Wednesday, March 2, 2011

Galileo and the collapse of Dante's Inferno

The young Galileo took part in a curious controversy which raged among Italian Renaissance intellectuals, and his participation would lead him to some vital insights about structure:
Ever since its 1314 publication, scholars had toiled to map the physical features of Dante’s Inferno — the blasted valleys and caverns, the roiling rivers of fire. What Galileo said, put simply, is that many commonly accepted dimensions did not stand up to mathematical scrutiny. Using complex geometrical analysis, he attacked a leading scholar’s version of the Inferno’s structure, pointing out that his description of the infernal architecture — such as the massive cylinders descending to the center of the Earth — would, in real life, collapse under their own weight. Later, Galileo realized the leading rival theory was wrong, too, and that even the greatest scholars of the time simply didn’t understand how real-world structures worked.
Debating the mechanics of the Inferno might sound like intellectual horseplay, the 16th-century equivalent of MIT cafeteria debates about the viability of “Star Trek” teleporters. But there was more to the lectures than this. The insights Galileo gleaned from analyzing Dante’s measurements in fact anticipated a vital principle of structural engineering.
Galileo defended the Infernal design of Antonio Manetti against that of Alessandro Vellutello:
The various levels of Manetti’s Inferno are regularly spaced, for the most part, with 1/8 the radius of the earth between each level and the next. In particular the first level, Limbo, is at a depth of 1/8 the radius of the earth below the surface, and the shell of material down to this depth forms a cap of this thickness over the whole of Hell. Vellutello’s Inferno, by contrast, is much smaller, located near the center of the earth, and only about 1/10 the radius of the earth in height, making it, as Galileo is quick to say, ridiculously small, only 1/1000 the volume of Manetti’s.
[Galileo describes] a scale model of the roof of the Inferno, including a certain anteroom hollowed out of it, at a scale of about 1 braccia [about 26 inches] to 100 miles. A normal man is 3 braccias tall, so the model suggests a large domed roof, somewhat smaller than the famous Brunelleschi dome of the Florentine cathedral which, as Galileo says, is less than 4 braccias thick and supports itself beautifully. This is a convincing argument that Manetti’s model can support itself – but only until you realize that the argument assumes scale invariance! Could you really scale it up by a factor of 100,000? Absolutely not! The scaled up version is effectively weaker by that enormous factor and would immediately collapse of its own weight.

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