A while back I shared some Perl code for calculating the circumference of a circle without knowing π. Just for fun, and due to my longtime infatuation with all things Schemish, I’ve written a little pi approximator in Scheme. It uses the idea that we can approximate a circle using smaller and smaller triangles stacked on top of each other. (See previously for a better explanation with a picture.)

And now, the code!

;;;; pi.scm -- Estimate the value of π using smaller and smaller
;;;; triangles.
;;; Call it like so: (pi-estimate n), where n is the number of
;;; iterations you'd like to go through. It doesn't take many to get
;;; pretty accurate.
(define reference-pi 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679)
(define (circumference radius)
(* 3.14159 2 radius))
(define (square x)
(* x x))
(define (hypotenuse a b)
(sqrt (+ (expt a 2) (expt b 2))))
(define (pi-iter radius a b count maxcount)
(let* ((hyp (hypotenuse a b))
(newbase (- radius (sqrt (- (square radius)
(square (/ hyp 2)))))))
(if (= count maxcount)
(/
(* 2 hyp (expt 2 (+ 1 count)))
(* 2 radius))
(pi-iter radius newbase (/ hyp 2) (+ count 1) maxcount))))
(define (pi-estimate iterations)
(pi-iter 128 128 128 0 iterations))

(Origami image courtesy Melisande under Creative Commons license.)

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