Problem 2

Let

S

be a finite set of at least two points in the plane. Assume that no three points of

S

are collinear. A windmill is a process that starts with a line going through a single point

P\in S

. The line rotates clockwise about the pivot

P

until the first time that the line meets some other point belonging to

S

. This point,

Q

, takes over as the new pivot, and the line now rotates clockwise about

Q

, until it next meets a point of

S

. This process continues indefinitely. Show that we can choose a point

P

S

and a line going through

P

such that the resulting windmill uses each point of

S

as a pivot infinitely many times.

Step 4 of 6: Every ordinary direction returns after each half-turn

Previous step Next step

In plain words

Pivot changes only ever happen at the finitely many directions determined by pairs of points, and turning the whole configuration by a half-turn brings it back to itself, so the same list of directions is visited again and again forever.

180^\circ

Detailed analysis

A pivot change can only occur when the rotating line's direction coincides with the direction of some segment joining two vertices, and there are only finitely many such directions. Rotating any line by a full $180^\circ$ returns it to the same undirected line through the same points, so the finite pattern of directions at which pivot changes occur is exactly repeated every $180^\circ$ of rotation. Since the windmill continues rotating clockwise forever, each of the finitely many ordinary directions that occurs at all must occur again after every subsequent $180^\circ$ , hence infinitely often.