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)