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        general >> truth >> Golden Ratio or Phi 
        
(Message started by: Mugwump101 on Dec 6th, 2006, 1:20am)
Title: Golden Ratio or Phi
Post by Mugwump101 on Dec 6th, 2006, 1:20am
I'm looking for more information of where Phi or the Golden Ratio is found (I.e. exotic or creative places not like the vitruvian man or obvious) and how it was found.

Any ideas?
Title: Re: Golden Ratio or Phi
Post by towr on Dec 6th, 2006, 1:48am
You can find it in a lot of ancient greek architecture. Various places in math (like geometry, e.g. pentagrams); if you look hard enough you can find it in pretty faces. If you really look hard enough, everywhere (But that's just the aneristic principle at work, really)

I'm not sure what exactly you want to know though. Of course wikipedia is a good start.
Title: Re: Golden Ratio or Phi
Post by rmsgrey on Dec 6th, 2006, 7:05am
It turns up (as an angle) in plants because it's very irrational - spacing shoots the golden angle apart means they overlap as little as possible.
Title: Re: Golden Ratio or Phi
Post by THUDandBLUNDER on Dec 6th, 2006, 7:37am
Here is a good source:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/
Title: Re: Golden Ratio or Phi
Post by Mugwump101 on Dec 8th, 2006, 3:32am
Actually more so, I'm creating a mail merge for an Excel Project and We're trying to send all the people a letter or basically advertisement for an Amusement Park that involves the Golden Ratio. Any ideas for Rides?

I started out with a DNA rollar coaster and a beach with seashells, waves, lanterns, chairs that embody phi.

Do any more brilliant ideas?
Title: Re: Golden Ratio or Phi
Post by Whiskey Tango Foxtrot on Dec 8th, 2006, 7:29am
Pretty much anything can implement the golden ratio if you choose it to do so.  Take a ferris wheel as an example.  The dimensions of the cars on a ferris wheel could be determined by the Golden Ratio.  The length and width of the spokes (or whatever they're called) could also do the same.  Even the queue lines to get on the rides could be constructed using the Golden Ratio.  There are an infinite number of applications here.
Title: Re: Golden Ratio or Phi
Post by Aurora on Mar 1st, 2008, 6:58am
I found  this website the other day whilst researching Phi for art coursework. There are quite a few examples of where it can be found.

http://goldennumber.net/
Title: Re: Golden Ratio or Phi
Post by Master of Everything Unknown on Mar 9th, 2008, 5:09pm
turns out its 42
Title: Re: Golden Ratio or Phi
Post by Roy on Mar 18th, 2008, 11:29pm
I have a book by Mario Livio about the history of the Golden Ratio, it's origin etc. it's quite good
Title: Re: Golden Ratio or Phi
Post by BenVitale on Jun 5th, 2009, 12:42pm

on 12/06/06 at 07:37:11, THUDandBLUNDER wrote:
Here is a good source:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/


Yes, I like this source. Then I went to another page of the same website:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html

See under "Similar Numbers"
about other numbers that have the Phi property that when you square them their decimal parts remain the same.

series of number here is 5, (9), 13, 17, 21, (25), 29, ... which are the numbers that are 1 more than the multiples of 4.

I searched for this series on the "The On-Line Encyclopedia of Integer Sequences" but couldn't find it.

Did I miss it?
Title: Re: Golden Ratio or Phi
Post by 0.999... on Jun 17th, 2009, 2:19pm

on 03/18/08 at 23:29:20, Roy wrote:
I have a book by Mario Livio about the history of the Golden Ratio, it's origin etc. it's quite good

I have it and agree.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 2nd, 2010, 1:24pm
Suppose a Fibonacci sequence starts with (a,b), that is to say:

(a, b, a+b, a+2b, 2a+3b, 3a+5b, ..., Fn-2 a + Fn-1 b, ...)

with

f0 = a,
f1 = b,
f2 = a+b,
f3 = a+2b,
f4 = 2a+3b,
f5 = 3a+5b,
....................................
fi = Fi-2 a + Fi-1 b

and the value of fi given, say, 104 = 24 * 54

What are the values of f0 and f1 ?
Title: Re: Golden Ratio or Phi
Post by towr on Feb 2nd, 2010, 2:40pm

on 02/02/10 at 13:24:33, BenVitale wrote:
and the value of fi given, say, 104 = 24 * 54

What are the values of f0 and f1 ?
There is no way to tell if you're only given one fi

For example, if f2=x, then for any a  f0=a, f1x-a works.
And obviously you can work backwards for later i in a similar way.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 2nd, 2010, 2:58pm

on 02/02/10 at 14:40:12, towr wrote:
There is no way to tell if you're only given one fi


Could we use the Index shift rule to determine the first two terms of this sequence?
I thought I could, but I got stuck ... so I posted this problem, here, requesting help.


Quote:
For example, if f2=x, then for any a  f0=a, f1x-a works.
And obviously you can work backwards for later i in a similar way.
Title: Re: Golden Ratio or Phi
Post by rmsgrey on Feb 2nd, 2010, 5:01pm

on 02/02/10 at 14:58:04, BenVitale wrote:
Could we use the Index shift rule to determine the first two terms of this sequence?
I thought I could, but I got stuck ... so I posted this problem, here, requesting help.

