out in this 'technique', is that it is a "look-see" device, or a sort of visual "check" fer whether a string of parentheses are properly 'balanced' or NOT! :-)
[

The simple heuristic/observation is that the brackets are 'okay' if the mountain-ranges reach 'back to ground-zero' i.e., the same height that they started out with/frem!

This assumes that the input-string begins with a "(" else, a trivial counter-example is : "())(" -- Can you see why..? :P

]

Any sort of 'imbalance' in the brackets, gives rise to a 'limping' Mountain-range...!:P

I request you to please check it out yourself!

The beauty of this is the utter simplicity; NO Need of sophisticated programs, nothing..!! :)

Even a child can doodle it & discover...& create a infinite number of mountain-ranges of different shapes & sizes...!

Note the following points:

• The deeper the nesting, the taller the mountains become.
• Also, YOU can 'guess &/or calculate & prove' the relationship between the TOTAL number of bracket-pairs & the TOTAL possible number of ways in which a 'proper' mountain range could be created.

Feel FREE to EXPLORE & share...!! <3 :)

Cheers, to the weird werld of Patterns around (which includes u (s) ! ;-) ...! ♥ :)

Merry X-mas & a Wonder-Filled Happy New Year to YOU n oLL your dear ones..!

Take care, & Keep Smiling, AL(L)-ways..!

Love & Best Wishes,

The boy in "rags" :-)

"Math is sometimes called the Science of Patterns."

: Ronald Graham

Randomness_001

Coins...Fer a Change...!! :)


Was feeling 'lost'...and then decided to pay ettention to my thoughts....

I was thinking randomly...a lot about random numbers... 
(Hmmmm....i've not yet stopped...nor will I ever...hahaha!)

You can't imagine the MAGICAL things that lurk behind a seemingly RANDOM Universe....!

As one of my thoughts encouraged me to do, I took out a coin...& flipped it...(yeah...Cricket -match coin-TOSS! :P )...

And as it turns out...fer a few tosses, it appeared to be biased towards HEADS or TAILS...

[Out of 10 chances...7 will be heads, or even 8 or 9 will be tails...!]

But then, I imagined....
WHAT if...what if I extend this tossing OVER & OVER & OVER egain....& egain..!?!?

In the short-term...we may feel the Universe/God is biased...but over a Loooong term...there is JUSTICE..n EQUALITY (of Probability..)..?? :P

So armed with this thought...and tonnes of emptiness....i sat blinking before my monitor....waiting....fer the magic to happen...

And happen it did...!!

Before long, between my work....I had whipped up a couple of lines to show me graphically the results of 1,000 coin tosses...to help me 'see'...

And the results...are (t)here fer YOU to see...! :)

• n(heads)  = 504 
• n(tails)    = 496

{ Click on the picture below !} 

Moral: EVEN a state of ORDER has CHAOS & Randomness underneath...! :) 
It is just amazing HOW a seemingly random set of events...can aggregate and 'appear' to be 'orderly' to our Mind/Brain...! 


Is it not pa..? :)


Phew..! :)

Love & Best Wishes,

The boy in "rags"

if

If. . . ! 

If you can keep your head when all about you 
Are losing theirs and blaming it on you; 
If you can trust yourself when all men doubt you, 
But make allowance for their doubting too;

 

If you can wait and not be tired by waiting,
Or, being lied about, don't deal in lies,
Or, being hated, don't give way to hating, 
And yet don't look too good, nor talk too wise;

If you can dream - and not make dreams your master; 

If you can think - and not make thoughts your aim; 

If you can meet with triumph and disaster 

And treat those two imposters just the same; 

If you can bear to hear the truth you've spoken
Twisted by knaves to make a trap for fools,
Or watch the things you gave your life to broken,
And stoop and build 'em up with wornout tools;

If you can make one heap of all your winnings
And risk it on one turn of pitch-and-toss, 
And lose, and start again at your beginnings
And never breath a word about your loss; 

If you can force your heart and nerve and sinew
To serve your turn long after they are gone, 
And so hold on when there is nothing in you 
Except the Will which says to them: "Hold on";

If you can talk with crowds and keep your virtue, 
Or walk with kings - nor lose the common touch; 
If neither foes nor loving friends can hurt you; 
If all men count with you, but none too much; 

If you can fill the unforgiving minute
With sixty seconds' worth of distance run - 
Yours is the Earth and everything that's in it,
And - which is more - you'll be a Man my son! 

