i just finish reading this wonderful math book from the library and i better type down whatever i remember....
i once receive an e-mail highlighting the striking parallels between Lincoln and Kennedy and was actually fooled...but in actual fact, its only normal for them to have those similarities, 1 men's lifetime is full of thousands, millions of data and for 2 men to share like only 10 of them? ITS PERFECTLY NORMAL, STATISTICALLY
just like how those psychologists do experiments by separating twins and see how they grow up, usually these twins might have some similarities which will amaze some people, but in actual fact they need not be twins or related to have those similarities...
same as those investment scams when they can actually predict correctly for 10 consecutive weeks....its all because they send all different kinds of mails to a large database, each week they just have to shave off half the population
chaos theory: in maths chaos theory is not referring to how chaotic the equations are, but actually how a minor change in the initial input can cause a dramatic change in the eventual outcome, thus in chaos theory we believe that the outcome depends greatly on the initial conditions and since a small change and interference somewhere can cause great changes...we aren't able to predict things in a long run...like weather
imagine you have a piece of A4 paper, each iteration you fold it into half, after 50 times, how high can you get...? a foot? a building?
You can reach the sun dude...
don't believe?
try punching 2^50 on your calculator
p.s: i can't guarantee smoke won't come out from your calculator
the Fibonacci's sequence, the entries in it is found everywhere in nature, the number of left or right spirals on a pineapple, a daisy flower, a pine cone,
for any three consecutive Fibonacci numbers X1, X2, X3
(X1)(X3) and (X2^2) always differ by 1
once there was an area paradoxical puzzle on all MRT trains...this explains it...there was a nearly invisible 1 unit square area in the triangle
this beautiful relationship is also true: X2/X1 - X3/X2 = +- (1/(X1X2))
golden ratio = 1 + 1/1+1/1+1/...forever
and each time if i were to shave off the one and 1/, i still get the same answer
thus
phi = 1 + 1/ phi
phi= (1 + 5^0.5)/2
there's also a sequence of numbers, the Lucas numbers, whose initial conditions are 2, 1
and they allow us to get the golden ratio too!!
thus the essence of Fibonacci lies not in its initial conditions, but its process
Divine proportions, the golden rectangle
how to draw one
draw a square, extend out the bottom line towards the left, mark a center point at the bottom of the square. using a compass, fixing it at the center point, starting from the top left corner of the square, draw a curve down to the bottom extended line.
link everything up, we have a golden rectangle...
i hope the rigorous proof of this stays with me forever...simply start by assuming that the square has length 2
the golden rectangle is the only rectangle that possess the same ratio property after squares are shaved off
if we take our image and place a quarter circle within each square, then we create a beautiful spiral. this spiral is close to a LOGARITHMIC spiral...the mathematical abstraction of the natural curve exhibited in the graceful nautilus shell.
the ratio of your height and that of your belly button is almost the golden ratio
we also have the golden triangle...its the triangle that arose during our construction of the golden rectangle...its a right angle triangle with one leg twice as long as the other leg
if we make four copies of the triangle, they can be assembled in a straightforward manner to produce a triangle having the same proportions as the first, but with twice the size of the original.
the golden triangle is also the only triangle for which five copies can be assembled to produce a larger copy of itself
YET
we can also repeat the regeneration process involving only four copies and we can produce a tiling of the entire plane...that is, an endless floor.
this pattern tiling...is known as PINWHEEL tiling, its aperiodic, which means it has no repeated pattern that would allow us to re-create the pattern after a translation. This chaotic or jumbled image does possess an attractive dissonance.
We do see it in some churches...