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a b a 2 + b 2 2 ,

which implies that:

a + b 2 2 a 2 + b 2 2 , as desired

Also, this averaging will not affect the integral length of the function from side to side.

Another useful result is a bound on the metric at the center of the Swiss Cross.

Consider a discrete metric on the Swiss Cross. Then create a square in the center of the Swiss Cross with side length ϵ , and other ϵ squares at the center of the boundary of each outer square. If the weight in the center is w c e n t e r , and the weight on the sides is w s i d e then we want to minimize:

Small Squares

4 w s i d e 2 ϵ 2 + w c e n t e r 2 ϵ 2 = A

subject to:

2 w s i d e ϵ + w c e n t e r ϵ = L

We can use Lagrange multipliers to solve this equation. So:

( 8 w s i d e ϵ 2 , 2 w c e n t e r ϵ 2 ) = λ ( 2 ϵ , ϵ )

When we solve for the ratio w c e n t e r w s i d e , we get:

w c e n t e r w s i d e = 2

In order to find a lower bound for the metric area of the Swiss Cross, we can consider the simplified case where Γ = { ( Straight lines connecting A and B ) ( Straight lines connecting C and D ) } . It is easy to verify that the extremal metric is as shown in Figure 3. This function has a metric area of 4.5, so the minimum metric area for the entire set of curves Γ will be greater than or equal to 4.5.

Swiss Cross Straight

In order to visualize a solution, we used a recursive Matlab program to create an approximation of the extremal metric. Using a modified version of Dijkstra's algorithm we were able to find the shortest path between two sides, and progressively lower the metric area. In order to find a global minimum instead of a local, we used the method of gradient descent. Briefly, by selectively raising certain areas and rerunning the recursion, we wound up with a better metric area.

A 36x36 matrix of a discrete approximation of the extremal metric on the Swiss Cross


Again, a prime candidate for extremal metric, ρ 1 seems reasonable which gives a metric area of π . However, it is not extremal as is demonstrated in Figure 4. The metric area of Figure 4 is about 2.95, and it satisfies the constraints.


Using a symmetry argument similar to the Swiss Cross, it is evident that the metric on the disk will be a radial function, and hence in polar coordinates will have no θ dependence.

At this point it became useful to find a lower bound for the metric area of the circle. However, the straight line case is degenerate as every line passes through one point[1]. Thus using other methods it is possible to show that

Δ ρ 2 r d r d θ = 2 π 0 1 ρ 2 r d r 2 π lim n 1 + n 0 1 log n ρ 2 r d r , using Jensen's Inequality = 2 π lim n 1 + n 2 0 1 log n ρ d r n 0 1 log n r d r = 2 π lim n 1 + n 2 0 1 log n ρ d r n 1 log n , since 0 1 log n r d r = - 1 log n 2 π lim n 1 + n 2 log n 2 π n 1 log n , since we have a global lower bound of 2 π = 2 π 2 π 2 = 8 π

Thus the extremal metric will have metric area between 8 π 2.54 and 2.95.

Future work

A primary goal of any future work is creating a continuous minimizing metric on the disk and Swiss Cross. Further, some of the methods developed here can be applied to similar problems in extremal length. Many related problems involve sets of curves which connect two boundary sets. Specifically, the hexagon or Swiss hexagon (i.e. the hexagon with unit squares attached to the sides) are two interesting cases that could be the subject of further research.


1. Ahlfors, Lars V. Conformal Invariants: Topics in Geometric Function Theory . New York: McGraw-Hill, 1973. Print.


We would like to thank Mike Wolf, Colin Carroll, Leo Rosales, Bob Hardt, Paul Munger, and Renee Laverdiere for their help and guidance. We would also like to acknowledge the Rice University VIGRE program. A credit is due to the VIGRE program for the Summary section.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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