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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Last modified: June 18, 2007