What's So Special About Special Triangles?

This chapter answers a question related to high school geometry, but whose answer extends into number theory, abstract algebra, and complex analysis. In Trigonometry, students learn about two special triangles: the 30–60–90 triangle and the 45–45–90 triangle. These triangles are useful in identifying coordinates on the unit circle, and thus deriving values for the sine and cosine functions, but they aren’t the only ones. Every constructible regular polygon corresponds to a special triangle. The 30–60–90 triangle corresponds to constructing an equilateral triangle or a regular hexagon, and the 45–45–90 triangle corresponds to constructing a square. So, the question about special triangles is really one about geometric construction, following Plato’s three rules. Answering that question leads to Fermat primes, field extensions, and the complex plane.