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SOAP BUBBLES make for mathematical wonders and excellent physics. One of the principal proponents of this was 19th-century Belgian physicist Joseph Plateau, whose name is associated with these ephemeral objects. He did other cool things as well.
One of his early studies concerned perception of a moving image, of what we now call movies. A result of this was Plateau’s phenakistoscope, an animation device that exploits the persistence of an image to trick the eye into seeing motion. The word comes from the Greek, φενακιζειν, phenakizein, meaning “to trick,” “to deceive.”
Plateau also studied our perception of sunlight and of colors. In particular, he identified the phenomenon of our mind retaining a color image in its complementary hues. A classic example of this is to stare at the dot in this image below for 20 seconds, then close your eyes and identify what you see.
The physics of soap bubbles also attracted Plateau’s attention. For instance, it’s clear that a single soap bubble is spherical simply because a sphere has the least surface area for that given volume. Nature chooses the most efficient enclosure.
This got Plateau thinking about something eventually called the Double Bubble Conjecture (now properly a theorem, as its full mathematical proof appeared in 2002). It states that, for any two separate volumes, a double bubble is the shape of minimal surface area enclosing them. Said another way, any other shape enclosing the same volumes has more surface area.
It turns out this theory has practical application in areas as varied as optics and electrical engineering. However, our interest here is purely geometrical, in describing the part separating the two volumes.
In 1873 (and remember the year), Plateau published his findings in Statique Expérimentale et Théorique des Liquides Soumis aux Seueles Forces Moléculaires. In it, he offered a neat geometrical construction by which the interface of any two dissimilar bubbles can be determined.
You’ll want a pair of compasses and a 30-60-90 triangle, the sort of thing left over from high-school geometry. (One assumes it’s still taught these days….) Otherwise, freehand will do.
From any starting point C, draw line f, then at 60 degrees from it, line g. Repeat the process as shown below with line h at 60 degrees from line g.
Next, on line f, mark the radius of the larger bubble from point C. Call this center A. Do the same for the radius of the smaller bubble on line g. Call this center B.
Now, as is well known to students of Euclid (see www.wp.me/p2ETap-Fg), two points determine a line. So draw this line j determined by A and B, and label as point K its intersection with line h.
Then use your compasses to draw the two bubbles as shown, one centered at A, the other at B.
Last, their interface is the arc of the circle with center K.
Clean up the construction details, and you have the two bubbles and their interface. This construction is so elegant that it brings tears to my eyes.
I conclude by citing that, traceable to his solar investigations in 1829, Joseph Plateau went totally blind in 1843. ds
© Dennis Simanaitis, SimanaitisSays.com, 2013