Hi pa !! :-)
HOW have you been...?
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.
[2D-Case...2 circles against the axes...! :)]
And now fer the 2 scenarios for pa(r/c)king "balls"....! :P :)
[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"
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...!!! :)
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...! :) ]
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
SCENARIO --> 003
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)
