In the right place at the right time

The River Hero is a comic feature film (silent) released in 1928. It is co-directed and starring Buster Keaton who plays William, the son of Bill Canfield. Canfield owns a riverboat that competes with businessman John James King for control of the transportation of goods on the Mississippi River. William and Kitty (King's daughter) are in love, causing the anger of their parents.

Near the end of the film a hurricane passes through the city. William fights against the wind that carries him incessantly. After receiving a blow to the head, disoriented, he stands stunned in front of a house. The facade falls on him, but a providentially open window saves him from being crushed to death (the scene can be seen at approximately minute 59:00 of the film ). It seems that the actor did not allow himself to be bent in stunts and, in this one, he took a huge risk. Would you do any rehearsals before filming it? Of course, the protagonist remains unchanged during the scene…

This sequence has been covered on numerous occasions from an episode of the series MacGyver to some film by Jackie Chan passing through Deadpan (1997) by the filmmaker Steve McQueen in which he himself reproduces the scene starring Keaton almost 70 years before. For almost 20 minutes, from different angles, you can see the facade of a house that falls on McQueen, who, like Keaton in 1928, shows no reaction.

I consult every day the magnificent page Futility Closet [19659007] who shares amazing stories on various topics, including recreational mathematics. In one of his entries, Greg Ross introduces an article by James Metz published in the journal Mathematics Teacher, in which the author proposes different and simple calculations based on this scene from The river hero. In other words, he carries out a mathematical study of the scene, looking at some of its frames.

Metz comments in his article that the space around Keaton when the window "goes through" is really scarce … and he's right. And it proposes a series of reflections and calculations to accurately understand the details of the scene. We reproduce some of his observations below:

Suppose Keaton stands on the ground which is 5 inches (1 inch = 2.54 centimeters) below the “hinge” of the falling wall. Keaton is 5 feet 5 inches (1 foot = 30.48 centimeters; that is, Keaton is approximately 165 centimeters). Leaving a margin of 3 inches (7.62 centimeters) above your head, the window of height h should pass over you 5.25 feet (1 foot = 12 inches) above the hinge line. Viewing the scene from The River Hero after the complete fall of the wall, the back of his feet is approximately 0.5 feet (15.24 centimeters) in front of the bottom of the window . Before the fall, the bottom of the window is at a distance D from the hinge of the house (above it, of course). We summarize the above in a diagram:

What is the relationship between h and D? It is a simple relationship between the hypotenuse and the legs of a right triangle:

(D + h) 2 = (D + 0.5) 2 + 5, 25 2 .

If Keaton's shoulder width is 1.5 feet (about 45.72 centimeters) and the sides of the window become 3 inches ( about 7.62 centimeters) on each side of Keaton, what is the width of the window? Since 3 inches is 0.25 feet, the window has a width of 1.5 + (2 x 0.25) = 2 feet.

Looking at the scene again, suppose that the relationship between the height and width of the window is 7/11. What is the height h of the window? Since 7/11 = 2 / h, solving for, it is h = 22/7, that is, about 3 feet (about 91.45 centimeters).

We can now calculate D using the relationship between h and D obtained earlier . Solving for, D = 3.76 feet (about 114.6 centimeters).

These are just a few small calculations that help us analyze the situation. Will they be similar to those made by Keaton's team before filming the scene? As Metz points out in his article, you can generalize this analysis and ask yourself different "what if?" Questions.

References:

About the author: Marta Macho Stadler is a professor of Topology in the Department of Mathematics of the UPV / EHU, and a regular contributor to ZTFNews, the blog of the Faculty of Science and Technology of this university

The article In the right place at the right time has been written in Cuaderno de Cultura Scientific .

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