Download You may have heard of the Golden Mean, the

You may have heard of the Golden Mean, the Golden Ration, or the Golden Section- they
refer to one thing that has been popular throughout history, used in mathematics, art,
and architecture. It is said to embody beauty, and is found in nature in the designs of
shells and the ratio of our finger joints.
What is it?
[Draw Segment]
Defined by Euclid as the ratio of the longer to shorter segment when the ratio of
shorter to longer equals ratio of longer to whole. (dividing a line in the extreme and mean
ratio)
[Split Segment]
What is it really?
[Assign a,b and set golden mean=b/a]
To make it the most convenient, we can use a line of length golden mean+1
[Assign x and 1]
x/(x+1)=1/x
x^2/(x+1)=1
x^2=x+1
x^2-x-1=0
So we can solve this using the quadratic equation
(1+sqrt(5))/2 is about 1.618 (03398875) and that’s the number that the golden mean
refers to, but there is a need for notation. The American mathematician Mark Barr came
up with the letter phi after the Greek sculptor Phidias who used the golden ratio in his
works.
This number has special algebraic properties. If you square it, you get phi+1, just coming
from the intermediate equation. And if you take the inverse of it you get phi-1, which is
the absolute value of the other root of the equation.
---Maybe some commentary here--As for the arithmetic properties, we can show that it is an irrational number:
Suppose phi=p/q in reduced form for integer p,q
(p/q)^2 – (p/q) – 1 = 0
p^2/q^2 – p/q = 1
multiplying by q^2: p^2 – pq = q^2
p(p-q)=q^2
basically by fundamental theorem of arithmetic p must be a factor of q, since p/q is in
reduced form p must be one, here it says that phi would have to be the inverse of an
integer, which is pretty clearly wrong since it is greater than one, but proceeding:
p^2 = q^2 + pq
p^2 = q(q+p) so now q must be a factor of p, which means by the conditions, q must be
1, but phi is definitely not 1/1 so we conclude in a contradiction and phi must be
irrational. If you didn’t buy it after seeing the expression with a radical in it, then you
should buy it now.
There are more special qualities of the golden mean. Dividing the second to last equation
by x yields: x = 1 + 1/x From this it is possible to generate a new expression by
plugging it into itself. x = 1 + 1/(1+ 1/x) And continuing this makes an infinite
fraction that is equal to phi (that doesn’t have to have an x at the bottom). It is important
to note that this is the simplest possible continued fraction. And that means that it is also
the slowest to converge. This again shows its universal significance.
---Transition to Fibonacci--So computing this continued fraction will give a good approximation of phi. Lets try it:
uh-oh, what is x? we cant use the answer in our calculation of it, oh well lets use 1 since
it’s the easy way out and come back later: 2/1, 3/2, 5/3, 8/5
A pattern emerges, the denominator is the previous numerator, and the numerator is the
sum of the previous numerator and denominator. Based on the first rule, the set of
denominators is just the set of numerators shifted back one. So whatever sequence that
emerges is the same for both of them. Time for algebra:
N_n=N_n-1+D_n-1 D_n=N_n-1, plugging in the D_n: N_n = D_n+D_n-1
So since D_n+1 = N_n:
D_n+1 = D_n + D_n-1, and shifting just to get a simpler expression:
D_n = D_n-1 + D_n-2 and we arrive at a second order recurrence relation
This recurrence simply means that the number at any position in the sequence is just the
sum of the last element and the element before that. You may recognize this sequence as
the Fibonacci Sequence. A correlation between the two has been found. From the
direction we came in it is hard to say what the correlation is, but looking at things from
the other direction it is more explicit : The ratio between consecutive elements of the
Fibonacci sequence equals the approximations to phi. Extending this concept, if the
approximations to phi tend to phi, then the ratio between consecutive elements of the
Fibonacci sequence tends to phi.
Lets take a look at this geometrically.
[rotate longer right side segment up and connect end to top]
This line has slope phi. If we had a grid with dots at each integer coordinate pair then this
line would never touch them- that was shown with the irrationality proof. But take a look
at where it is closest. It is the Fibonacci numbers again! They oscillate over and under the
line, but always get closer.
So what about that arbitrary decision to use 1 in the continued fraction? Well, using 1
yields the Fibonacci Sequence. Using pretty much any other number would give entirely
different values and never be the same. ( can you spot the exceptions? (any of the
numbers in the last set, because it just skips to that point) ) However, the recurrence
relation still holds and the properties of the sequence are the same, it even converges to
phi still, no matter what number you pick. These other sequences are called Lucas
Sequences and they are the general class that the Fibonacci numbers are a specific type
of. The Lucas Sequences are simply generated from different initial two numbers.
But why is it that phi should be the convergence of the ratio of the terms of all Lucas
Sequences?
Phi^2 = phi +1 multiply by phi: phi^3 = phi^2 + phi
phi^4 = phi^3 + phi^2
It has this property that the power of phi obey the recurrence relation property- meaning
the exponents of phi generate a lucas sequence. In this sequence let P_n = phi^n so the
ratio of two large terms is P_n/P_n-1=phi^n/phi^(n-1)=phi
But an equation for the exact values of the elements of the Fibonacci Sequence can also
be generated: [phi^n – (1-phi)^n]/sqrt(5) and as the exponent gets very large, only the
first term will become significant so the ratio of terms will be about phi^n/phi^(n-1)
which is just phi. All Lucas sequences have ratios of consecutive elements that converge
to phi, just at different rates, initial two numbers 1 and phi does so instantly.
---End Fibonacci Section--Lets look at some more appearances of phi.
[Draw a 36,72,72 isosceles triangle]
Bisecting the left base angle, we get two new isosceles triangles. AB=DB and DB=CD so
AB=CD. Also, since BDC is similar to ACD: CD/BC = AC/CD, and we can add to this
=AB/CD from the last statement. Lets say all these are equal to x, to solve for x, lets say
we have a triangle that has BC=1 [Label 1 and 3 x’s] now x/1=(x+1)/x so the x’s here are
the golden mean again. The lengths of the sides may change but the ratio AB/BC is
always phi.
This leads to a decagon because 360/10 is 36, so the radius of a circle circumscribing a
decagon is phi times the length of a side of the decagon.
[Dotted line for altitude]
The altitude of this triangle bisects angle BDC. The distance from A to the bottom of the
altitude is phi+1/2. This is equivalent to phi+1/2(phi^2-phi) which is also equivalent to
½(phi^2 + phi). This is interesting because cos(36)= adjacent/hypotenuse=
1/2(phi^2+phi)/phi^2 = ½(1+1/phi) = ½(1+phi-1) = ½(phi). So a relationship to
trigonometry has been established.
But that’s not the only one: Tan(x) = sqrt(phi-1) where tan(x)=cos(x)