Graphic Convergence
The point (x , y) moves to a new position (x’ , y’) by the rule :
eg the point (2, 10) moves to (5 , 6) because x = 2 and y = 10 gives:
The Problems : A Start with the point (0 , 1) and repeat the rule 4 times B Start with the point (5 , 6) and repeat the rule 4 times C Plot the points on graph paper, and comment on your results so far. D Can you guess a point that will not move when the rule is applied ? This point is described as an 'invariant' of the process.
A slightly different rule is :
E Start with the point (1 , 1) and see where it goes after 4 repeats of this rule F Do the same for the point (5 , 4) G What did you notice about the way the 2 different points behaved ? Can you find the
'invariant' under this rule?
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