1.35 Exercise 1.35

We can see that the golden ratio is the fixed point of the function f(x) = x + 1/x simply by evaluating:

> (define (golden-transform x) (+ 1 (/ 1 x)))
> (define golden-start 1.61803398875)
> (golden-transform golden-start)
1.6180339887498547
> (golden-transform (golden-transform golden-start))
1.6180339887499102
> (golden-transform (golden-transform (golden-transform golden-start)))
1.6180339887498891
> (golden-transform (golden-transform (golden-transform (golden-transform golden-start))))
1.6180339887498971

This can also be derived simply from the definition of the golden ratio, which relates a to b by (a + b) / a = a / b = phi. The left side can be expanded to a / a + b / a, or 1 + 1 / (a / b) = 1 + 1 / phi.

We can then write a procedure to compute the golden ratio using fixed-point as such:

(define golden-ratio
(fixed-point golden-transform 1))

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1	Chapter 1
2	Chapter 2
3	Chapter 3
4	Chapter 4
5	Chapter 5

1.1	Exercise 1.1
1.2	Exercise 1.2
1.3	Exercise 1.3
1.4	Exercise 1.4
1.5	Exercise 1.5
1.6	Exercise 1.6
1.7	Exercise 1.7
1.8	Exercise 1.8
1.9	Exercise 1.9
1.10	Exercise 1.10
1.11	Exercise 1.11
1.12	Exercise 1.12
1.13	Exercise 1.13
1.14	Exercise 1.14
1.15	Exercise 1.15
1.16	Exercise 1.16
1.17	Exercise 1.17
1.18	Exercise 1.18
1.19	Exercise 1.19
1.20	Exercise 1.20
1.21	Exercise 1.21
1.22	Exercise 1.22
1.23	Exercise 1.23
1.24	Exercise 1.24
1.25	Exercise 1.25
1.26	Exercise 1.26
1.27	Exercise 1.27
1.28	Exercise 1.28
1.29	Exercise 1.29
1.30	Exercise 1.30
1.31	Exercise 1.31
1.32	Exercise 1.32
1.33	Exercise 1.33
1.34	Exercise 1.34
1.35	Exercise 1.35
1.36	Exercise 1.36
1.37	Exercise 1.37
1.38	Exercise 1.38
1.39	Exercise 1.39
1.40	Exercise 1.40
1.41	Exercise 1.41
1.42	Exercise 1.42
1.43	Exercise 1.43
1.44	Exercise 1.44
1.45	Exercise 1.45
1.46	Exercise 1.46