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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 .

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.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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