[FoRK] weighed curve computation

[Stephen Williams]

Just a few days ago, I was reading about soap bubble simulation 
algorithms.  Along similar lines, I found this which was pretty cool:
http://www.tomnoddy.com/science.html

> Bubbles have fascinated physicists and mathematicians for centuries. 
> Sir Isaac Newton made his own bubble formulas, others preceded and 
> followed him in efforts to understand their nature. One of the most 
> useful ways that they have served science is in showing minimal areas 
> when applied to various geometric frames. Oddly, there is no good 
> mathematics for solving these very basic questions.
>
> If you were to take a ring or hoop and ask what shape would be 
> necessary to fill the space within the round ring ... you may 
> intuitively guess at a flat round disc. Any hills or valleys within 
> that disc would cause the form to have extra surface not needed to 
> fill the ring. If you were to dip that ring or hoop into soapy water 
> you would get a film that is a flat round disc.
>
> Now, suppose that you bent the ring here and there into some oddly 
> curvy shape (but still closed like a ring). Now what is the minimal 
> shape needed to fill that new wavy ring? We don't know ... and there 
> is no mathematics to help you settle the question.
>
> This is part of an old mathematics puzzle called the Plateau Problems. 
> Plateau, a Belgian physicists asked questions like these in the 1880s 
> and many of his questions are still unanswered. It is known that if 
> you dip that wavy ring into soapy water you would pick up a film that 
> is the most minimal shape possible ... soap films must minimize, they 
> have no choice.
>
> An architect named Otto Frei used this principle to design some 
> beautiful buildings using the soap films ability to show him minima. 
> He was therefore able to construct buildings that did not need extra 
> pillars or other help to hold up the walls or ceiling because he knew 
> that he was using the most minimal shapes for the construction of his 
> light weight materials. He knew it was the most minimal because he 
> tested it with soap films ... mathematics alone was not enough, he 
> needed the soap bubbles.
>
Wire up a shape, create a soap bubble, then just measure the shape and 
position of the bubble.  Repeatable, known to be ideal, and simple.  
Heck, you could create a machine to do just that in an automated fashion 
with arrays of movable 'wire' segments in a full 3D sphere and a laser 
digitizer.

And if any of you have opinions on the best, fastest, most flexible 
(i.e. with real or artificial constraints) bubble computation 
algorithms, please let me know.  Phun's methods of handling similar 
"integration" problems are among the few that have been designed for and 
implemented for real time performance.
http://www.phun.at/

sdw

Dave Long wrote:
> Despite having used planimeters, which seem related to the first 
> device, and having been guilty of accounting by weight (as well as -- 
> after Ford -- by height), I would never, in this era of cheap 
> transistors, have thought of applying a M-T to questions of area...
>
> research!rsc, "Computing History at Bell Labs"
> <http://research.swtch.com/2008/04/computing-history-at-bell-labs.html>
>> When Bell Labs was founded, it had of course some calculating 
>> machines, and it had one wonderful computer. This. That was bought in 
>> 1918. There's almost no other computing equipment from any time prior 
>> to ten years ago that still exists in Bell Labs. This is an 
>> integraph. It has two styluses. You trace a curve on a piece of paper 
>> with one stylus and the other stylus draws the indefinite integral 
>> here. There was somebody in the math department who gave this service 
>> to the whole company, with about 24 hours turnaround time, 
>> calculating integrals. Our recent vice president Arno Penzias 
>> actually did, he calculated integrals differently, with a different 
>> background. He had a chemical balance, and he cut the curves out of 
>> the paper and weighed them.
>
>
> -Dave
>
> (it appears M.V. Lomonosov used this massively parallel approach to 
> numerical integration a few centuries before Penzias)
>
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