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Minimal surfaces A mathematical concept linking black holes to bubbles
A minimal surface is a surface that is described to locally minimise its area. As confusing as the definition is for minimal surfaces, it can be more easily understood by comparing it to shapes we are familiar with, such as soap bubbles. When blowing soap bubbles, a loop is dipped into soapy water and a flat film can be seen stretching across the loop when lifted from the bubble solution. This film takes the simplest shape possible, a perfectly flat surface with no bumps. It naturally minimises surface tension and also minimises area. Imagine if the flat circle gained bulges, each bulge would add unnecessary surface area. Once blown, a perfect sphere is created. Here we once again witness the tendency for nature to form efficient shapes. A sphere has the least surface area for a given volume.
surface shapes were called the catenoid (left) and the helicoid (right) The catenoid can be formed by dipping two rings in soapy water and joining the films to form a tube shape. One might initially assume that a cylinder will be formed, since at any point on a ring to another identical ring, a straight line would be the shortest length required to cross that distance. However, the catenoid is formed,
This is the starting point for understanding minimal
which is essentially a cylinder which tapers in at the
surfaces, though a soap bubble is still not a minimal
center before broadening back out symmetrically.
surface, since the air pressure within the bubble stops it from collapsing from the opposite force of
Now, finally going back to the original definition of
its surface tension, which wants to form a flat plane.
locally minimising area, a minimal surface is
In comparison, the flat plane of the dipped loop is a
essentially a surface that whenever you define a
minimal surface, as it does not need to account for
boundary upon it, the area within that boundary
any additional force a spherical bubble has that
has the minimal surface it can have. Another
prevents it from collapsing into the minimal surface
definition explains that a minimal surface, like a
area possible. The effect of air pressure can be seen
catenoid, occupies the least area when bounded by a
by gently blowing on the flat surface of the dipped
closed space. The thin middle allows a catenoid to
loop. The added force disrupts the surface but as
occupy less space than a cylinder.
soon as you stop, it goes back to the flat plane. It was Leonardo da Vinci who first started studying soap films but it was mathematicians of the 18th century,
Leonhard
Euler
and
Joseph-Louis
Lagrange who had taken up the subject of minimal surfaces and discovered two others from the original, trivial plane. The two new minimal
