Golden Ratio 'definition'
cr
> The formal definition of the Golden Mean is (1 + SQRT(2)/2.
> ...
Why? I thought the Golden Ratio (assuming it's the same as the Golden Mean),
is defined to be the ratio of the longer of two parts of a line segment to
the other part, when it is the same as the ratio of the whole line segment
to the longer part. No doubt that can be stated more elegantly, but isn't
that it?
If the whole line segment in question is of length 1, and the two parts are
x (longer) and y (shorter), then
1 is to x as x is to y
or
1 is to x as x is to 1-x
or
1/x=x/(1-x)
thus
1-x=x2.
Maybe this works out to the 'formal definition' above, and thus shows the
irrationality of phi, but I don't understand why a concept such as the
Golden Ratio should be assigned a 'definition' which has no intuitive relation
to the concept behind it.
-cr
flip
news:22680de.03030...@posting.google.com...
> Frank C. Post (fcp...@medea.gp.usm.edu) wrote:
> > The formal definition of the Golden Mean is (1 + SQRT(2)/2.
Mean),
> is defined to be the ratio of the longer of two parts of a line segment to
> the other part, when it is the same as the ratio of the whole line segment
> to the longer part. No doubt that can be stated more elegantly, but isn't
> that it?
> irrationality of phi, but I don't understand why a concept such as the
> Golden Ratio should be assigned a 'definition' which has no intuitive
relation
> to the concept behind it.
I like the following sites and think they answer your question.
1. http://mathworld.wolfram.com/GoldenRatio.html
2.
http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Golden_ratio.html
3. http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
4. http://galaxy.cau.edu/tsmith/KW/golden.html
5. http://brian.quentin.students.noctrl.edu/webproject.htm
6. http://mathforum.org/dr.math/faq/faq.golden.ratio.html
7. http://astronomy.swin.edu.au/~pbourke/analysis/phi/
Not sure that helps answer your question, but think it does.
Flip.
George Cox
>
> Frank C. Post (fcp...@medea.gp.usm.edu) wrote:
> > The formal definition of the Golden Mean is (1 + SQRT(2)/2.
> > ...
>
> Why? I thought the Golden Ratio (assuming it's the same as the Golden Mean),
> is defined to be the ratio of the longer of two parts of a line segment to
> the other part, when it is the same as the ratio of the whole line segment
> to the longer part. No doubt that can be stated more elegantly, but isn't
> that it?
>
> If the whole line segment in question is of length 1, and the two parts are
> x (longer) and y (shorter), then
>
> 1 is to x as x is to y
>
> or
>
> 1 is to x as x is to 1-x
>
> or
>
> 1/x=x/(1-x)
>
> thus
>
> 1-x=x2.
>
> Maybe this works out to the 'formal definition' above, and thus shows the
Well, let's see if it does. Your equation is x^2 + x - 1 = 0 which has
the solutions:
-1 +/- sqrt(1^2 - 4*1*(-1)) -1 +/- sqrt(5) -1.6180...
x = --------------------------- = -------------- = or
2 2 0.6180...
So, you or Frank got it wrong. The geometric definition (a perfecty
good one) leads to an equation which leads to a couple of numbers (no
less good). Question for you: why _two_ numbers?
GC
cr
> Well, let's see if it does. Your equation is x^2 + x - 1 = 0 which has
> the solutions:
> -1 +/- sqrt(1^2 - 4*1*(-1)) -1 +/- sqrt(5) -1.6180...
> x = --------------------------- = -------------- = or
> 2 2 0.6180...
> So, you or Frank got it wrong. The geometric definition (a perfecty
> good one) leads to an equation which leads to a couple of numbers (no
> less good). Question for you: why _two_ numbers?
I suppose because the general form of the quadratic equasion has two solutions.
Intuitively, -1.618... is the negative of the ratio of x to 1-x, and
0.618... is the ratio of x to 1, by my original definition of x.
-cr
cr
was a typo:
Frank C. Post (fcp...@medea.gp.usm.edu) wrote:
> The formal definition of the Golden Mean is (1 + SQRT(2)/2.
> ...
Probably he meant (1 + SQRT(5))/2. But my point was not whether the
formula was correct or not. My point was that I think it should have been
called the -value- of phi, not the 'formal definition' of phi. The fact that
it was a typo that I had not detected merely confused the issue.
-cr