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Surface Areas of Soap Bubbles


Date: 11/30/1999 at 19:00:53
From: Alex
Subject: Bubble behavior

If you build a frame shaped like a tetrahedron and dip it in bubble 
solution, why do all of the faces of the bubble collapse to a point in 
the middle of the tetrahedron? I would like a VERY technical answer 
please.


Date: 11/30/1999 at 22:36:57
From: Doctor Douglas
Subject: Re: Bubble behavior

Hi Alex,

The reason that this happens has to do with a physical property called 
"surface tension."  Many substances like to minimize their surface 
area. For example, a drop of water on a piece of waxed paper will roll 
up into a little bead. And the bubble that you blow from a piece of 
bubble gum has roughly a round, or spherical shape. If you pop the 
bubble with a pin, you can see the surface contract away from the 
pricked point, indicating that the surface is under tension, or 
stretched like a sheet of rubber. Unless something (such as your 
breath) keeps it stretched out, it will contract to decrease the total 
surface area.

Now, the soap bubbles on the tetrahedron frame obey the same 
principle: what shape gives the smallest surface area, subject to the 
constraint that all the surfaces are spanned? One possibility is to 
cover each of the four triangles with a flat plane, but the surface 
area of those four triangles is larger than the shape where the faces 
get pushed inward slightly so that they collapse in the middle. 
Instead of having four large triangles, we have six much smaller 
triangles, each of which is composed of the plane that connects two of 
the tetrahedron vertices and one at the tetrahedron center. You might 
enjoy trying to calculate the surface area of these six isosceles 
triangles and comparing the total with the four large triangles.

Soap bubbles on wire frames represent a wonderful area of mathematics, 
full of beautiful shapes and surprises.

Hope that helps,

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
High School Physics/Chemistry
Middle School Geometry
Middle School Higher-Dimensional Geometry

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