Gauss: Construction of the Heptadecagon

Carl Friedrich Gauss not only solved the seventeenth root of unity in terms of radicals but also realized that his solution indicated that a heptadecagon (a regular, seventeen-sided polygon) could be constructed by compass and ruler.

In my previous blog, I showed Gauss's solution of the seventeenth root of unity. In today's blog, I will show how this solution provides a recipe for constructing a heptadecagon.

For those ready to cut to the chase, a YouTube video of construction of the heptadecagon can be found here.

The algorithm below is a bit more complicated than the above YouTube video. It is taken from Hardy and Wright's Introduction to the Theory of Numbers (see the reference below).

Algorithm: Construction of a heptadecagon using compass and ruler

























































































(1) Draw a line connecting point A and P1
(2) Let C be the midpoint
(3) Draw a circle with C as the center and the radius equal to CP0
(4) Let DE be a line perpendicular to CP0 which passes through C.
(5) Let F be the midpoint on CD and G be be the midpoint on CF.
(6) Draw a line connecting G and P0.
(7) Find a point H on GE where GH = GP0
(8) Draw an arc with center G and radius GP0 from P0 to H.
(9) Draw a line connecting P0 to H
(10) Let I be the midpoint on HP0.
(11) Draw a line passing from G through point I onto the arc and label the point on the arc J.
(12) Draw a line connecting J and H.
(13) Let K be the midpoint on JH.
(14) Draw a line from G to K and let L be the point where GK intersects with CP0.
(15) Draw a line from point G that is perpendicular to GL and label the point N where this line intersects with AC.
(16) Bisect the angle ∠ NGL and label P the point where this angle bisector intersects with AC.
(17) Let Q be the midpoint between P and P0.
(18) Draw a circle with center Q and radius QP0.
(19) Label R the point where this circle intersects with CD
(20) Draw a circle with center L and radius LR
(21) Label S the point where this circle intersects with AC.
(22) Draw a line from S that is perpendicular to AC and label P5 the point where this line intersects with the top of the circle.
(23) Draw a line from Q that is perpendicular to CP1 and label P3 the point where this line intersects with the top of the circle.
(24) Draw a line connecting P3 and P5 and label the midpoint W.
(25) Draw a line from C to W and label the point P4 where CW intersects with the top of the circle.
(26) Now, all the rest of the points can be completed in the following way:
(a) Draw a circle with center P3 and radius P3P4.
(b) Label P2 where this circle intersects with the larger circle.
(27) Repeat this same step for all missing remaining points.

In my next blog, I will show the proof that the above construction really works. This proof is based on the solution for the seventeenth root of unity.

References

• "The 17-gon", Jim Loy's Home Page
  
• G. H. Hardy & E. M. Wright, An Introduction to the Theory of Numbers, Oxford Science Publications, 1979.