Formulae

I'm sure somebody will criticize me for putting Mathematics on dA, but you have to understand... math geeks like myself see math very differently.  It makes me wish other people could.  I look at a geometric concept and see endless planes and twisted realities, and bending suns and infinity and the like.  Really.  I look at an integral and have a sense of wild completion as I know that every infintessimally inexistant sliver had been ultimately and undeniably accounted for, in an infinite and unending sum which I can conduct with the stroke of a pen.

So, before the "art" part of my journal, I'd like to share something of a different "art" that a friend of mine and I came up with.  It looks very similar to binomial expansion: indeed, there are proportionally increasing and decreasing powers along with a striking utilization of pascal's triangle.

                n        n
x^n  =  { sum  (      )  (x-k)^(n-j) * k^j
              j = 0      j

for {x,n} E all positive integers
and given that k is any positive integer < x.
I hope that ends up looking okay in your browser.  In words, it says that x to the n equals the sum from j equals 0 to n of (whatever number is in the nth row, jth column of pascals triangle where the first row and column are referred to by 0) times (the quantity x minus k to the n minux j) times (k to the j).

It may not look very cool, and you may not understand where I saw anything cool in this, but the coolness came in the derivation.  I drew the derivation.  I first noticed the correlation in Chemistry class.  I was trying to figure out 17 squared in my head.  I knew that 16 squared (16 x 16) equaled 256.  I then figured that if I added 16 to that number, I'd have (16 x 17); adding 17 to that would yield (17 x 17).  Thus, 17^2 = 16^2 + 17 + 16, or 16^2 + (2 x 16) + 1.  Striking.  Then I drew it in a figure:


n^2 =
                       n-k             k
                   ________     ___                  _
                  |                |   |     |                     
  n-k           | (n-k)^2    |   |     |                      |
                  |________|   | __|                      |  n-k + k = n
                  __________  __                        |
     k           |_________| | __|     k            _ /
                        n-k            k                
                  ____________/
                           n-k+k = n

From my terrible ascii art, you can see how I easily verifyed that
n^2 = (area of a smaller square) + (two times what I called "shields" in the upper right and lower left corners of the diagram) + the square of whatever number I brought down the original square by, which i call the cornerstone.
OR    
n^2 = (n-somenumberk)^2 + 2*(n-k)*(k) + k^2

Easily and graphically verifiable in 2 dimensions, i.e. x^2

x^3 was harder.  It's a pretty cool drawing a did, but I'm not going to try it in ascii art.  I'll scan it in and deviate a nice version.  Looks pretty cool.  Same concept, though.  If you can visualize it, we basically construct an (n x n x n) cube from a smaller cube, 3 "shields" that go on 3 adjacent faces, 3 "columns which fill the space between any two shields, and a corner stone where every other piece intersects.  The important thing I realized was that I have these pieces:

(where n = x -k)
cube (n x n x n)
3 shields (n x n x k)
3 columns (n x k x k)
cornerstone (k x k x k)

As you can see, starting with the cube and going down, we lose an 'n' dimension everytime and replace it with a k.  Also note that we have 3 sheilds and 3 columns.  In the 2d case, we had 2 shields.

I thus generalized that x^n could be found by the same formula I listed above, but moving (x-k)^n and k^n to the outside and bringing each limit of the sum inward by one to account.  I also generalized that we should use "n" instead of nCj (the pascal's triangle number), because for 2d it was 2, 3d it was 3, etc.

And this kept not working for things beyond 3d.  It took until Mark Gomer was showing me something totally different to realize that the 2s and 3s weren't there because that was the dimension, they just happened to be in the same rows of this 2d pascal's simplex...

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 5 1
.               .
.                  .
.                     .

... also known as pascal's triangle.

Using triangular numbers worked.

Unfortunately, these numbers are all the result of integer calculations, so I can't extend my formula beyond the set of all positive integers (yet).  Hopefully I'll figure something out.

Anyway, the 3d drawing is cool.  I'll draw another one to put up.

So, hopefully you're not reading anymore and haven't realized that this post is where you sign up for my 10 deviations.  

1. madhatterzwei [not sure if she actually wanted something, she just commented]
2. Misplaced-Karma [is getting an epic poem]
3. immobileFreedom [is getting either an epic poem or moccassins]
4. musicwhorestolen [is getting some sort of poem]
5. WrittenMemory [is getting prose]
6. Livingadejavu [is getting earings]
7.
8.
9.
10.

Bring it on, though.  It'll be good for both of us if you do.

I love you all.

-- Matthew


----------------
"Take all away from me, but leave me Ecstasy;
and I am richer, then, than all my fellow men;
ill it becometh me, to dwell so wealthily,
when at my very door, are those possessing more,
in abject poverty..."
(Dickinson, who I love but can never get over the childishly perverted dissection of her last name.)