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madmike159

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madmike159

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symbolipoint

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matt grime

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The golden ratio a/b is defined by the rule that a/b = (a+b)/a. If we set b=1, then a=(a+1)/a, and we see that it is the larger of the two roots of x^2-x-1.

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Continuing with that idea, then you can re-write this: " a+b is to segment a, as a is to the shorter segment b."

As

The total lineament is to the longer sub line-segment, as the longer sub line-segment is to the the shorter sub line-segment.

I hope that made sense.

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madmike159

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Thanks I understand now. I was mostly confused how they got from (a+b)/a to a/b.

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phi-1 = 1/phi.

Now, thats all fine and good, but something interesting happens when you take the continued fraction expansion:

phi = 1+1/phi = 1+1/(1+1/phi) = 1+1/(1+1/(1+1/phi)) = 1+1/(1+1/(1+1/(1+1/(...))))

It is the only number whose continued fraction expression is 1 1 1 1 1 1 1 1....

This can be used to show that phi's rational approximation converges as slowly as possible, meaning that in some sense that phi is "the most irrational number". Now that is interesting!

http://www.ams.org/featurecolumn/archive/irrational1.html

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Find two numbers that have a difference of 1, and when multiplied together equal 1.

The numbers are 1.61803.... and .61803.... ; the golden ratio being former.

Another way to find the golden ratio is to use the Fibinachi sequance, where every term is the sum of the previous two terms, starting with 1,1

1,1,2,3,5,8,13,21,34,55.......

The golden ration can be approximated by the ratio of any two succesive numbers in this sequance. The approximation gets better the larger numbers you use.

ie. 55/34 = 1.617....

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HallsofIvy

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The "height to width" ratio of the first rectangle is h/w, the "height to width" ratio of the second is w/(w-h). If those are the same h/w= w/(w-h). Taking a= w-h and b= h, then a+ b= w-h+ h= w so h/w= (a+b)/b and w/(x-h)= b/a: the proportion h/w= w/(w-h) is (a+b)/b= b/a.

From h/w= w/(w-h) we can multiply both sides by w(w-h) and get h(w-h)= w

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madmike159

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Wow that is confusing. I'm going to have another look in the moarning =D. Thanks for the help every one. I think I know enough for what I need to do, but any more places where the GR appears would be nice to know (wiki article said something about the golden ratio in trees and humans =S).rectangle" (it is said that the side view of the Parthenon in Athens is a golden rectangle and that DaVinci's painting "The Last Supper" is in the proportions of a golden rectangle. More generally it is claimed that the golden rectangle has the "most elegant" proportion of any rectangle. That is, of course,nota mathematical claim.) is a rectangle, with width w and height h, such that, if you mark off distance h from one corner of the width and draw a perpendicular to make a new rectangle; that is, construct a new rectangle having width w-h and height w, the ratio of "height to width" is still the same: you have constructed a new golden rectangle.

The "height to width" ratio of the first rectangle is h/w, the "height to width" ratio of the second is w/(w-h). If those are the same h/w= w/(w-h). Taking a= w-h and b= h, then a+ b= w-h+ h= w so h/w= (a+b)/b and w/(x-h)= b/a: the proportion h/w= w/(w-h) is (a+b)/b= b/a.

From h/w= w/(w-h) we can multiply both sides by w(w-h) and get h(w-h)= w^{2}or hw- h^{2}= w^{2}. Dividing both sides by h^{2}, (w/h)- 1= (w/h)^{2}. (w/h)^{2}- (w/h)+ 1= 0. Using the quadratic formula to solve that equation gives phi as the positive solution for the ratio w/h.

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It's actually very simple. It's a rectangle which you try to make square by cutting out the excess material. Try to picture:Wow that is confusing.

- a square of dimensions a x a;

- a rectangle which is (a+b) x a (so that the square fits in);

- the remainder of the rectangle, after you cut out the square, which is a smaller rectangle of dimensions a x b.

Try this:

http://golden-rectangle.lcpdesign.com/construct.htm [Broken]

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madmike159

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Yea it is really easy. Best not to do maths late at night though. Thanks for the help everyone.

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