The Monster and The Woman

In the 1953 film, "The Monster and the Woman" (a.k.a "The Four-Sided Triangle"), friends Bill and Robin collaborate to develop a technology that replicates objects. When Robin and their friend/lab assistant, Lena, get married, an envious Bill makes some improvements to their "Reproducer" and tries it on a guinea pig, a bunny, and then on Lena herself. The result of the latter effort is Helen--Lena's doppelganger down to the last detail. Bill is overjoyed, but it turns out that the replica has a mind of her own (or, at least, Lena's…)

The film's more commonly used title, "The Four-Sided Triangle", refers to the old drama standby we know as "the love triangle." In a love triangle, two people are in love with one other person. Since the assumption of these stories tends to be that monogamy is the route to a happy ending, the triangle generates tension and forces the story to get more interesting than it might if everyone just got along. In a drama, to resolve the love triangle you have to dissolve the love triangle.

Storytellers also use the concept of a triangle to describe the structure--or arc--of many stories. The 19th century dramatist, Gustav Freytag, developed a diagram to describe the plots of ancient Greek tragedies. Dubbed "Freytag's Pyramid" or "Freytag's Triangle," the diagram features an upward pointing triangle. The left side represents a work's "exposition" and "rising action," the top angle is its "turning point," and the right side its "falling action" and "resolution." Though Freytag intended to describe what he saw in Greek drama, his diagram has been used for years as a prescription for countless storytellers ever since.

But writers and filmmakers aren't the only ones to have fun with this shape. Mathematicians, philosophers, and theologians have argued over the question "Can a four-sided triangle exist?" If you've learned that a triangle is a polygon with three corners and three straight sides, the answer might seem simple: No. But if you consider that this definition came into being after debates between heavy-duty thinkers, you might come to see this fundamental unit of geometry in a whole new light. Proclus Diadochus (410 – 485 A.D.), for instance, allowed that there could be a four-sided triangle if one of the sides had an external angle (think of something like a chevron).

The loopholes in the definition of "triangle" that might allow this argument to stand were eventually closed by the addition of the "three sides" bit, but it's likely Proclus wouldn't have minded the change. Just as the entertainment value of a Love Triangle is in the journey it forces the characters to take, Proclus considered the primary value of geometry to be its power to exercise philosophers' minds.

Kevin Wortman

Mathematician Kevin Wortman of the University of Utah discusses geometry and the film “The Monster and the Woman” (a.k.a., "The Four-Sided Triangle”).

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