### Archive

Posts Tagged ‘polygon’

## Shrinkable polygons

Here’s a nice problem inspired from a post on MathOverflow: link

We call a polygon shrinkable if any scaling of itself with a factor ${0<\lambda<1}$ can be translated into itself. Characterize all shrinkable polygons.

It is easy to see that any star-convex polygon is shrinkable. Pick the point ${x_0}$ in the definition of star-convex, and any contraction of the polygon by a homothety of center ${x_0}$ lies inside the polygon.

## Numerical method – minimizing eigenvalues on polygons

December 23, 2013 1 comment

I will present here an algorithm to find numerically the polygon ${P_n}$ with ${n}$ sides which minimizes the ${k}$-th eigenvalue of the Dirichlet Laplacian with a volume constraint.

The first question is: how do we calculate the eigenvalues of a polygon? I adapted a variant of the method of fundamental solutions (see the general 2D case here) for the polygonal case. The method of fundamental solutions consists in finding a function which already satisfies the equation ${-\Delta u=\lambda u}$ on the whole plane, and see that it is zero on the boundary of the desired shape. We choose the fundamental solutions as being the radial functions ${\Phi_n^\omega(x)=\frac{i}{4}H_0(\omega|x-y_n|)}$ where ${y_1,..,y_m}$ are some well chosen source points and ${\omega^2=\lambda}$. We search our solution as a linear combination of the functions ${\Phi_n}$, so we will have to solve a system of the form

$\displaystyle \sum \alpha_i \Phi_i^\omega(x) =0 , \text{ on }\partial \Omega$

in order to find the desired eigenfunction. Since we cannot solve numerically this system for every ${x \in \partial \Omega}$ we choose a discretization ${(x_i)_{i=1..m}}$ on the boundary of ${\Omega}$ and we arrive at a system of equations like:

$\displaystyle \sum \alpha_i \Phi_i^\omega(x_j) = 0$

and this system has a nontrivial solution if and only if the matrix ${A^\omega = (\Phi_i^\omega(x_j))}$ is singular. The values ${\omega}$ for which ${A^\omega}$ is singular are exactly the square roots of the eigenvalues of our domain ${\Omega}$.

## Cyclic polygon has largest area

Suppose ${a_1,..,a_n}$ are positive real numbers. Then the following statements are equivalent:

1. ${a_1,...,a_n}$ satisfy the inequalities

$\displaystyle 2a_i < a_1+...+a_n,\ i=1..n$

2. There exists a polygon with sides ${a_1,..,a_n}$.

3. There exists a convex polygon with sides ${a_1,..,a_n}$.

4. There exists a cyclic polygon with sides ${a_1,..,a_n}$.

Of all these, the cyclic polygon has the maximal area.

Categories: shape optimization

## Isoperimetric inequality for polygons

Among polygons with ${n \geq 3}$ sides with given perimeter ${P}$, the one which maximizes the area is regular.
Se say that a point in the plane is a lattice point if both its coordinates are integers. Prove that for $n\neq 4$ we can’t find a regular polygon with $n$ egdes, its vertices being lattice points.