Is Phi More Than Just a Fun Mathematical Curiosity?

[tony873004]

Is there anything significant about phi, other than 1 + phi = phi^2 ?

Is this just fun number trivia, or is phi actually useful to science, etc...?

 

[Hootenanny]

Phi, search for it, the 'divine proportion' apparently. If you draw a regular pentangle, the lengths of the sides are divided by the intersections are exactly in the ratio of phi. Objects made in proportion using phi are apparently the most aesthetically pleasing.

 

[arunbg]

Read up the Da Vinci Code .

 

[DeadWolfe]

Read up the Da Vinci Code .

No. You might as well recommend that he watch the movie "Pi"

If you wish to learn about math, then indeed, there really is nothing special about phi except what you mentioned.

 

[matt grime]

There is nothing at all that supports the assertion that things in the proportion of phi are aesthetically pleasing more than any other ratio that is close to 3/2.

Oh, and don't read the crap novel of dan brown.

Finally, yes, phi is important in nature. It is related to the Fibonacci sequnce and has the slowest converging continued fraction (making it mathematically distinguished) explaining its appearance in sunflower seeds and distributions of leaves on stems.

 

[fourier jr]

it has obvious importance for artists & architects, or anyone else interested in harmony or proportion. I'm not sure how important it is to math though, except it has a place in math history. i know a pentagram (aka pentacle) used to mean good fortune or protection in ancient times & was also the 'secret sign' of the pythagoreans because the golden ratio/proportion occurs all throughout it. practically everywhere two segments intersect they divide each other in the divine proportion.

gyorgy doczi (an architect) wrote a good book called the power of limits which has good stuff on phi. matila ghyka's geometry of art & life is also a classic.

 

[tony873004]

Thanks! I'm glad I asked. This thread is far more informative than the Google's top 10 links searching for phi.

 

[matt grime]

it has obvious importance for artists & architects, or anyone else interested in harmony or proportion.

why? prove it (the harmony part)

i'm not sure how important it is to math though, except it has a place in math history.

reasonably, thought not as important as pi or e.

 

[fourier jr]

why? prove it (the harmony part)

i don't know what exactly what I'm supposed to 'prove'... i didn't mean harmony in music specifically, but the general idea of things "looking right" or "sounding right". good ol wikipedia has some stuff about the uses of phi in aesthetics
http://en.wikipedia.org/wiki/Golden_ratio

 

[matt grime]

Yes, I know what you meant, I am asking you to justify why you said it.

A list of examples would suffice, or some reasonable links (and no wikipedia is not reasonable) to support the position. Evidence in other words. Ideally a statistically significant sample to show that humans have a preference for this. I've often heard this stated but never seen any justification of it and indeed writers I would trust more than wikipedia have stated that there is no reason to think that this ratio was preferred over others.

Here for instance (the MAA server isn't responding, here's a cached version

http://66.249.93.104/search?q=cache...+devlin+golden+ratio&hl=en&gl=uk&ct=clnk&cd=1

http://www.dur.ac.uk/bob.johnson/fibonacci/miscons.pdf is a link to the article Devlin mentions, sorry it's in pdf - it is the first hit in googlewhen searching for the author and article name so you can convert it to html if you prefer)

Here for instance is a quote to support my position taken from p 13 in the pdf

Schifmann and Bobko find

"Research on the golden proportion as an empirically demonstrable preference has mostoften been applied to the rectangle where the results, on the whole, are negative"

 

[James R]

Phi has lots of interesting properties. For example:

$$\phi = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + ...}}}$$

 

[arunbg]

I agree Dan Brown's novels are crap, but still that's where I first came to know of phi and the convergence in the Fibonacci series, though there is little else regarding mathematics . What was I thinking !
I apologise ( hangs head in shame) .

 

[phibonacci]

This sites has some data on phi: http://goldennumber.net/

 

[physmike]

Phi

Is there anything significant about phi, other than 1 + phi = phi^2 ?

Is this just fun number trivia, or is phi actually useful to science, etc...?

tony, from another PF thread,

fine-structure constant,
...with fibonacci and golden ratio.


α = 7.297 352 568 x 10-3 [1]


(α/3)^-12 = 2.33061803... x 10^31

(α/3)^-12 = (233 x 10^29) + (61803... x 10^23)


This result was inspired by the harmonic system of William B. Conner [2].
A harmonic scale of 12 "musical intervals" from 144, ..., 233, ...270.
Fibonacci number, 233, is in the 13th place of the series that begins with
1,1,2,3,..., and 89 + 144 = 233. 233 is the tone SE in the harmonic system.
144 is the fundamental tone DO, light harmonic, and a decagon
angle. 144^1/2 = 12 & 27^1/3 = 3 . 270 is the tone of "action" TI,
and 27 is the "time" harmonic. Inverse golden ratio, φ^-1 = 0.61803...


