Introduction

In the Calculus of Variations PFUG, we studied Melzak's Problem using variational methods. These methods involve the study of geometric objects by defining a functional, which quantitatively measures those objects. We then seek to describe properties of objects which optimize this functional.

For the problem studied, the functional involved the edge length and volume of polyhedra. For a polyhedron P,P, let V(P)V(P) be its volume and E(P)E(P) be its total edge length. Let PP be the set of all polyhedra PP with V(P)=1.V(P)=1. Then Melzak's Problem, which is also referred to as the Waste Storage Problem, asks if there exists a polyhedron P0∈PP0∈P with E(P0)≤E(P)E(P0)≤E(P) for all P∈P.P∈P.

As a practical consideration, it is inconvenient to consider polyhedra PP with V(P)=1,V(P)=1, namely because easy examples are unwieldy to construct. For example, it is computationally much easier to work with the tetrahedron with unit length edges, having volume 212.212. In order to circumvent this problem, we define

for PP any polyhedron. It follows that a polyhedron PP minimizes the quantity F(P)F(P) amongst all polyhedra if and only if the rescaled polyhedron P˜,P˜, given by taking the edges eiei of PP and setting the new corresponding edge of P˜P˜ to be e˜i=eiV(P)1/3,e˜i=eiV(P)1/3, solves Melzak's Problem (that is, V(P˜)=1V(P˜)=1 and P˜P˜ minimizes edge length over all polyhedra with unit volume). We therefore seek a polyhedron PP which minimizes F(P).F(P).

The problem is mentioned in several sources, most notably as Problem 13 in [M65]. Although not much work has been done to solve the problem, the Regular Triangular Prism is conjectured to be the minimizer. This polyhedron is constructed from three squares and two equilateral triangles:

Figure 1

When the edges are chosen to have unit length, then for the Regular Triangular Prism we have F(RTP)=9723.F(RTP)=9723. The work contained in [B01] gives evidence to believe that the Regular Triangular Prism is the minimizer.

Known Results

First, calculations show that the Regular Triangular Prism has a smaller value of FF than the Platonic Solids. These are the Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron with FF values 12962,1728,25922,1728005(1+5)-4,12962,1728,25922,1728005(1+5)-4, and 129600(1+5)-2129600(1+5)-2 respectively.

Second, we define a prism to be a polygon in the plane translated in the direction of a vector not in the plane. Then [B01] shows that Right Regular Prisms, prisms formed by translating a polygon with equal sides vertically in the perpendicular direction so that the vertical sides are squares, minimize the value of FF over all prisms. This is done by writing down the formula for the volume of a prism depending on the area of the polygonal base, the height of the prism, and the angle between the base and the generating vector. Given a base, then differentiating this function shows that the volume is minimized, while edge length is preserved, when the generating vector is perpendicular to the base. Then, the 2-Dimensional Isoperimetric Inequality for Polygons shows that the volume can be decreased, while the edge length kept fixed, if the base is regular. Finally, a calculation shows that the Regular Triangular Prism has smallest value of FF amongst all Right Regular Prisms.

Third, the Regular Triangular Prism is shown in [B01] to have smaller FF 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 FF is minimized over pyramids with equal-sided bases when the pyramid is a Right Regular Pyramid. A calculation then shows FF is minimized over all pyramids for the Tetrahedron, but on the other hand the value of FF 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 FF 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 FF for any equal-faced polyhedron with ten or more faces using the Isoperimetric Inequality, a lower bound which is greater than F(RTP).F(RTP).

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 FF over all prisms. More specifically, given a polyhedron PP we define a variationPtPt of PP to be a continuous deformation of PP so that F(Pt)F(Pt) is a differentiable function of tt with P0=P.P0=P. We then calculate

in order to see whether F(Pt)F(Pt) has a critical point, and possibly a minimum, at t=0.t=0.

There are some difficulties with this approach. First, as a theoretical matter, although it is true that if PP is the solution to Melzak's Problem then F(Pt)F(Pt) achieves a minimum at t=0t=0 for any variation PtPt of P,P, we may encounter a pseudo-minimizer, a polyhedron PP which has ddtF(Pt)|t=0=0ddtF(Pt)|t=0=0 for all variations PtPt of PP 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 ddtF(RTPt)|t=0ddtF(RTPt)|t=0 for any variation RTPtRTPt 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 Pθ,Pθ, where θθ is the angle between the old triangular top and the new one:

Figure 2

A calculation shows that

which attains its minimum at θ=0.θ=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:

Figure 3

Taking the base to be equilateral triangle, then if 1+n1+n was the length of an edge of the base, then we have for the variation PnPn

which has minimum at n=0,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:

Figure 4

When θθ is the angle between the old top and the new one, then for the variation PθPθ we have

which has a minimum at θ=0.θ=0.

Next, we varied the cube by fixing a vertex, lifting the diagonal vertex a certain height and the two adjacent vertices by half that height:

Figure 5

In this case, when θθ was the angle between the old top and the new one, then for the variation PθPθ we have:

which has minimum at θ=0.θ=0.

The next variation of the cube is collapsing two of the sides together, so as to form a shape approaching the Regular Triangular Prism:

Figure 6

When θθ is the angle of collapse, then

which has minimum at θ=0,θ=0, when the figure is a cube. Observe that although the figure with θ=π6θ=π6 is the Regular Triangular Prism, F(Pθ)|π6F(Pθ)|π6 with the formula given above is not F(RTP).F(RTP). This is because when θ=π6,θ=π6, two vertical edges have collapsed into one, and our formula above double counts these edges at θ=π6.θ=π6.

