Circle and a point

Here is a puzzle that consists of three pieces and that you can assemble in two ways: you can make a circle or you can make a circle plus one extra point. You create it as follows: take the circle x2+y2=1 in the plane and let p be the point with coordinates (1,0). Our first piece Q1 consists of just the point p. The scond piece Q2 we make by rotating p through 1, 2, 3, ... radians, so Q2={( cos n, sin n) : n=1,2,3,4,...}. The third piece Q3 is simply what is left of the circle.

Clearly Q1, Q2 and Q3 can be put together to form exactly the circle. If we rotate Q2 clcokwise through 1 radian then we have covered the circle exactly but using Q2 and Q3 only, the piece Q1 remains. This works because all points (cos n,sin n) are different (in particular p does not belong to Q2); because of this Q2 basically plays the role of P1 from the previous puzzle.

The form of this puzzle makes clear that you will not find it in the stores: no knife is sharp enough to separate Q2 from the rest of the circle. Try and plot Q2 on a graphic calculator by having it plot the points ( cos 1, sin 1) ... ( cos 100, sin 100); you will see that Q2 is everywhere dense on the circle.

We can improve this puzzle so that even Q2 can be missed. We do this by splitting Q3 into two pieces Q3a and Q3b. First we make Q3a by rotating Q2 through sqrt2 , 2sqrt2 , 3sqrt2 , ... radians and Q3b is what remains of Q3. We have covered the circle with Q1, Q2, Q3a and Q3b but we can also do this with Q1 , Q3b by rotating Q3a clockwise through sqrt2 radians; Q2 is then no longer needed.

We see that in some puzzles we can put some pieces aside and still complete it. With the circle this is the best you can expect; because the circle can be rotated in one direction only we will not be able to double it. The ball and the sphere are different; by mixing two directions you can create the puzzle pieces of Hausdorff, Banach and Tarski.

\includegraphics{banach-tarski.1} \includegraphics{banach-tarski.2} \includegraphics{banach-tarski.3}
Last modified: Wednesday 26-02-2003 at 10:35:46 (CET)