Posts Tagged ‘polygon’

Shrinkable polygons

October 7, 2014 Leave a comment

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.

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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}.

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Cyclic polygon has largest area

April 28, 2012 Leave a comment

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.

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Isoperimetric inequality for polygons

April 27, 2012 3 comments

Among polygons with {n \geq 3} sides with given perimeter {P}, the one which maximizes the area is regular.

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Regular polygons and lattice points.

October 28, 2009 2 comments

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.
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