Here's a cute procedure you might enjoy if you haven't seen it before.
We want to write a fixpoint operator, such that

        ((fix G) x y z ...) = ((G (fix G)) x y z ...)

That is, such that

        (apply (fix G) (list x y z ...)) = (apply (G (fix G)) (list x y z ...))

So using unrestricted lambda, one simply writes down the right hand side,
naming the list args (or whatever):

(define fix
  (lambda (G)
    (lambda args
      (apply (G (fix G)) args))))

Then you can use this to do the usual stuff you can do with a fix
point combinator.  For example...

(define fact-gen
  (lambda (fact)
    (lambda (n)
      (if (zero? n)
          1
          (* n (fact (- n 1)))))))

(define factorial (fix fact-gen))

Of course, you can write fix without using recursion
(the famous Y combinator).  But if you're just interested
in taking fixpoints for computation, and not in explaining
the semantics of features like define and letrec,
then the above definition of fix is adequate.

        Gary Leavens

--
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