Phi (fie) is the number 1.618033.... it is like pi but it is totally different numbers. They both never end.
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The main use for the golden ratio is its aesthetic appeal - in art and architecture. Rectangles with the golden ratio as their aspect appeal to the human mind (for some reason). So various aspects of the Parthenon in Athens, for example, have dimensions whose ratio is phi. Phi is closely related to the Fibonacci sequence: the ratio of successive terms of the sequence approaches phi and so, just like the Fibonacci sequence, phi appears in many natural situations. However, there is no particular application based on phi.
It is the set comprising the following 4 elements:phi,{phi},{{phi}} and{phi, {phi}}
cos(phi - 1) = cos(phi)cos(1) + sin(phi)sin(1)
(a+b)/a=a/b=phi (the golden ratio, as defined) (a+b)/a=phi (we'll solve this equation) 1+b/a=phi (just changing the form of the left side a little) 1+1/phi=phi (a/b=phi so b/a=1/phi) phi+1=phi2 (multiply both sides by phi) phi2-phi-1=0 (rearrange) From here, we can use the quadratic equation to find the positive solution: phi=(-b+√(b2-4ac))/(2a) phi=(1+√(1+4))/2 phi=(1+√5)/2≈1.618
I don't know the exact numbers without using a calculator but they aproach phi, 1.6180339..., if that helps you at all. All sequences that use the formula Xn-1+Xn-2= Xn approach phi.