Symmetry, a Math Film by Charles and Ray Eames
In their 1961 short film Symmetry, Charles and Ray Eames explore the mathematical properties of, you guessed it, symmetry. Using animation, they show how mathematicians can measure the relative amount of symmetry any given shape can have. What's amusing to me today about this film is that many of their examples are best known as multi-sided dice used in Dungeons & Dragons and similar games!
Charles Eames discussed the film in a 1974 paper in the Bulletin of the American Academy of Arts and Sciences. Here's a snippet (emphasis added):
Another film directly addressed itself to the definition of “symmetry.” It was done with animated drawings. The narrator says, “When we think of symmetry we usually think of a design balanced around a center line. . . . We think of man being symmetrical. . . . There are many kinds of symmetry and some things can be shown to be more symmetrical than others. One test for this is to count the number of positions that an object can take in a box that fits it perfectly.” At this point in the film, cartoons of a man sitting in a chair and a dog beside him are suddenly encased in boxes and their asymmetry is quickly apparent. The narrator says, “A man can fit in a man box only one way, but a card can fit in a card box four ways—front upside down, back up, back upside down, as well as face up.” This sort of explanation continues as the film progresses through objects which are increasingly more symmetrical until it shows a sphere which fits in its box an infinite number of ways. Although the narrator mentions that a mathematician determines symmetry with “a form of algebra called group structure,” the main purpose of the film is to communicate a direct understanding of and feeling for the basic concept of symmetry.