I am looking for someone who would know about the "golden mean"?? I am guessing
that, when evaluating complex fractions, there is a pattern that would lead you
to the "golden mean"?? If anyone can help me answer this question, I would
appreciate it.
Articles on "the golden mean", "the golden section", and "the golden ratio" are
all referring to the same thing. You can find lots of articles on the
internet, using Google, about this golden "thing". I prefer to call it the
golden ratio, because it really is a ratio. The golden ratio can be found in
art and nature.
If two numbers x and y are such that x is to y as y is to x+y, then
they are said to be in the golden ratio.
Suppose x and y are in golden ratio.
x y
=
y x + y
x(x + y) = y˛
x˛ + xy = y˛
x˛ + yx - y˛ = 0
Use the quadratic formula:
Use the quadratic formula:
______
-b ą Öb˛-4ac
x =
2a
where a = 1; b = y; c = -y˛
_____________
-(y) ą Ö(y)˛-4(1)(-y˛)
x =
2(1)
_____
-y ą Öy˛+4y˛
x =
2
___
-y ą Ö5y˛
x =
2
__
-y ą yÖ5
x =
2
Factor out y
_
y(-1 ą Ö5)
x =
2
We will only use the + symbol, because the - sign
gives a negative number.
_
y(-1 + Ö5)
x =
2
Divide both sides by y
_
x -1 + Ö5
=
y 2
So the right side is the golden ratio.
The golden ratio is .6180339887
Notice that this number is the only number whose
reciprocal can be found by simply adding 1.
The reciprocal of .6180339887 is 1.6180339887
Each of the 5 concruent isosceles triangles forming
the 5 points of an ordinary 5-pointed star like this:
«
are such that each of their two equal sides are
in golden ratio with its base.
If you take the Fibbonacci sequence which begins
with the first two terms being 1 and each term
beginning with the third consisting of the sum
of the two preceding terms, i.e.,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
have these successive ratios:
1/1 = 1, which is more than the golden ratio, .6180339887
1/2 = .5, which is less than the golden ratio, .6180339887
2/3 = .6666666667, which is more than the golden ratio, .6180339887
3/5 = .6, which is less than the golden ratio, .6180339887
5/8 = .625, which is more than the golden ratio, .6180339887
8/13 = .6153846154, which is less than the golden ratio, .6180339887
13/21 = .619047619, which is more than the golden ratio, .6180339887
21/34 = .6176470588, which is less than the golden ratio, .6180339887
34/55 = .6181818182, which is more than the golden ratio, .6180339887
55/89 = .6179775281, which is less than the golden ratio, .6180339887
89/144 = .6180555556, which is more than the golden ratio, .6180339887
etc., etc., etc.
Notice how the ratios go above then below the golden ratio,
but getting closer and closer to it each time. The ratios will
never reach the golden ratio exactly, but will eventually become
closer than any tolerance you choose.
Edwin