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AlbertJayNock

The Golden Ratio and Egyption Fractions

All true geeks know about the golden ratio, or the number (1+sqrt(5))/2. The Greeks believed it  to be the ratio of sides of the most beautiful possible rectangle. Many think  it is useful in the technical analysis of stocks (in so called Fibonacci retracements). It was featured prominently in the movie Pi. It is America's most beloved non-transcendental irrational number.

A popular way of approximating the golden ratio is with successive ratios of consecutive  fibonacci numbers. For the uninitiated, fibonacci numbers are the following sequence

1,1,2,3,5,8......

where each fibonacci number is the sum of the two previous fibonacci numbers.

So the sequence 2/3,3/5,5/8 gets closer and closer to the golden ratio. Note the second two numbers. These are the ratios of sides of index cards. I don't think that the inventors of index cards said "hey let's make the sides so they approximate the golden ratio." What I believe is that these just naturally looked good. This would lend credence to the Greeks beliefs about the golden ratio.

The reason that ratios of fibonacci numbers are good approximators is because they are obtained by the following continued fraction, which converges to the golden ratio:

1/(1+1/(1+....)

or more compactly

rk+1=1/(1+rk+1)

so you have

1/(1+1)=1/2
1/(1+1/2)=2/3
1/(1+2/3)=3/5

etc.

In the interest of not losing anymore readers, I present all of the above without proof.


The use of continued fractions to approximate the golden ratio is fairly well known. The other day I came across a method for using Egyption fractions to represent ratios of consecutive fibonacci numbers, and thus approximate the golden mean.

Egyptian fractions are representations of fractions as sums of unit fractions (fractions where the numerator is one). So 2/3 in Egyptian fractions is 1/2+1/6. 3/7 would be 1/3+1/15+1/35. The Egyptians represented fractions this way because they were so busy building the pyramids they forgot to invent numerators of fractions.

It is very easy to represent ratios of consecutive fibonacci numbers as Egyptian fractions, because of the following identity

f2k*f2k-1-f2k+1f2k-2=1

From this  it is easy to derive that

f2k/f2k+1=f2k-2/f2k-1+1/f2k-1*f2k+1


so 3/5=1/2+1/2*5=1/2+1/10
and 8/13=3/5+1/5*13=1/2+1/10+1/65

etc.

You can use this method to represent any ratio of the form f2k/f2k+1 this way just by using the above formula and working backwards. (There is also a way to do it with ratios of the form f2k-1/f2k, but the margin of my blog is too small to contain it).

I should say that although I came up with this myself, I would be absolutely shocked if no one else has. It easily derivative from a lot well known formulas. Still, it's fun to discover things even when you're not the first. And it also gave me a chance to try out the subscript feature of the rich text editor, which of course was the principal motivation for this entry.
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Comments

cool

I need to print this entry to show to people when thay ask me about the tattoos on my arms.

Re: cool

And they say there's no real life application of this stuff.
Ha ha. Great entry. It's interesting the way the golden ratio pops up all over nature. Pretty interesting... not getting a lot of math practice up here in VA so it's nice to see some fibonacci sequences. Love the fermat reference.
I knew I could count on you to catch the Fermat reference. I'm pleased to see that nobody said "hey blog's don't have margins".