According to Conner, 233 represents, among other things here;
the minimal compression density of the formative forces
in the quadrispiral cycle of interlocking compressive/expansive
vortices.

And the inverse golden ratio reflects the spiral geometry.


[1] http://physics.nist.gov/cgi-bin/cuu/Value?alph|search_for=abbr_in!

[2] Conner, William B. Harmonic Mathematics: A Phi-Ratioed Universe as
Seen through Tone-Number Harmonics. Chula Vista, CA: Tesla Book Company, 1982

https://www.physicsforums.com/showpost.php?p=970914&postcount=220

 

[0rthodontist]

I once read an article in a popular science magazine (Discover or SciAm) about a study of human faces. In the study, subjects viewed various male and female faces and rated them according to attractiveness. The faces most preferred corresponded to a physiological "average" face, which turns out to be one in which the height to width ratio is roughly phi.

 

[Hubert]

Is there anything significant about phi, other than 1 + phi = phi^2 ?

Is this just fun number trivia, or is phi actually useful to science, etc...?

If you're interested in learning about phi, try picking up Mario Livio's excellent book The Golden Ratio. In it, he describes all of the properties/patterns associated with the number (there are a lot more than those already mentioned) and he also dispels some of those rumors relating to phi's "aesthetic properties" and its supposed inclusion in ancient buildings and monuments.

 

[BoTemp]

Careful

I once read an article in a popular science magazine (Discover or SciAm) about a study of human faces. In the study, subjects viewed various male and female faces and rated them according to attractiveness. The faces most preferred corresponded to a physiological "average" face, which turns out to be one in which the height to width ratio is roughly phi.

Don't trust that unless you read the study carefully. People love to attribute mathematical ratios, especially the "golden ratio" to things for no good reason.

The ratio of successive terms in any Fibonnaci sequence tends to phi, and there is actually a closed-form solution for the most famous sequence (1,1,2,3,...) involving phi (can't find a link at the moment).

It does show up in nature sometimes, which is interesting. Sunflower seeds grow in a sunflower in a logarithmic spiral, which is related to the golden ratio. For that reason, the number of seeds in a spiral is always a fibonnaci number. There are other examples; Livio, Mario. "The Golden Ratio: The Story of Phi, the World's Most Astonishing Number." is a good book on the subject.

Really, though, it's more of a mathematical curiosity than anything terribly important. I believe it was the first number to be proven irrational. The greeks proved phi irrational geometrically (how else?), starting with the fact that certain ratios in a regular pentagon are phi. Whether this came before or after proving sqrt(2) irrational I'm not sure.

 

[MeJennifer]

The faces most preferred corresponded to a physiological "average" face, which turns out to be one in which the height to width ratio is roughly phi.

Roughly? That says it all to me and that clearly indicates the predisposition of the "study".

Perhaps the next book from Brown could be something like "The secret magic of Phi".

 

[physmike]

Bode's Law and Phi

Fascinating link on Bode's Law, planetary distances, and phi.

http://www.spirasolaris.ca/sbb4d.html

PART IV. SPIRA SOLARIS ARCHYTAS-MIRABILIS

A. THE FIBONACCI SERIES, PHI-SERIES AND SYNODIC MEAN

 

[shmoe]

The greeks proved phi irrational geometrically (how else?), starting with the fact that certain ratios in a regular pentagon are phi. Whether this came before or after proving sqrt(2) irrational I'm not sure.

I've always thought sqrt(2) was first, but I'm not at all positive.A proven fact about phi- you can "find" it anywhere you like if you start out determined to find it there.

 

[CRGreathouse]

Base-phi numbers are interesting, at least to me. I don't think phi is that important, overall, but it does have some curious properties.

 

[physmike]

Looking for Phi ...

This Week's Finds in Mathematical Physics (Week 203)
John Baez

"The golden number is a great favorite among amateur mathematicians, because it has a flashy sort of charm. You can find it all over the place if you look hard enough - and if you look too hard, you'll find it even in places where it's not.
...

The golden number also shows up in the theory of quantum groups. I talked about this in "week22" so I won't explain it again here. But, I can't resist mentioning that Freedman, Larsen and Wang have subsequently shown that a certain topological quantum field theory called Chern-Simons theory, built using the quantum group SUq(2), can serve as a universal quantum computer when the parameter q is a fifth root of unity. And, this is exactly the case where the spin-1/2 representation of SUq(2) has quantum dimension equal to the golden number!"

math.ucr.edu/home/baez/week203.html