:~ Rudyard Kipling

----------------------------

One of my favourite poems---it truly inspires me in moments of despair & loneliness...!
Hope you can FEEL it too pa..??

Please DO let me know...IF you like...! :-)

Take care, & Keep SmilinG...!

Love & Best Wishes,

The moron in "rags"

Planet Earth

A RuN to Remember...

Stairs to Heaven...!


June-17, 2011. 
It was the birthday of Miss Sinda & Dinsa, my dear friends at the office...
Apart frem being identical twins their names are anagrams of each other, & their b'day is a Prime (17) , with their star-sign's constellation being Gemini which also stands fer twins..!! This really made me ponder & smile a bit..


June-15, 2011. 
Another good friend of mine Amod Pandey, had posted a pic of him completing a marathon in the city of Bangalore. But due to heavy work at the office, I hadn't noticed it at till, late in the night of June-17, 2011(Friday), while was doing some official work at home, I just see it...& smile...!
...but was feeling a bit feverish & dizzy.


The number on his shirt, caught my attention---3649.
Well, suffice it to say that it drove my impending sleep away, as I began to find patterns in it, & ask my-self a barrage of questions like what is the smallest number with such a pattern, etc...!


June 18, 2011.
Sunday. Got reminded by mom that it was Fathers' day ! 
By now, I had devised a series of numbers with such a pattern...as I saw last night.
I secretly wanted to gift my Pappa something he'll feel happy about--because, he's always stressed the importance of Math ever since I was a child.


And the previous night, I had also submitted my series of numbers to an extensive online search program which would apply all kinds of heuristics & algorithms to check & ensure if this series that I had stumbled upon, was indeed a NEW one, or else the proverbial, old-wine in a new bottle...! Luckily, I received confirmation that the series was indeed NEW..!


Along with this confirmation came the need for lots of formalities---I was asked to fill detailed forms about myself, & provide description about the series, & even give a program to compute the first 'n' terms of the series...[Else, anyone could reel off some random numbers & name the resulting series fer him/her-self... :)) ]


Anyways, I did that, & was asked to make some subtle changes in my documentation, which I did too.
The series I had submitted was now in the 'proposed' state---awaiting detailed analysis by REAL mathematicians & scientists behind the Online Encyclopedia of Integer Sequences, who would take the final decision...


Anyway, I immediately fergot all about it---and with absolutely no expectations about it, went to sleep....


The next morning, at office, I was very pleased to find a mail stating that my series had indeed been APPROVED & ALLOCATED...!!!


My lit'l Discovery had changed its status frem PROPOSED to APPROVED...!!


Mine is the 1,91,867'th series 'registered'---though it is actually a kind of generating function for an infinite(?) number of new similar series...! [I only considered n=2 fer now, since the square has always had mysterious applications w.r.t. the Universe around us...!]


That said, there are TONS of questions to be explored like the following :


a) The beautiful stair-case-like graph(s) of the members of this series...
Click on the below graph if you want to see a sample..! :)

{ Notice the unusual structures on the graph ?? Well, i purposely strove hard to make it look like the stamen (with those peculiar anthers !)of a Hibiscus flower ! :-) How ?? By using the Greek small letter 'theta' as the plot 'marker'...! ;)) }




b) Is the series INFINITE...???? Fer ANY value of n ??
[My conjecture is that it IS...at-least fer n=2]


c) Any 'weird' places where this series 'pops-up'...??


Well, those r interesting points to ponder about....


And the below lines express my feelings very aptly..

Love & Best Wishes,

The boy in "rags"

Some Reflection. . .!

What a love-ly SONG....! 

{I'm mailing this to SOME of my FrienDs right away...!!}

Really touched my heart...& made me wonder HOW many such lit'princesses abound...stifling their DReaMs....!!! :-(

Applicable to SO many of US on soo many levels......!!

