Heptadecagon: fact and fiction

Euclid developed his geometry using a straightedge and compass and believed that every number could be constructed with these instruments alone. This prompted a long search to determine which numbers were actually constructible, a quest that culminated only in the 19^th century with the research of Pierre-Laurent Wantzel (1814–1848).

In 1837, Wantzel proved, in particular, the converse of a theorem by Carl Friedrich Gauss (1777–1855), published in 1801 in his Disquisitiones arithmeticae. It stated that "if n = 2*^k p*₁ p₂…*p_q, where k or q may be zero and the p_i are distinct Fermat numbers, then the regular n*-gon is constructible with a straightedge and compass," where a Fermat number is a prime number of the form $F_r=2^{2^r}+1.F\_r\; =\; 2^\{2^r\}\; +1.$

Gauss's exact construction -----------------------------------

This theorem, with its dual parentage, is known as the "Gauss–Wantzel theorem."

Since the perpendicular bisector of a line segment can be constructed with a straightedge and compass, the number of sides of a polygon can always be doubled. The first Fermat numbers are F₀ = 3 and F₁ = 5, so regular polygons with 3, 4, 5, 6, 8, 10, 12, 15 or 16 sides can be constructed.