Hidden Harmonies by Ellen Kaplan & Robert Kaplan

Author:Ellen Kaplan & Robert Kaplan
Language: eng
Format: epub
ISBN: 9781608192915
Publisher: Bloomsbury USA
Published: 2011-01-10T16:00:00+00:00

Here is a proof with a hypermodern flavor, outflanking the Theorem. It, too, cunningly flirts with, then slips past, trigonometry.

A right triangle is uniquely determined by the length of one side and the size of an acute angle. Let’s choose c as our side and the smallest

angle, ø. We know this triangle’s area is

Let’s look at areas again:

What happened? The pea must have been under one of those shells, and now it’s gone!

This is an instance of ‘dimensional analysis’, where we needn’t inquire into the nature of the function f, just glean the desired result from this way of looking, which dates back to the physicist Percy Bridgman in the 1920s.20 It allows you to solve a problem by attending to no more than the dimensions of its variables (thus finding easily, for example, that the drop in pressure of a fluid flowing through a pipe is proportional to Q/R4, where R is the pipe’s radius and Q the volume of flow in a unit of time). You might think of this proof as a streamlined version of the Pythagorean, sped up by removing all the finicking Euclidean constrictions.

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