Pressing Cos Until the Stick Comes Home

Set a calculator to radians, type any number, and press cos over and over. The display wanders for a moment, then locks: 0.7390851… It doesn’t matter where you started. You’ve found a value the function no longer moves — a value where cos(x) = x. A fixed point.

I kept circling that on the bench today, sanding a hair of dihedral into a birch boomerang. A tuned returning boomerang is a stick whose flight brings it back to the hand that threw it. The launch point is, near enough, a spot the flight leaves unmoved: throw, loop, return, and you’re standing where you began. An untuned one sails straight off over the fence — the calculator equivalent of a function that never settles.

(defn fixed-point [f x0 tol]
  (->> (iterate f x0)
       (partition 2 1)
       (drop-while (fn [[a b]] (> (Math/abs (- a b)) tol)))
       ffirst))

(println (fixed-point #(Math/cos %) 1.0 1e-10))
;; => 0.7390851331706995

iterate builds the lazy infinite chain x, f(x), f(f(x)), … and I walk it in overlapping pairs until two neighbours agree to ten decimals. Ada, born the same decade this idea got its canonical classroom treatment, writes the honest imperative version — hold the previous value, overwrite, stop once the change drops under tolerance:

with Ada.Text_IO; use Ada.Text_IO;
with Ada.Numerics.Elementary_Functions;
use Ada.Numerics.Elementary_Functions;

procedure Fixed_Point is
   X    : Float := 1.0;
   Prev : Float;
begin
   loop
      Prev := X;
      X    := Cos (X);
      exit when abs (X - Prev) < 1.0e-6;
   end loop;
   Put_Line (Float'Image (X));  --  7.39085E-01
end Fixed_Point;

Same attractor, reached from the same start, by two very different machines.

The parallel runs past the landing spot. Carving the boomerang is fixed-point iteration on a slower clock: throw, watch it fall short of the loop, bend the arm up two millimetres, throw again. Each pass applies one correction, and I’m hunting the shape where the correction stops asking for more. The calculator converged in six presses. The birch has taken all afternoon and still wants another millimetre.