<< Chapter < Page Chapter >> Page >
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 around the study of a common problem. In this PFUG, we study Melzak's Problem, which asks if we can find a polyhedron which minimizes the edge length amongst all polyhedra with unit volume. This work was studied in the Rice University VIGRE class MATH499 in the Fall of 2008.

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 , let V ( P ) be its volume and E ( P ) be its total edge length. Let P be the set of all polyhedra P with V ( P ) = 1 . Then Melzak's Problem, which is also referred to as the Waste Storage Problem, asks if there exists a polyhedron P 0 P with E ( P 0 ) E ( P ) for all P P .

As a practical consideration, it is inconvenient to consider polyhedra P with 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 2 12 . In order to circumvent this problem, we define

F ( P ) = E ( P ) 3 V ( P )

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

When the edges are chosen to have unit length, then for the Regular Triangular Prism we have F ( R T P ) = 972 3 . 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 F than the Platonic Solids . These are the Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron with F values 1296 2 , 1728 , 2592 2 , 172800 5 ( 1 + 5 ) - 4 , and 129600 ( 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 F 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 F amongst all Right Regular Prisms.

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'The art of the pfug' conversation and receive update notifications?

Ask