The "golden 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, not a 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)= w2 or hw- h2= w2. Dividing both sides by h2, (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.
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.