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Tuesday, November 30, 2010

Pascal's Triangle and Expanding Polynomial Powers

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Note: In Pascal's Triangle, the key to figuring out a given number is to add the left and the right numbers of the row immediately above. The sum of all the numbers in each row is equivalent to 2^row#. The individual numbers in each of the rows divided by the total of the row it is in shows a probability of getting that combination of heads or tails if a coin is flipped row# of times (row1 - 1H, 1T), (row 2 - 2H, 1H1T, 2T), etc.

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Update 11/30/2010 - Here demonstrates how the Fibonacci Sequence (1,1,2,3,5,8,13,21...) is the summation of the oblique diagonals of Pascal's Triangle.

The Golden Spiral/Rectangle/Ratio/Number

(1) In the first picture below, if you follow the spiral with your finger from the outermost point inwards, you will notice that the squares keep getting smaller. You could, theoretically, follow the spiral infinitely inwards or follow it infinitely outwards too.

(2) You will also notice that the spiral always intersects the corners of squares too. If you follow the spiral inward through any 4 consecutive intersections, you will notice that each of these 4 intersection points is on a side that makes up a rectangle.

(3) In these recangles, the ratio of the length of the short side to the long side is always

1 : (1+√5)/2 --> 1 : 1.618 --> 1 : Phi

... where "Phi" is the "golden number" which is the irrational constant "1.618033...".

(4) The key to understanding the Golden Spiral is that each larger square along the spiral path has a greater sidelength than the previous square by a factor of "Phi", ergo L(i+1)=L(i)*1.618033, and also that the sidelengths of any given square is equivalent to the net summation of the sidelengths of the previous two squares, ergo L(i) = L(i-1)+L(i-2).

(5) And hence, you have the most important ratio in ancient Greek architecture. They used this kind of golden rectangle in building the Parthenon as you can see in the front elevation in the picture below.

Update: Fibonacci Numbers & Spiral approximation:

(6) The Fibonacci Sequence can be used as an approximation for generating the Golden Spiral. The sequence is generated by starting with {1, 1, ...} and then adding those together 1+1=2, then 1+2 = 3, 2+3 = 5, 5+3 = 8, and so on and so forth. Thereby you get the Fibonacci Sequence: {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... }

(7) Using the Fibonacci Sequence you can generate the Fibonacci Spiral, which is similar to the Golden Spiral in that it is composed of squares whose sidelengths are equivalent to the net summation of the sidelengths of the previous two squares along the spiral path, such that L(i)=L(i-1)+L(i-2) holds true:

(8) Phi can be obtained from the Fibonacci Sequence, i.e. {1, 1, 2, 3, 5, 8, 13, 21, ...}, by taking the limit of the ratio of L(i)/L(i-1) as i->infinite, ergo:

Fibonacci Ratios -> 1/1 = 1.000, 2/1=2.000, 3/2=1.500, 5/3=1.666..., 8/5=1.600, 13/8=1.625, 21/13=1.615..., 34/21=1.619..., 55/34=1.617..., 89/55=1.618..., etc, ...Lim{i->inf, L(i)/L(i-1)}=Phi=1.6180339887...