All About (regular) n-gons—session at the Berkeley Math Circle

December 21, 2008

Last Tuesday I gave a talk at the Berkeley Math Circle titled “All About (regular) n-gons”. This was a sort of problem seminar that covered a mixed bag of mathematical tricks, but one focus was the use of complex numbers in geometry.

You can download my handout here. The questions range from fairly basic geometry to research questions—the question “How many intersection points are formed when we draw all the diagonals of a regular $n$-gon?” is much more difficult than it appears at first; it was first answered in closed form by Poonen and Rubenstein in 1997.

There was a follow-up problem session this morning, which discussed some of the harder questions. (though I have yet to see solutions to either of the Miklós Schweitzer problems) I was impressed to see more than half a dozen students, many of them in junior high, spend two hours on Saturday morning like this.

The last time I talked at Berkeley Math Circle was in 2002, on generating functions.

Here are some of the more fun problems from my session this week:

• What are all the values of $n$ for which you can tile the plane with regular $n$-gons of varying size?
• (Romania 1995) Find the number of ways of coloring the vertices of a regular $n$-gon with $p$ colors, such that no two adjacent vertices have the same color.
• Does there exist a regular $n$-gon such that exactly half of its diagonals are parallel to one of its sides?
• Show that the sum of the squares of the lengths of all sides and diagonals emanating from a vertex of a regular $n$-gon inscribed in the unit circle is $2n$.
• (USAMO 1997) To clip a convex $n$-gon means to choose a pair of consecutive sides $AB$, $BC$ and to replace them with three segments $AM$, $MN$, $NC$, where $M$ is the midpoint of $AB$ and $N$ is the midpoint of $BC$. In other words, one cuts off the triangle $MBN$ to obtain a convex $(n+1)$-gon. A regular hexagon $P_6$ of area $1$ is clipped to obtain a heptagon $P_7$. Then $P_7$ is clipped (in one of the seven possible ways) to obtain an octagon $P_8$ and so on. Prove that no matter how the clippings are done, the area of $P_n$ is greater than $1/3$ for all $n\geq 6$.
• (USAMO 2008) Let $P$ be a convex polygon with $n$ sides, $n\geq 3$. Any set of $n-3$ diagonals that do not intersect in the interior of a polygon determine a triangulation of $P$ into $n-2$ triangles. If $P$ is regular and there is a triangulation of $P$ consisting only of isoceles triangles, find all possible values of $n$.