Symmetrization of Polyhedra

The next method we considered is derived from the Isoperimetric Problem, which asks to find the figure in the plane with unit area and whose boundary curve has the smallest length. The solution, of course, is the disk of unit area. An argument for this is via symmetrization. That is, take a region AA in the plane and draw a line ℓℓ which cuts AA into two connected regions A1,A2A1,A2 both with area 1/2.1/2. Suppose then that the part of the boundary curve of AA corresponding to A1A1 has smaller length than that of A2,A2, that is

Consider the new region A1∪Rℓ(A1)A1∪Rℓ(A1) where Rℓ(A1)Rℓ(A1) is the reflection of A1A1 across the line ℓ.ℓ. Then A1∪Rℓ(A1)A1∪Rℓ(A1) has unit area, but the length of the boundary curve of A1∪Rℓ(A1)A1∪Rℓ(A1) is less than or equal to that of A.A. Hence, if AA is a solution to the Isoperimetric Problem, then the claim is that AA must be symmetric across every line which slices AA into two regions of equal area, and so AA must be a disk.

Although the solution to the Isoperimetric Problem is far more subtle than the argument given above, we nonetheless take the argument above as an analogy. That is, given a polyhedra P,P, we slice PP via a plane HH which cuts PP into two polyhedra P1,P2P1,P2 with equal volume. Then letting P1P1 have the smaller total edge length, where we only sum the edges which correspond to edges of P,P, then the polyhedron P1∪RH(P1),P1∪RH(P1), where RH(P1)RH(P1) is the reflection of P1P1 across H,H, will have V(P1∪RH(P1))=V(P)V(P1∪RH(P1))=V(P) and E(P1∪RH(P1))≤E(P),E(P1∪RH(P1))≤E(P), assuming that no new edges have been introduced when we took the reflection of P1P1 across HH and merged it with P1.P1.

The difficulty is finding planes given a polyhedron PP which halve the volume and which produce new polyhedra with no new edges introduced at the plane HH when we reflect the two halves across H.H. We call such a plane a bisecting plane of P.P. If PP and HH intersect along a face of PP which is not perpendicular to HH, then a new edge will be introduced when we reflect. For the Regular Triangular Prism, there are only two such planes, and the Regular Triangular Prism is symmetric under the reflection across these planes:

Figure 7

There are only two such planes for the cube

Figure 8

In general, the platonic solids are symmetric across their bisecting planes.

If we can show that the Regular Triangular Prism can be obtained from an arbitrary polyhedron with unit volume through this symmetrization with bisecting planes, then we would show that the Regular Triangular Prism is a solution to Melzak's Problem, given the existence of a solution. Hence, it is unfortunate that the Platonic Solids are already symmetric. We thereby consider the following operation on a polyhedron PP with unit volume. Take a plane HH that bisect the volume of P.P. Then choosing one of the two halves P1P1 and P2P2 of P,P, we merge PjPj with itself along any face that does not create new edges along the merging face. This method allows one to create distinct polyhedra even if PP is already symmetric about all bisecting planes. For example, the cube can be made into a triangular prism consisting of three squares and an isosceles triangle:

Figure 9

Figure 10

Future Work

The method of symmetrization remains to be fully explored. Specifically, given a polyhedra PP which is symmetric across a plane H,H, we ask if a comparison of the edge length of PP can be made to the prism formed by the cross section P∩H.P∩H. This may involve introducing variations of polyhedra in order to show that near the slice, a prism is optimal.

For the method of variations of polyhedra, we must investigate whether pseudo-minimizers have any special properties, or perhaps show that the only pseudo-minimizers are known polyhedra such as the platonic solids and the right regular prisms. One may show that prisms or right prisms are more efficient that all other figures, since it has already been shown that the Regular Triangular Prism is best among prisms. However, it would be easier to show that the Regular Triangular Prism in particular is more efficient than all other figures instead of considering an arbitrary right prism; hence the classical approach of pointing out improvements to large classes of polyhedra (aside from prisms) may not be effective. A non-variational approach may be more promising.

A computational approach using technology also remains to be taken. We seek to write a computer program to bisect and then reflect polyhedra, and then perhaps take variations. Thus far, our only use of computers has been to plot the graphs of F(Pt)F(Pt) for a variation PtPt for a given polyhedron P.P.

Summary

This report summarizes work done as part of the Calculus of Variations PFUG under Rice University's VIGRE program. VIGRE is a program of Vertically Integrated Grants for Research and Education in the Mathematical Sciences under the direction of the National Science Foundation. A PFUG is a group of Postdocs, Faculty, Undergraduates and Graduate students formed round the study of a common problem.

This module introduces an overview of Melzak's Problem, and discusses two methods for studying the problem: the method of Variations and the method of Symmetrization. Examples of each method applied to the cube and the Regular Triangular Prism are presented, and a discussion on future directions is provided.

Acknowledgements

This Connexions module describes work conducted as part of Rice University's VIGRE program, supported by National Science Foundation grant DMS-0739420. We would like to thank Professor Bob Hardt for leading our PFUG, and we thank the undergraduate members Siegfried Bilstein, Kirby Fears, Michael Jauch, James Katz, and Caroline Nganga.

Bibliography

[B01] S. Berger, Edge Length Minimizing Polyhedra, Thesis, Rice University, (2001)

[M65] Z.A. Melzack, Problems connected with convexity, Canad. Math. Bull. 8 , (1965), 565-573.