Roger's Equations Blog

### Roger's Equations

This blog is all about science and technology (with occasional math thrown in for fun). The goal of this blog is to try and pass on the sense of excitement and wonder I feel when I read about these topics. I hope you enjoy the posts.

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# Phi - The Golden Ratio Part I

Posted January 10, 2008 12:00 AM by Bayes

Derivation of Phi

Phi , in the words of Euclid:

"A straight line is said to have been cut in extreme and mean ratio (ratio of Phi) when, as the whole line is to the greater segment, so is the greater to the less"

Hopefully a picture will help us visualize what Euclid is saying.

Basically Euclid is saying that if you were to cut the line segment A above at the point, line segments B and C are created. Then for the special case where the ratio of line segments are related such that A/B = B/C, the ratio is Φ (Phi). So lets solve the problem algebraically to find the value of Φ.

A/B = B/C

noting that A=B+C (see diagram above) and substituting we get

(B+C)/B = B/C

which can be rewritten as

B/B + C/B = B/C

1 + C/B = B/C

for simplicity, lets call the ratio B/C = Φ, which gives us

1 + 1/Φ =Φ

solving,

1/Φ = Φ - 1

1 = Φ2 - Φ

0 = Φ2 - Φ - 1

which gives roots

Φ= (1 ± √5)/2 (I used the quadratic equation to solve this with a=1, b=-1, c=-1)

The positive root is traditionally taken to be the value of Phi

Φ=1.61803398874989.....

though the negative root is just as valid a solution

Φ= - 0.618033988749894....

Here is a link that gives phi to 50,000 places

http://www.cs.arizona.edu/icon/oddsends/phi.htm

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#1

### Re: Phi - The Golden Ratio Part I

01/10/2008 9:13 AM

Isn't it amazing how things fall in the right places? The same ratio that you have just mentioned is called "golden ratio" and is obtained by dividing any number in a Fibonacci series to the immediate preceding one. More, in the fundamental analysis of the stock market, the Elliott Waves theory is based on Fibonacci numbers and ratios.

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#2

### Re: Phi - The Golden Ratio Part I

01/11/2008 11:12 AM

"Isn't it amazing how things fall in the right places?" Yep!

Also observe that the Feigenbaum Constant is very nearly the ratio of (10/(Pi-1)). Chaos Theory is involved in many areas of study, and the Feigenbaum Constant is fundamental.

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#8

### Re: Phi - The Golden Ratio Part I

01/16/2008 3:13 PM

I'll talk about Fibonacci Series in part II of this blog.

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#3

### Re: Phi - The Golden Ratio Part I

01/11/2008 4:13 PM

"A straight line is said to have been cut in extreme and mean ratio (ratio of Phi) when, as the whole line is to the greater segment, so is the greater to the less"

Φ= (1 ± √5)/2

The positive root is traditionally taken to be the value of Phi

Can we say that the numeerical value of phi (as given in the link to the U of Arizona) is a nonterminating decimal number? Is there a theorem to prove that (if true), besides just turning a computer loose on it?

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#4

### Re: Phi - The Golden Ratio Part I

01/12/2008 1:03 AM

we use to use rho to represent it.

it seems to go back to middle school class!

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#6

### Re: Phi - The Golden Ratio Part I

01/16/2008 3:08 PM

Molefex,

Yes, Φ is an irrational (nonterminating as you put it) number, just like π and e.

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#5

### Re: Phi - The Golden Ratio Part I

01/12/2008 3:27 PM

Hi Roger,

This blog reminded me of a document I wrote about number theory. It includes some history of mathematics and the following topics:

Prime Numbers

The Golden Ratio

Fibonacci Numbers

Calculations of pi

The Uniqueness of e

The Moebius Strip

I don't want to steal your thunder, so I thought I would ask if you want me to share any of it. If so, how is the best way to do it? I could submit the golder ratio part here if you want, but I will wait until you respond. Each section is quite short.

Regards,

S

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#7

### Re: Phi - The Golden Ratio Part I

01/16/2008 3:12 PM

StandardsGuy,

Please feel free to add whatever you like. At the very least you should provide links to these pages so I can see them. I don't mind you stealing my thunder, but getting me interested and not providing links so I can give it a read is not right ;)

Roger

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#9

### Re: Phi - The Golden Ratio Part I

01/16/2008 7:53 PM

Roger, I said I'd wait until you wanted it, but the formulas didn't translate so I will have to print it out then re-do the formulas. I'll get to it as soon as I can. There are no links, as this is not on the web yet. I will start posting some in a separate discussion.

S

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#10

### Re: Phi - The Golden Ratio Part I

01/17/2008 10:18 AM

I look forward to it. Just so you know, I was kidding in my previous comment, thus the ;) at the end.