sol-1.35.tex

By definition, the golden ration $\phi$ satisfies the equation $\phi^2 = \phi + 1$, and hence the equivalent equation $\phi = 1 + 1/\phi$.  This means that $\phi$ is a fixed point of the transformation $x \mapsto 1 + 1/x$.
Here is the result of applying the `fixed-point` procedure to this transformation, starting with a first guess of~$1$:
\begtt\scm
(fixed-point (lambda (x) (+ 1 (/ 1 x))) 1.0)
;Value: 1.6180327868852458
\endtt