Fibonacci Phi - The Golden Ratio

1. Oct 11, 2005

Fractal Freak

Anyone else fascinated with the Golden Ratio (Phi)? It seems that there is an underlying principle with everything that is in this world that has some sort of aspect related to Phi. From artwork, proportions of the human body, to the growth rate of biological cells. Everything seems to have this infinite spiraling activity. Just curious if anyone else is as interested in it as I am.

2. Oct 11, 2005

Zurtex

A lot of people really like it, my favourite constant is by far e though.

3. Oct 11, 2005

robert Ihnot

I found it interesting that the phi is related to the 36-72-72 degree triangle. And also, the pentagram: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi2DGeomTrig.html#363672
Once we can get the EDITED: cos(36) = $$\frac{1+\sqrt{5}}{4}$$, which is of quadratic form, we can find the sin(36) and then find the values for 6 degrees, and then using the half-angle formula find 3 degrees.
In general, that is as far as we can go in term of surds and whole degrees.
Gauss, of course, found the 17-gon sides in terms of surds, (http://www.absoluteastronomy.com/encyclopedia/h/he/heptadecagon.htm) but that is very difficult to write out (doesn't evenly go into 360) and in one of his masterpieces, Disquisitiones Arithmeticaes published in 1801, he never went so far as to actually reach the final step in resolving it.

Last edited by a moderator: Apr 21, 2017 at 8:44 PM
4. Oct 12, 2005

Tide

That can't be true ... unless someone added an imaginary part to 36 when no one was looking! ;)

5. Oct 12, 2005

Tzar

Why e? I think pi has to win hands down!

But phi is pretty cool too. Whats the weirdest place that anyone knows where phi comes up in?

6. Oct 12, 2005

robert Ihnot

Tide: Quote:
Originally Posted by robert Ihnot
Once we can get the cos(36) = ...
That can't be true ... unless someone added an imaginary part to 36 when no one was looking!

Sorry! It was divided by 4: $$cos(36)=\frac{1+\sqrt5}{4}$$ (I made the change in the original entry too.)

Last edited: Oct 12, 2005
7. Oct 12, 2005

Zurtex

I just think e is a lot prettier than pi, I also love all the different places it comes up in in basic calculus.

8. Oct 12, 2005

Fractal Freak

OK.....I tried to go to the link that I posted and I don't know where the hell that came from....anyway, this is the actual link.

Last edited by a moderator: Oct 12, 2005
9. Oct 12, 2005

Fractal Freak

By the way, I think one of the most interesting places that phi is found is in the structure of DNA. It measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.
DNA in the cell appears as a double-stranded helix referred to as B-DNA.
This form of DNA has a two groove in its spirals, with a ratio of phi in the proportion of the major groove to the minor groove, or roughly 21 angstroms to 13 angstroms. (Cited from the Phi website)

The Fibonacci series is as follows: 0,1,1,2,3,5,8,13,21,34, etc.. which corresponds to Phi.

10. Oct 12, 2005

tomfitzyuk

Yeah e is much more fascinating, it appears in all sort of places. Most amazingly is places such as expotential growth and when you differentiate logorithmic functions. I suppose those two are similar but it's still amazing.

sin72/sin36 = Phi
2sin54 = Phi

These are things I found out myself :O

11. Oct 12, 2005

Gokul43201

Staff Emeritus
That's a crackpot religious website. Links to such sites at not allowed in these Forums.

Last edited by a moderator: Oct 12, 2005
12. Oct 13, 2005

robert Ihnot

In case one has not gone through with it: 1,1,2,3,5,8,13,21... We look at the formula: $$S_n+S_(n+1)=S_(n+2)$$

This yields: $$\frac{S_n}{S_(n+1)}+1=\frac{S_(n+2)}{S_(n+1)}$$

Or: 1/X+1 =X implies $$X=\frac{1\pm\sqrt5}{2}$$
Taking the positive value,1.618...this ratio is so good that it easily finds the next Fibonacci number with round off, even starting with the second 1: 1xPhi=1.62=2, 2xPhi =3.236=3, 3xPhi=4.85=5, etc. And 21xPhi = 33.9=34, the next Fibonacci number.

We also have Binet's Formula for the Nth Fibonacci number:

$$F(N) =\frac{Phi^N (-1)^{N+1} Phi^{-N}}{\sqrt5}$$

Last edited: Oct 13, 2005
13. Oct 13, 2005

Tide

You left out a + sign after the $\phi^n$ term.

14. Oct 13, 2005

robert Ihnot

Tide: You left out a + sign...

It seems clearer this way, thank you!

$$F(N) =\frac{Phi^N+ (-1)^{N+1} Phi^{-N}}{\sqrt5}$$

Last edited: Oct 13, 2005
15. Oct 13, 2005

Moo Of Doom

It would probably look even better if you used \phi:
$$F(N) =\frac{\phi^N+ (-1)^{N+1} \phi^{-N}}{\sqrt5}$$
:)

16. Oct 13, 2005

robert Ihnot

Moo of Doom: It would probably look even better if you used \phi:

That's a good idea. Live and learn I always say!

$$F(N) =\frac{\phi^N+ (-1)^{N+1} \phi^{-N}}{\sqrt5}$$

17. Oct 14, 2005

Tide

I think it looks better this way:

$$F_n = \frac {\phi^n - (-\phi)^{-n}}{\sqrt 5}$$

;)

18. Oct 14, 2005

robert Ihnot

Tide: I think it looks better this way:

Certainly does. Actually my form came about late at night when I was afraid that a lot of latex would be written if I had to invert Phi and rationalize the denominator, etc. So I looked for a way out.

In case of further difficulity over the question of capitalizing Phi, I will quote:
"We will call the Golden Ratio (or Golden number) after a greek letter,Phi () here, although some writers and mathematicians use another Greek letter, tau (). Also, we shall use phi (note the lower case p) for a closely related value." http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#golden

Last edited by a moderator: Apr 21, 2017 at 8:52 PM
19. Oct 14, 2005

matt grime

Whilst phi certainly does have things to do with nature, whcih are well known and understood (there is a notion of speed of convergence and phi has some special properties there that mean sunflower seeds for example are dstributed approximately as a fibonnacci sequence; this ensures best exposrure for least coverage) but most of what you cite as frequent occurences are famously red herrings. There is no use of phi in art or sculpture by the greeks: pots came in many sizes, and there is no reason to suppose that rations of phi have ever been thought "more attractive". Anatomical references are not accurate either. It so happens that if you divide something into two roughly equal parts then they may lbe close to phi, an number that is close to 1.

20. Oct 22, 2005

Mike AkA Don

I too find it fascinating. http://goldennumber.net/ has quite a lot on it and relates it to art it really is incredible. I did quite a lot of research on it after reading The Da Vinci Code.