Iterated Circumcentres - Experiment Room

Here you may experiment with the behaviour of a sequence P0, P1, P2, P3, ... of points in the plane, where each point P(i) is the circumcentre of points P(i-1), P(i-2), P(i-3).

You will find a protractor marked in 15-degree increments to help you measure angles. You may drag the protractor around.

Click on the applet window and press "u" to get a separate resizable window.

Here are some questions you might play with.

Is there a pattern to the resulting sequence?
When does the sequence diverge to infinity?
When does the sequence converge to a single point? If so, then what is the location of this special point, and how does it depend on P0, P1, P2?
Can the sequence get "stuck" (when three of the points become collinear)?
Can you make the sequence of points repeat in a cyclic (periodic) fashion? What positions of P0, P1, P2 cause such periodic behaviour?
If the sequence is periodic, then is what is the period length?
Without moving P0 and P1, can you find four places for P2 such that there is a cluster of points near P0?

The above applet is found at http://aleph0.clarku.edu/~djoyce/java/Geometry/Geometry.html. A users manual can be found there.

The following key shorcuts may help: r=reset, u=lift off, d=reset Back to Page 2