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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 P with unit volume. Take a plane H that bisect the volume of P . Then choosing one of the two halves P 1 and P 2 of P , we merge P j 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 P 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:

Future work

The method of symmetrization remains to be fully explored. Specifically, given a polyhedra P which is symmetric across a plane H , we ask if a comparison of the edge length of P can be made to the prism formed by the cross section 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 ( P t ) for a variation P t for a given polyhedron P .


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.


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.


[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.

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
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