The regular polygon of 17 sides is called the heptadecagon, or sometimes the heptakaidecagon. Gauss proved in 1796 (when he was
19 years old) that the heptadecagon is constructible
with a compass and straightedge.
Gauss's proof appears in his monumental work Disquisitiones Arithmeticae.
The proof relies on the property of irreducible polynomial
equations that roots composed of a finite number of square
root extractions only exist when the order of the equation is a product of
the form ,
where the
are distinct primesof
the form
(1)
known as Fermat primes. Constructions for the regular triangle (), square (), pentagon (), hexagon (), etc., had been given by Euclid, but constructions based
on the Fermat primes were unknown to the ancients. The first explicit construction
of a heptadecagon was given by Erchinger in about 1800.
The trigonometric functions and are both algebraic
numbers of degree 8 given respectively by
14. Use
and
to get the remaining 15 points of the heptadecagon around the original circle
by constructing ,
, , , , [filled circles], , , , , , [single-ringed filled circles], , , , , and [double-ringed filled circles].
15. Connect the adjacent points for to 17, forming the heptadecagon.
This construction, when suitably streamlined, has simplicity 53. The construction of Smith (1920) has a greater simplicity
of 58. Another construction due to Tietze (1965) and reproduced in Hall (1970) has
a simplicity of 50. However, neither Tietze (1965)
nor Hall (1970) provides a proof that this construction is correct. Both Richmond's
and Tietze's constructions require extensive calculations to prove their validity.
DeTemple (1991) gives an elegant construction involving the Carlyle
circles which has geometrography symbol and simplicity
45. The construction problem has now been automated to some extent (Bishop 1978).