MATHS BITE: Heptadecagon

A heptadecagon (or a 17-gon) is a seventeen sided polygon.

File:Regular polygon 17 annotated.svg
Regular Heptadecagon | Wikipedia

Constructing the Heptadecagon

In 1796, Gauss proved, at the age of 19 (let that sink in…) that the heptadecagon is constructible with a compass and a straightedge, such as a ruler. His proof of the constructibility of an n-gon relies on two things:

  • the fact that “constructibility is equivalent to expressibility of the trigonometric functions of the common angle in terms of arithmetic operations and square root extractions“;
  • the odd prime factors of n are distinct Fermat primes.

Constructing the regular heptadecagon involves finding the expression for the cosine of  2\pi /17 in terms of square roots, which Gauss gave in his book Disquistiones Arithmeticae:

{\displaystyle {\begin{aligned}16\,\cos {\frac {2\pi }{17}}=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {34-2{\sqrt {17}}}}-2{\sqrt {34+2{\sqrt {17}}}}}}\\=&-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+\\&2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}.\end{aligned}}}
Source: Wikipdia

An explicit construction was given by Herbert Willian Richmond in 1893.

Regular Heptadecagon Using Carlyle Circle.gif
Source: Wikipedia

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