Spinning Away... Topic: Complex Numbers. Level: AIME/Olympiad.

Problem: (2004 Putnam - B4) Let $ n $ be a positive integer, $ n \ge 2 $, and put $ \theta = 2 \pi /n $. Define points $ P_k = (k,0) $ in the $ xy $- plane, for $ k = 1, 2, \ldots, n $. Let $ R_k $ be the map that rotates the plane counterclockwise by the angle $ \theta $ about the point $ P_k $. Let $ R $ denote the map obtained by applying, in order, $ R_1 $, then $ R_2 $, $ \ldots $, then $ R_n $. For any arbitrary point $ (x,y) $, find, and simplify, the coordinates of $ R(x,y) $.

Solution: Hmm, rotations, and lots of them. That's a big hint to use complex numbers! Recall that the rotation of a point $ z_1 $ around a point $ z_2 $ counterclockwise by an angle of $ \alpha $ is $ z_2+(z_1-z_2)e^{i \alpha} $. Let's use this formula over and over again.

Start with the point $ x+yi $, but write it as $ r e^{i \phi} $, which is much more convenient notation for rotating. Then, if we rotate around $ P_1 = 1 $, we get

$ re^{i \phi} \rightarrow 1+(re^{i \phi}-1)e^{i \theta} = 1-e^{i \theta}+re^{i(\phi+\theta)} $.

Continuing this, rotating around $ P_2 = 2, P_3 = 3, \ldots, P_n = n $, we find that the points are

$ 2-(e^{i \theta}+e^{i 2\theta})+re^{i(\phi+2\theta)} $, $ 3-(e^{i \theta}+e^{i 2\theta}+e^{i 3\theta})+re^{i(\phi+3\theta)} $, $ \ldots $.

An easy induction gives us the final position as

$ \displaystyle n - \sum_{k=1}^n e^{i k \theta} + re^{i(\phi+n\theta)} $.

However, we know that $ e^{i k \theta} $ for $ k = 1, 2, \ldots, n $ are the roots of the polynomial $ x^n-1 = 0 $, so their sum is zero. Also, $ n\theta = 2 \pi $ so $ re^{i(\phi+n\theta)} = re^{i\phi} $. Thus the result is

$ n+r^{i \phi} = (n,0)+(x,y) = (x+n, y) $.

QED.

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Comment: This is a really neat application of the power of complex numbers in rotation problems. Basically, rotating in the plane means use complex numbers to reduce everything to algebra (which is infinitely better than geometry...). Not bad at all for a B4. Note another solution that I found to be quite interesting. Take a regular $ n $-gon of side length $ 1 $ and top edge with vertices $ (0, 0) $ and $ (1, 0) $. The map $ R $ corresponds to a "rolling" of the $ n $-gon along the $ x $-axis. Since that translates the $ n $-gon $ n $ units in the positive $ x $ direction, it can be argued that it does the same for all points in the plane.

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Practice Problem: Given three points $ A, B, C $, let $ \theta_1, \theta_2, \theta_3 $ be the smallest angles that $ A $ must be rotated to lie on line $ BC $, $ B $ must be rotated to lie on line $ AC $, and $ C $ must be rotated to lie on $ AB $, respectively. Find the maximum value of $ \theta_1+\theta_2+\theta_3 $.