Rice university vigre: edge length minimizing polyhedra  (Page 2/4)

Third, the Regular Triangular Prism is shown in [B01] to have smaller $\mathcal{F}$ value than any Right Regular Pyramid . A pyramid is a polygon in the plane, together with an apex at a point above the polygon, and a pyramid is regular if the base is regular and the triangles forming the sides of the pyramid are equilateral. Using techniques similar to the case of a prism, it is shown that $\mathcal{F}$ is minimized over pyramids with equal-sided bases when the pyramid is a Right Regular Pyramid. A calculation then shows $\mathcal{F}$ is minimized over all pyramids for the Tetrahedron, but on the other hand the value of $\mathcal{F}$ for the Regular Triangular Prism is smaller than that of the Tetrahedron.

Fourth, an Equal-Faced Polyhedron is a polyhedron where the faces have equal area. The Regular Triangular Prism has smaller $\mathcal{F}$ value than any equal-faced polyhedron with ten or more faces. This is shown in [B01] by proving a lower estimate for the value of $\mathcal{F}$ for any equal-faced polyhedron with ten or more faces using the Isoperimetric Inequality, a lower bound which is greater than $\mathcal{F}\left(RTP\right).$

Variations of polyhedra

Our first approach for solving Melzak's Problem is derived by analogy from the proofs provided in [B01] to show that Right Regular Prisms minimize $\mathcal{F}$ over all prisms. More specifically, given a polyhedron P we define a variation P _t of P to be a continuous deformation of P so that $\mathcal{F}\left(P_t\right)$ is a differentiable function of t with $P_0=P.$ We then calculate

in order to see whether $\mathcal{F}\left(P_t\right)$ has a critical point, and possibly a minimum, at $t=0.$

There are some difficulties with this approach. First, as a theoretical matter, although it is true that if P is the solution to Melzak's Problem then $\mathcal{F}\left(P_t\right)$ achieves a minimum at $t=0$ for any variation P _t of $P,$ we may encounter a pseudo-minimizer , a polyhedron P which has $\frac{d}{dt}\mathcal{F}\left(P_t\right){|}_{t=0}=0$ for all variations P _t of P which is only locally a solution to Melzak's Problem. Second, as a practical matter, finding variations of a given polyhedron is typically delicate, as edges tend to appear and disappear as we deform polyhedra.

Given the idea of taking variations of polyhedra, the first task was to verify that $\frac{d}{dt}\mathcal{F}\left(RT{P}_t\right){|}_{t=0}$ for any variation $RT{P}_t$ of the Regular Triangular Prism. We checked this for two variations, which are not merely deformations on the Regular Triangular Prism into other prisms. The first variation, consisted of lifting a vertex of the triangular top, to get a polyhedron $P_{\theta },$ where θ is the angle between the old triangular top and the new one:

A calculation shows that

which attains its minimum at $\theta =0.$

Next, we varied the Regular Triangular Prism by taking the triangular base and expanding it, to form a tetrahedron with the top sliced off:

Taking the base to be equilateral triangle, then if $1+n$ was the length of an edge of the base, then we have for the variation P _n

which has minimum at $n=0,$ when the figure is a Regular Triangular Prism.

These calculations are further evidence that the Regular Triangular Prism is the proposed minimizer. On the other hand, taking variations of the cube showed that it is a pseudo-minimizer. First, we vary the cube by lifting the top via two vertices: