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Three Coalescing Soap Bubbles

​
3D view
show construction
zoom
6
bubble radii
a
3.5
b
2.5
c
1.5
Using the calculus of variations, the Belgian mathematician Joseph Plateau deduced that the interfaces between adhering soap films represent localized minimal surfaces, which occupy minimum areas around a set of constrained geometric regions. Of course, this is not a global minimum, which would consist of a single spherical bubble. This Demonstration considers the case of three connected bubbles with variable radii, centered at
A
,
B
, and
C
. You can also choose to see the details of Plateau's construction.
Every intersection of two or three bubbles makes angles of exactly 120°. These interfacial angles are maintained even if any of the radii are changed. The radii
F
,
G
, and
H
of the three interface arcs lie on a straight line, as shown in the construction diagram. In turn, these are connected to the three centers by a network of straight lines. The interface arcs are given by relations similar to those for the focal lengths of thin lenses:
1
b
-
1
a
=
1
ab
int
,
1
c
-
1
a
=
1
ac
int
,
1
c
-
1
b
=
1
bc
int
.​
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