Minimal surface

 
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nextstep
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PostPosted: Sun Oct 05, 2014 2:59 pm    Post subject: Minimal surface Reply with quote

Hi all,
An introduction to the "minimal surfaces" subject and it's physical model application Cool

From soap films to roof structures:
Quote:
Soap films are obtained by dipping a wire frame in soap solution and
they are a physical model of a complex mathematical concept : the minimal surfaces.
In mathematics, a minimal surface is a surface that locally minimizes its area.
Due to these qualities, soap bubbles films have been used with practical problem solving application.
Structural engineer Frei Paul Otto (31 May 1925), a German architect and structural engineer used soap bubble films to determine the geometry of a sheet of least surface area that spreads between several points, and translated this geometry into revolutionary "tensile roof structures".
A famous example is his West German Pavilion at Expo 67 in Montreal.
For more information on
"tensile roof structures":http://en.wikipedia.org/wiki/Tensile_structure
Frei Paul Otto : http://en.wikipedia.org/wiki/Frei_Otto
In the images, two examples of tensile structures :
1) A cooling tower as an hyperboloid structure
2) The roof tensile structures by Frei Otto of the Olympiapark (Munich), with a minimal surface geometry


Hyperboloid by taha_ab, on Flickr

Olympic_park_12 by taha_ab, on Flickr[/quote]
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Abderrahman
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 538

PostPosted: Sun Oct 05, 2014 3:35 pm    Post subject: Reply with quote

Hi all,
Ken Brakke, is the Susquehanna University mathematics professor who co-authored "The Physics of Microdroplets" and developed "Surface Evolver", a free interactive program for modeling liquid surfaces shaped by various forces and constraints.
His web site (http://www.susqu.edu/brakke/) gives an idea of his astonishing and fruitful work in the mathematical area!
In the "Triply Periodic Minimal Surfaces" website's section there is a description of the "minimal surface" along with a large set of examples:
"A minimal surface is a surface that is locally area-minimizing, that is, a small piece has the smallest possible area for a surface spanning the boundary of that piece. Soap films are minimal surfaces. Minimal surfaces necessarily have zero mean curvature, i.e. the sum of the principal curvatures at each point is zero. Particularly fascinating are minimal surfaces that have a crystalline structure, in the sense of repeating themselves in three dimensions, in other words being triply periodic. Many triply periodic minimal surfaces are known, some of which are pictured on this page"

Numerous minimal surfaces, such as the "Lidinoid Surface", can be described or approximated implicitly and/or parametrically:
MathMod script:
{
"Iso3D": {
"Component": [
"Lidinoid"
],
"Description": [
"Sven Lidin and Stefan Larsson"
],
"Fxyz": [
"-((1/2)*(sin(2*x)*cos(y)*sin(z)+sin(2*y)*cos(z)*sin(x)+sin(2*z)*cos(x)*sin(y))-(1/2)*(cos(2*x)*cos(2*y)+cos(2*y)*cos(2*z)+cos(2*z)*cos(2*x))+0.15)"
],
"Name": [
"Lidinoid"
],
"Xmax": [
"3"
],
"Xmin": [
"-3"
],
"Ymax": [
"3"
],
"Ymin": [
"-3"
],
"Zmax": [
"3"
],
"Zmin": [
"-3"
]
}
}

Lidinoid by taha_ab, on Flickr
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Abderrahman
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nextstep
Site Admin


Joined: 06 Jan 2007
Posts: 538

PostPosted: Sat Oct 11, 2014 11:32 pm    Post subject: Reply with quote

Hi all,
This document describe another interesting use of Soap films :

Quote:
Introduction
Some Motivating Scenarios
1. A new sub-marine data cable needs to be laid across the Pacific Ocean, connecting several major cities
in Asia, North America, and South America. The longer the cable, the more it will cost, so what is the
cheapest way to lay the cable?
2. A new road network needs to be built between several cities in your state. Every city must be a part of
the road network. Since building roads costs money, what is the cheapest way to build the network?
3. A new electrical grid needs to be built for your town. Due to the budget, however, you need to use the
existing electrical substations and power plants. Electrical cable costs money. How do you most efficiently
connect the substations and power plants together?

There is also a description of how to build your own materials to experiment with...
Have a fun Smile
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Abderrahman
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