page_title: Heptadecagon
section_title: Construction
hierarchical_section_title: Heptadecagon / Regular heptadecagon / Construction
context_page_description: In geometry, a heptadecagon or 17-gon is a seventeen-sided polygon.
context_section_description: As 17 is a Fermat prime, the regular heptadecagon is a constructible polygon (that is, one that can be constructed using a compass and unmarked straightedge): this was shown by Carl Friedrich Gauss in 1796 at the age of 19. This proof represented the first progress in regular polygon construction in over 2000 years. Gauss's proof relies firstly on 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, and secondly on his proof that this can be done if the odd prime factors of , the number of sides of the regular polygon, are distinct Fermat primes, which are of the form  for some nonnegative integer . Constructing a regular heptadecagon thus involves finding the cosine of  in terms of square roots, which involves an equation of degree 17—a Fermat prime.  Gauss's book Disquisitiones Arithmeticae gives this as (in modern notation):
  Constructions for the regular triangle, pentagon, pentadecagon, and polygons with 2ʰ times as many sides had been given by Euclid, but constructions based on the Fermat primes other than 3 and 5 were unknown to the ancients. (The only known Fermat primes are Fₙ for n = 0, 1, 2, 3, 4. They are 3, 5, 17, 257, and 65537.)
  The explicit construction of a heptadecagon was given by Herbert William Richmond in 1893. The following method of construction uses Carlyle circles, as shown below. Based on the construction of the regular 17-gon, one can readily construct n-gons with n being the product of 17 with 3 or 5 (or both) and any power of 2: a regular 51-gon, 85-gon or 255-gon and any regular n-gon with 2ʰ times as many sides.
  Another construction of the regular heptadecagon using straightedge and compass is the following:
  T. P. Stowell of Rochester, N. Y., responded to Query, by W.E. Heal, Wheeling, Indiana in The Analyst in the year 1874:
  "To construct a regular polygon of seventeen sides in a circle.Draw the radius CO at right-angles to the diameter AB: On OC and OB, take OQ  equal to the half, and OD equal to the eighth part of the radius: Make DE and DF each equal to DQ and EG and FH respectively equal to EQ and FQ; take OK a mean proportional  between OH and OQ, and through K, draw KM parallel to AB, meeting the semicircle described on OG in M; draw MN parallel to OC, cutting the given circle in N – the arc AN is the seventeenth part of the whole circumference."
  The following simple design comes from Herbert William Richmond from the year 1893:
  "LET OA, OB (fig. 6) be two perpendicular radii of a circle. Make OI one-fourth of OB, and the angle OIE one-fourth of OIA; also find in OA produced a point F such that EIF is 45°. Let the circle on AF as diameter cut OB in K, and let the circle whose centre is E and radius EK cut OA in N₃ and N₅; then if ordinates N₃P₃, N₅P₅ are drawn to the circle, the arcs AP₃, AP₅ will be 3/17 and 5/17 of the circumference."
  The point N₃ is very close to the center point of Thales' theorem over AF.
  The following construction is a variation of H. W. Richmond's construction.
  The differences to the original:
  The circle k₂ determines the point H instead of the bisector w₃.
  The circle k₄ around the point G' (reflection of the point G at m) yields the point N, which is no longer so close to M, for the construction of the tangent.
  Some names have been changed.
  Another more recent construction is given by Callagy.
caption: Heptadecagon in principle according to H.W. Richmond, a variation of the design regarding to point N