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l33tminion) wrote2004-07-19 09:58 pm
l33tminion) wrote2004-07-19 09:58 pm
Soap Bubble Science
Yesterday's lecture on chaos theory was interesting, but today's lecture, which was on the physical properties of soap films, was by far the best lecture this session. The lecture showed how properties of soap bubbles can be used to demonstrate solutions to minimization problems.
More Details of the Lecture:
Soap films have the following interesting properties:
1. A soap film will always seek to be at a minimum energy, and thus a minimum area.
2. That means a soap film will always be at a relative minimum area when it is stable.
3. Any configuration that has a bubble will never be an absolute minimum area.
The professor giving the lecture demonstrated these properties in simplified (2D) situations by using nails going between two parallel layers of plastic.
4. In 2D figures, the number of stable configurations without bubbles is determined by the number of points. By measuring the total length (area) or each solution, you can find the absolute minimum solution.
5. In 2D figures, soap films will always meet at a 120°.
6. The number of points at which films meet will always be n-2 or less, where n is the number of points (nails). In figures made entirely of bubbles the same principle applies for the number of internal intersections, where n is the number of bubbles.
7. For 3D configurations, films always meet at a tetrahedral angle (four lines meeting at 109.5° approx.)
The bubble solutions have practical applications in any problem that requires minimization of length (such as in making phone networks) or in minimizing area while maintaining connectivity.
The professor also gave the following problem. How can you construct a square of side length a given two points a units apart using only a compass (no straight edge)? I've thought about this for a little, but so far I haven't figured it out. I might work on it a little later.
Today in lab, I finished working on the first draft for our final report. Tomorrow we will finish the report and begin work on our presentation.
More Details of the Lecture:
Soap films have the following interesting properties:
1. A soap film will always seek to be at a minimum energy, and thus a minimum area.
2. That means a soap film will always be at a relative minimum area when it is stable.
3. Any configuration that has a bubble will never be an absolute minimum area.
The professor giving the lecture demonstrated these properties in simplified (2D) situations by using nails going between two parallel layers of plastic.
4. In 2D figures, the number of stable configurations without bubbles is determined by the number of points. By measuring the total length (area) or each solution, you can find the absolute minimum solution.
5. In 2D figures, soap films will always meet at a 120°.
6. The number of points at which films meet will always be n-2 or less, where n is the number of points (nails). In figures made entirely of bubbles the same principle applies for the number of internal intersections, where n is the number of bubbles.
7. For 3D configurations, films always meet at a tetrahedral angle (four lines meeting at 109.5° approx.)
The bubble solutions have practical applications in any problem that requires minimization of length (such as in making phone networks) or in minimizing area while maintaining connectivity.
The professor also gave the following problem. How can you construct a square of side length a given two points a units apart using only a compass (no straight edge)? I've thought about this for a little, but so far I haven't figured it out. I might work on it a little later.
Today in lab, I finished working on the first draft for our final report. Tomorrow we will finish the report and begin work on our presentation.
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You have a compass but no straight edge.
That means you can draw circles if you know the points that determine the length and the center, but you cannot draw (straight) lines.
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There even might be a straight edge on the pencil, unless it's one of those cylindrical ones.
Think outside of the bubble...
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you could make circles of radius a, on centered on each of the two points. that way, you know that your square has to have two other corners one that is on each cricle. I didnt think how to find where on each circle, but you can figure that out, I'm sure.
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Folding the paper might be allowed, though. I don't know.
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I also have that the intersection of those circles gives you equilateral triangles.
I don't have the rest, though.
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Found a solution on this site.
Here is the solution.
The solution builds on the solution to this problem
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*points to different comment on post*
Did you have anything about soap bubble freezing?
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