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GOLDEN TRIANGLES AND RECTANGLES

Beauty is truth, truth beauty, that is all

Ye know on earth, and all ye need to know.

–John Keats, Ode on a Grecian Urn

According to Martin Gardner (1914–2010), a popular Scientific American columnist, “Pi (c017-math-001) is the best known of all irrational numbers. The irrational number c017-math-002 is not so well-known, but it expresses a fundamental ratio that is almost as ubiquitous as pi, and it has the same amusing habit of popping up where least expected.” Gardner made this trenchant observation in 1959. In the preceding chapter, we found that the ubiquitous number c017-math-003 makes spectacular appearances in plane geometry. As can be predicted, it does also in solid geometry; see Chapter 18.

Some triangles, such as the golden triangle, are linked to the golden ratio in a mysterious way. We begin our pursuit with a simple definition.

17.1 GOLDEN TRIANGLE

An isosceles triangle is a golden triangle if the ratio of a lateral side to the base is c017-math-004.

We now pursue some interesting properties ...

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