Ball Park-ing ! :)

Hi pa !! :-)


HOW have you been...?

Well, yester-night,  me stayed over with my dear friends (Amod & Vijay)...since the weather (& my health! :-( My Fever...! :) !) was not very good...!

But we slept at dunno...bout 3 am or so purr-haps...!!! :)

Not fer werk...(we'd have dozed off sooner if it was jussst our stupid werk! In fact, we did some good progress on our work first...)

But...discussing & doodling on a problem I had preposed while I was in the UK to my friends...
 

I asked my friends again if we could...try to see what is the MAX size of a smaller ball that will fit in the space left when a Bigger ball rests in a cubical enclosure..

[NOTE that this could be a standard, "solved" problem elsewhere in Mathematics, but looking up too much of already "done" stuff, is what usually try to avoid...since it might (& does!) influence our "original" thoughts. Whatever we get should be our own "flowering"...]

• ...initially, it was bout packing circles ( 2 Dimensions)...
• ...then, I took it 'higher'...to 3 Dimensions...!!
• ....and there. started the FUN! Because it is NOT easy to imagine 3-D spatial relationships...We 3 friends were like 3 Blind POETS/MEN trying to describe the proverbial ELEPHANT, gesturing, doodling, & doing our best to get a "handle" on our intuitive "grasp" of what could really happen in different scenarios...! Our heads got scrambled when we realized that in a 3-D cube, the "space" available at the CORNER is MORE compared to that at the meeting of just 2 faces...!!

Walking deeper based on some great li'l insights that my friend Amod stumbled upon, we were able to "break-ice, make in-roads" & slowly began to see some beautiful results.
 

We also began wondering HOW amazing the patterns are in Nature...in multiple-dimensions [right frem ZERO to THREE...it also got me back to wondering...as to what exactly is "space"..."dimensions", etc...! Hmmm...]..!! :)

I could DREAM of OLL sorts of NUMBERS n EQUATIONS n maddening PATTERNS jumping out o' nowhere & asserting their existence(?!?) to me..!!
And I went on "venting" them out to my dear friends, who were more than willing to listen, & even contribute to oLL that....!!

In the end, it was deeep into the night (or MORNING!? :P ) when we "fainted" & "slipped" into the valley of Sleeeep...

I used our 2D result that we got thanks to the work by Amod, & intuitively guessed the pattern fer 3-D...
We could even "verify" it experimentally, but our fatigued, stressed, THINKING Minds could NOT find the equation Formally, TRY as WE MIGHT...!!! :-(

AND I woke up EARLY in the MORNING, n sat Listening to SONGS....without THINKING about it, & LO & BEHOLD..!!!!!
I suddenly "got" the way to find the equation fer the 3D case that we were SO desperately TRYING to "derive"!  ["The 'EUREKA' PHENOMENON" :-) ]

Anyways, Don't think that OLL we did was some crazy doodling in our scrap-books, & write some "imaginary" equations & symbols...!!
The beautiful lit'l patterns we discovered have kept us animated n all awake....!!!! <3 :)

I enclose below few pics of some Mathematica-l  "doodles" I did immediately yester-night in order to "visualize" & "verify" our Mathematical results. :)
And I must say, I'm pleased as punch with them all! :)
[Coz these drawings are "driven" by the mad doodling n equations we friends scribbled all night...!! :P :) And the results just made us smile with joy !!  

The Very Spirit of SCIENCE...! <3 --Prediction("Guesses"), Observation/Experimentation, and Refinement of the Prediction/Guess...! :) ]

Click below on the pics to see clearly:

SCENARIO --> 001

[2D-Case...2 circles against the axes...! :)]  

 

And now fer the 2 scenarios for pa(r/c)king "balls"....! :P :)

SCENARIO --> 002

[Pa(r/c)king "on the Edge"]

SCENARIO --> 003

[Pa(r/c)king in the Corner] 

 

At-LAST, we could say, Ball-e Ball-e [fer the un-initiated, this is an oft-used phrase used in Punjabi songs which are known fer their peppy music..!]...!! :P

...Or...should i call it "BALL-PARK-ing" ?!?! ;-) 

LOL..!!!

TC, n Keep Smilin'....

Love n Best Wishes,
The boy in "rags"

 

Planet (H)eart(H)

Poe-tic Lie-scents ! :)

Thanks to a funny article shared by my friend Roopa D'Almeida about pony-tails & an equation describing them, I suddenly had this hunch to scribble a couple of lines modeled on the famous (& one of my favourites, fer its ability to create 'atmosphere' with words!) poems of Edgar Allan Poe, "The Raven".

