the tetrahedral angle

Joseph Plateau, blind for the last forty years of his life, dipped wire frames into soapy water and asked his son, his son-in-law, his colleagues to describe what they saw. He could not see the films himself. In the summer of 1829, he had stared at the burning sun for twenty-five seconds without blinking — an experiment to learn how an image fixes itself onto the retina. Over the following decade his sight failed. By the time he formulated the laws that bear his name, he was conducting an orchestra of describers.

The orchestra reported back in unison. The films always meet in threes. Always at 120 degrees. Where four films come together at a single point, they meet in fours, at an angle of about 109 degrees and a half.

That second angle is the tetrahedral angle. arccos(−1/3). It is the angle between any two bonds in a methane molecule. Between any two oxygen atoms in a silicate. Between the corners of a regular tetrahedron seen from its center. It shows up wherever four equal forces pull on a single point and find their balance.

The films do not know about methane. They are doing what soap films do, which is shrink, which is minimize their own surface area, which is settle into the configuration that has nothing left to relax toward. The 120 degrees and the 109.47 are not aesthetic choices. They are what falls out when surface tension is asked to hold still.

Jean Taylor proved in 1976, using geometric measure theory, that no other configuration is possible. The angles are not observed regularities. They are necessities. Any other geometry stores free energy that wants to escape; the system will rearrange until it doesn’t. Only the 120s and the 109.47s sit still.

Plateau died in 1883, having described the geometry of a thing he could not see for thirty years on the unanimous testimony of people who could. The wire frames are in a museum in Belgium. The angles are still 120.