A fixed point of a function f (x) is a solution of the equation f (x) = x. For example, the function cos x has one fixed point, approximately .73908, which can be found graphically by intersecting the graphs of y = x and y = cos x; at the intersection point (x, cos x) we have cos x = x: The fixed point of cosine can also be found numerically by hitting the cosine button (in radians, please) on your calculator repeatedly starting from any initial value you wish: x₀, cos x₀, cos(cos x₀), cos(cos(cos x₀)),... will always tend to the fixed point .73908... (try it!).

We will indicate why the concept of a fixed point is important in mathematics, and in particular see how a fractal like the Sierpinski gasket is a “fixed point” which can be approximated by iteration starting from any initial set in the plane. For instance, here are iterations starting from three different sets which in all cases appear to “converge” to the same Sierpinski gasket: