Tap any number into your calculator. Yes, any number at all, plus or minus, big or small. Now tap the cosine button. You will get a number in the range [ -1, +1 ]. Now tap “cos” again and again, and keep tapping it repeatedly (make sure that angles are set to radians and not degrees). The result is a sequence of numbers that converge towards the value** 0.739085 …** .

What you are calculating is the iterated cosine function

cos ( cos ( cos ( … cos ( *x*0 ) … ) ) ).

where x0 is the starting value. As the process converges, you have found the solution of the equation

cos( *x *) = *x*

This is called a *fixed point* of the cosine mapping. If we plot the two functions *y* = *x* and *y* = cos( *x *), we see that they intersect in just one point, x=0.739085 … . **Cobweb Plot** We can represent the iterative process of repeating cosine calculations using a diagram called a cobweb plot. On this visual plot, a stable fixed point gives rise to an inward spiral. For a function *f *( *x *), we plot both the diagonal *y* = *x* and the curve *y* = *f *( *x *). We are looking for a point where the two graphs cross. Let us start with a point (*x*0, *x*0) on the diagonal. For the picture below, we choose ( 0, 0 ). Moving vertically to the curve y = f ( x ), we get the point ( *x*0, *f *( *x*0 ) ). Now move horizontally to the diagonal again, to ( *f *( *x*0 ),* f *( *x*0 ) ). Moving vertically and horizontally again between the curves, we get (*f *( *x*0 ), *f *( *f *( *x*0 ) ) ) and ( *f *( *f *( *x*0 ) ), *f *( *f *( *x*0 ) ) ). By means of these alternately horizontal and vertical moves, we generate the sequence

{ *x*0, *f *( *x*0 ), *f *( *f *( *x*0 ) ), *f *( *f *( *f *( *x*0 ) ) ), … }

Under favourable circumstances, the sequence converges to the fixed point of the mapping *y* =* f *( *x *). The process is illustrated in the following diagram, starting at ( 0, 0 ) and homing in on the fixed point. Fixed point theorems (FPTs) give conditions under which a function* f *( *x *) has a point such that* f* ( *x* ) = *x*. FPTs are useful in many branches of mathematics. Amongst the most important examples are **Brouwer’s FPT**. This states that for any continuous function *f *( *x* ) mapping a compact convex set into itself, there is a point *x*0 such that *f* ( *x*0 ) =* x*0. The simplest example is for a continuous function from a closed interval *I* on the real line to itself. More generally, Brouwer’s theorem holds for continuous functions from a convex compact subset K of Euclidean space to itself. Another FPT is that of Stefan Banach. In technical terms, this states that a contraction mapping on a complete metric space has a unique fixed point.

[Next week: Brouwer’s Fixed Point Theorem]

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Peter Lynch’s book about walking around the coastal counties of Ireland is now available as an ebook (at a very low price!). For more information and photographs go to http://www.ramblingroundireland.com/

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