Without much ado, I present to you the work below (please click on the image to see/read better!)

Note:

Another peculiar thing fer my dear friends to observe is the 'dimensions' of the 'FRAME' in which I have 'composed' this; the white image originally is of the size 1034 X 639 pixels. 

Why..??

Any guesses..?


Yes !!!


The ratio 1034/639 approaches the value of 1.618, the Golden Ratio-'Phi'...

At-least some food fer thought, in my "Phi-night" existence...?!?!

Hope this poem makes you smile...fer a while pa..!??!

Liked it..!?!?

Ok then...give me a High-Phi...!!! :-)

[Incidentally, the pun was on High-Five, and Phi plays a vital role in all things five sided, from the Pentagon to the Star-shape...Hmmm.....! :)]

Love & Best Wishes,

 

the boy in "rags". . .

Planet Earth

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

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"

A pithy lit'l ge(R?)m..!! <3 :)

A R[ae]gex quote...!! :)

Recently, this quote "came to me" out of nowhere, in response to a cute little cartoon shared by my dear friend Arun Kapoor...

And i felt it deserve(s/d) to be shared...with YOU too/esp....!! J

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

"....OLL i need....is a 

((CARTOON)?(ist)?) (FRienD)?...

Hmmm...."

:~ me-(mo)R(on)

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

Yes !!

I couldn't believe it...!

It was a unique, REGEX quote...where (expr)? denotes 0 or 1 occurrences of expr --TRY it..! :)

And then, I spontaneously thought (& wrote!) aloud (on FaceBook). . .

that 

since there aRe 3 ? symbols, (0 or 1 occurrences!), there aRe 2^3 = 8 possible combinations & hence quotes possible...! 

Now, that's pithy....ain't it..!?! ;-) :P
[

Yeah...what a PiTy...!! :( 

Indeed it is...that the language has to suffer a fool like me gladly !:D

]

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* * (000)

"....OLL i need....is a...Hmmm..."
--Sadness...un-expressed emotions..!! :-(::

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

* * (001)

"....OLL i need....is a FRienD...Hmmm...."
--Loneliness...

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

Soon, I also observed that (010) & (011) r not meaningful...

* * (010)  "....OLL i need....is a...ist...Hmmm..."
&
* * (011)  "....OLL i need....is a ist FRienD...Hmmm..."

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

But the REMAINING were oLL fine :)

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

‎* * (100)

"....OLL i need....is a CARTOON...Hmmm...." --again, someone lonely...but who loves cartoons..! :P [who else..! :P]

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

* * (110)
"....OLL i need....is a CARTOONist...Hmmm...." --Purrhaps..the editor of a newspaper/magazine..!? :P

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

‎* * (111)
"....OLL i need....is a CARTOONist FRienD...Hmmm...." --straight-forward... :)

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

Sorry fer irritating oll by stating the 'obvious' (?!)...but ONLY becoz i tried it did i realize that 2 of them were invalid..!

But as I've quoted E.T.Bell in my blog ,

"Obvious, is the most dangerous word in mathematics." 

Thanks fer bearing with me...!

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

The Boy in 'rags'

Planet Earth

Keep smiling...!. . .

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:-)..!

A strange Black Swan. . . ! :)

Hi pa!

i know you may laugh at it; i too did initially, till... 
...till. . .i began hearing it laugh at me..!! :)

Here it is...a pattern that emerged by sheer chance...or in short, serendipity..!

I was shocked & amazed at HOW it looked like a mysterious, tantalizingly beautiful SWAN...!! :)

And before you laugh it off as a random cellection of numbers & symbols, let me assure you otherwise...!

It is actually the exact expression, or, a short-cut if you want, to compute the sum of the first 'n' natural numbers, with each raised to the power of 100. 

Yes, you 'heard' it right...it is the value of : {Click to see it as a larger image}

And you might be surprised to know that I had to toil more to get the above summation equation than the below expression...!! :)

Anyways, here she is. . . my beautiful Black Swan...! :)

"Why can't I  too recognize/find visual patterns in beautiful numbers..?? :-/

Certainly, if our ancestors could find our so-called 'constellations' by sheer co-incidence like the Big Bear(Ursa Major), the Hunter (Orion), Scorpius(Scorpion), The Bull(Taurus) et al, then, I too can ( & shall)...! :)"

~R. . . 

[Click on it, to view it better...]

Love & Best wishes,

the boy in "rags"

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