The main idea here is that if you want to find a root of some function

*f*, then you start from a guess

*a*

_{0}; then compute

*a*

_{1}=

*a*

_{0}-

*f*(

*a*

_{0})/

*f*'(

*a*

_{0}); geometrically this corresponds to replacing the graph of the function with its tangent line at (

*a*

_{0}, f(

*a*)

_{0}) and finding the root of the tangent line. Then starting from

*a*

_{1}we find

*a*

_{2}and so on. If you're already close to a root you'll get closer. But if you're far away from a root unexpected things can happen; the set of all starting points

*a*

_{0}for which the sequence (

*a*

_{0},

*a*

_{1},

*a*

_{@},

*a*

_{3}, ...) converges to a given root of

*f*is fractal.

(I've mentioned this before, and so has John Armstrong, but Tatham's pictures are better.)

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