What’s Between Fibonacci and Golden Ratio?
Golden ratio and Fibonacci sequence are well known “entities”. Whether you are a mathematician or an artist or just a curious person you probably met them somewhere. In this post I will present a well known property that ties them together.
What is Fibonacci Sequence?
I guess every one knows what is Fibonacci sequence, but since I wish to hope that even creatures from Mars reading my posts so I will give a brief description:
Fibonacci sequence is a sequence that is determined by two real numbers a,b
Where a is the first element than b and then it follows the formula :
If a and b are both 1 we get the following sequence:
1,1,2,3,5,8,13,21,34…
Which is in this post the Basic Fibonacci Sequence
Golden Ratio
Golden ratio (g.r.) is the following number
Among its other names we can find the “divine proportion” and “golden cut” The Greeks studied it discussing the golden rectangle (with a and a+b sides) .Some artist found it astatically fascinating due to the proportions that it offers, It was studied in architecture as well.
It’s All about a Quadratic Equation
The golden ratio is the solution of the innocent quadratic equation:
You may suspect that this equation has another solution… and you are right
It is
This polynomial and his roots will be key players in the following sections we will denote it as the quadratic eq,
The Quotient Sequence
We move our focus to the quotient sequence. Consider a Fibonacci sequence. If an is Fibonacci’s nth element, We define the following sequence
We are interested in the sequence
Claim: If the sequence of r’s has a limit it is a solution of the quadratic eq.
Proof:
Where the third equality is since we assume it is a limit . We can change sides to obtain the quadratic eq.
We have proved that if the quotient sequence has a limit it is a solution of the quadratic eq , now we need to show that this sequence converges
Claim: if an element in the quotient sequence is positive then the entire sequence that follows is positive
Proof: It can be seen that if X is positive than its successor quotient
is positive too. Hence the quotient sequences are either consists of only positive numbers (besides a few numbers in the beginning) or only negative numbers
Positive Quotients Sequence
In this section we consider only positive quotient sequences. Hence the only relevant solution of the quadratic eq is the g.r. Let ri be an element in this sequence
So we observe that if one element is bigger than the g.r. then its follower is smaller and vice versa. (g.l means g.r.)
We can use the same methods to prove the same property for the decreasing sequence. It can be easily seen that the limit of those sequences is a solution of the quadratic eq. Since these sequence are positive it leaves us wilt the g.l.
As we wished.
Negative Ratios
In this section we discuss cases where all the ri are negative. We wish to answer on two questions :
- Do we have negative quotient sequences?
- What are the required conditions for such sequences?
- What about their “convergence manners”?
Now we will construct a Fibonacci sequence that its quotients are always negative.
We define
These are the first elements in the Fibonacci sequences . We can observe that:
- The magnitude of a must be bigger than the magnitude of b (otherwise the quotient (a+b)/b is positive)
- Clearly a*b <0
Lets construct the sequences
Now let’s see the nature of quotient sequence
Since every element has a different sign than its neighbors we have w.l.o.g
Let’s write equations that satisfy these inequalities for the indices 2,4,. 2k
Namely b/a is an upper bound for the a subsequence of quotients in the Basic Fibonacci Sequence. In the same way we can show it is a lower bound for the odd indices
The corollary is that in order to construct a negative quotient sequence of a Fibonacci sequence the two early elements must satisfy the ratio
Which is itself a solution of
In all other cases Fibonacci sequence has positive quotients ,thus converges to the g.r.