(frem the Disney movie --- Mulan --it's beautiful..! Sung by Christina Aguilera)

Hope we Don't Just see Animation movies jusst fer entertainment...!

And nourish & cherish An(y)-Emotion that such gems stir in our Hearts...!!

Love,

the boy in "rags"

Planet (H)Eart(h). . .

------------------------------------------------------------

Reflection. . .

Look at me---
i will never pass for a perfect bride, or a perfect daughter
Can it be i'm not meant to play this part...??
Now i see that if i were truly to be myself 
i would break my family's heart

Who is that girl I see. . .
. . .staring straight back at me?

Why is my reflection someone i dont know...?? 
Some how i can not hide who i am, though i've tried. . .

When will my reflection show who i am inside...??!!!
When will my reflection show who i am inside...??!!!  

-------------------------------------------------------

Unlearning misht-lakes..!! :-/ :)

I pursed my eyes, observed deeply, & realized a shocking truth, that I was mis-spelling the name of the great Hungarian genius Pál Erdős...!!! :-/ :-(

You too are mistaken, if you thought that the o in his name has an ordinary 'umlaut' over its head ! :P Decided to type it out & share...! :)

Also, thanks to my teacher Mrs.Sanhita Mahindrakar (who was such a great friend & guide & changed my life forever, by gifting me this book I didn't read for the 1st 2 years---"A Brief History of Time" by Prof.Stephen W Hawking ! :) ) at high-school, I learnt very late in my life about 2 more typos in my brain...'vaccum' (should be 'vacuum'!), & 'seperate' (should be 'separate' !)

Here's the not-right way...which is easy to assume...hahaha...

[Click on it..!]

and here's the right way, with what is called a Hungarian umlaut accent mark on the letter "o" : [Click on it...to see it better..]

Sigh...our mind/intellect, takes too many things fer granted...we must try and be... awake,...alert,...and aware...of the sheer emtpiness inside..!! 


Keep Smiling...! :)

Lots of LOVE & Bessst Wishes...AL(L)ways...as always...! :)

The Boy in 'rags'

Planet Earth

=================================

“How very little does the amateur, dwelling at home at 

ease, comprehend the labours and perils of the 


author.”

-- R.L.Stevenson and L.Osbourne, "The Wrong Box" 

=================================

A Font-astic quote on Fractals..!

Fractal Universe! :)

Hi pa..!!

Please click on the below image, download it if you please,  observe it...after reading it..!

Would you believe that it took almost 45 minutes to carefully "type" out the above paragraph..?? :) You may find the book mentioned above, but you will never find it in a way presented above.

Just notice the minute details, like for example, the way the words "cold" & "dry"  (the quotes around them rather !) have been 'rendered', as also the fact that "i" is/am wearing a "hat" too...!!! ;-) :)

This is in fact, the opening paragraph of the magnum-opus of Benoit Mandelbrot (notice the right way to spell his name)--"The Fractal Geometry of Nature", which took the world by storm, & almost single-handedly elevated Fractals into the realm of the believable, rather than being mocked upon as 'silly program artefacts' that have no practical applications or implications what-so-ever...! It now turns out that Fractals & Chaos are at the 'heart' of almost every Physical phenomenon or manifestations of the Universe...!!

I am really shocked to find people who routinely ask me --"WHAT use is OLL this mathematics of?? It does not have any practical use..." That is the farthest from the TruTh anyone could ever get! :-)

Anyway, I just wanted to share this fantastic piece of quote by 'dressing it up' with an amount of care perhaps never given to it before..! Hope you liked it, & it makes you (p/w)onder pa...?

I should soon make some FRESH starts...of discoveries, & this has really inspired me indeed...! :-)

I wonder if...if...

"our universe is itself, a fractal...?!??" :)


Love & Best Wishes,

The boy in 'rags'
Planet Earth
----------------------------
'Begin at the beginning,' the King said gravely, 'and go on till you come to the end: then stop.'
:- Lewis Carroll (Alice in Wonderland)
----------------------------