What towr was trying to convey is that for any given fi, you can choose any value you want for fi-1 and that will give you a different (but still valid) sequence.


Another way of looking at it is that you have one equation in two unknowns:
fi = Fi-2a + Fi-1b
where everything but a and b is known.

Adding in an expression for any other term of the sequence adds another equation and another unknown (that term of the sequence) so doesn't help make the system of equations any more solvable.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 2nd, 2010, 5:49pm
Oh, I see. I'm trying to be creative with the Fibonacci series. And, I was trying to figure out a formula to test a number with a Fibo that starts with (a,b)

We know that in the Fibo series that starts with (1,1), that is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ...

N is a Fibonacci number if and only if 5N2 + 4 or 5N2 – 4 is a square number

What would be the formula to test numbers to see if they belong in Fibo (a,b)?
Title: Re: Golden Ratio or Phi
Post by JohanC on Feb 3rd, 2010, 3:22am

on 02/02/10 at 13:24:33, BenVitale wrote:
Suppose a Fibonacci sequence starts with (a,b), that is to say:

(a, b, a+b, a+2b, 2a+3b, 3a+5b, ..., Fn-2 a + Fn-1 b, ...)

with

f0 = a,
f1 = b,
f2 = a+b,
f3 = a+2b,
f4 = 2a+3b,
f5 = 3a+5b,
....................................
fi = Fi-2 a + Fi-1 b

and the value of fi given, say, 104 = 24 * 54

What are the values of f0 and f1 ?

A more tricky variant on this question would be:
what is the largest i for which the series exists entirely of positive numbers?
Title: Re: Golden Ratio or Phi
Post by pex on Feb 3rd, 2010, 3:48am

on 02/03/10 at 03:22:51, JohanC wrote:
A more tricky variant on this question would be:
what is the largest i for which the series exists entirely of positive numbers?

For fi = 10000, I find i = [hide]13[/hide] for (a, b) = [hide](80, 20)[/hide] by a simple exhaustive search. I don't think it's a coincidence that for these (a, b), [hide]fi-1 is approximately 10000 / Phi[/hide].
Title: Re: Golden Ratio or Phi
Post by towr on Feb 3rd, 2010, 4:23am

on 02/02/10 at 17:49:19, BenVitale wrote:
What would be the formula to test numbers to see if they belong in Fibo (a,b)?

fn ~= (a+bhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gif)/sqrt(5) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/phi.gifn-2, so if a,b are given it's simple enough.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 3rd, 2010, 1:18pm
Thanks to all of you for the contributions.

post deleted

Reason: Basically, I asked how was the formula (5N2 + 4 or 5N2 – 4 is a square number) constructed?

I found the construction of the formula.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 20th, 2010, 2:53pm
This site suggests that there is a relationship between Fibonacci series and Stock Market prices

http://goldennumber.net/stocks.htm

What do you think?
Title: Re: Golden Ratio or Phi
Post by towr on Feb 20th, 2010, 3:05pm

on 02/20/10 at 14:53:13, BenVitale wrote:
What do you think?
I think they're mad.
Title: Re: Golden Ratio or Phi
Post by Grimbal on Feb 23rd, 2010, 2:14am
Not mad, just salespeople.

The madmen are those who buy from them.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Feb 23rd, 2010, 10:51am
Yes, I agree. It is a mad attempt. It is the behavior of a snake oil salesmen.

This shows our deep need for control. We are in a deep recession, and we feel out of control. We feel fear.

From an evolutionary standpoint, if we are in control of our environment, then we have a far better chance of survival.

The owners of that website are selling a software. Either they believe in their product or are just dishonest. They know that the stock market is driven by fear and greed.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Sep 3rd, 2011, 2:06pm
Mathworld gives the following Golden Ratio Approximations

http://mathworld.wolfram.com/GoldenRatioApproximations.html

The first two approximations:
(5*pi/6)^.5
http://www.wolframalpha.com/input/?i=%285*pi%2F6%29^.5
or here http://tinyurl.com/4y66rat

(7*pi/5*e)
http://www.wolframalpha.com/input/?i=%287*pi%29%2F%285*e%29

I found a curious approximation where only the digit 5 is used:

5^.5*.5+.5

http://www.wolframalpha.com/input/?i=+5^.5*.5%2B.5
or here http://tinyurl.com/3as4uau
Title: Re: Golden Ratio or Phi
Post by towr on Sep 3rd, 2011, 2:21pm

on 09/03/11 at 14:06:33, BenVitale wrote:
I found a curious approximation where only the digit 5 is used:

5^.5*.5+.5
That's not an approximation, that's sqrt(5)/2+1/2 = phi, exactly.
Title: Re: Golden Ratio or Phi
Post by BenVitale on Sep 3rd, 2011, 3:05pm

on 09/03/11 at 14:21:28, towr wrote:
That's not an approximation, that's sqrt(5)/2+1/2 = phi, exactly.


Right. Sorry. Silly of me! I don't know what happened... perhaps I spent too much time playing on wolframAlpha trying to find something


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