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Q&A #3565


17-gon with straight-edge and compass

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From: Pat Ballew (for Teacher2Teacher Service)
Date: Apr 01, 2000 at 23:58:07
Subject: Re: 17-gon with straight-edge and compass

Francois,
  Let me instead, call on the great geometer, John H Conway who wrote the
following...

     The neatest construction I know is due to Richmond - I call it the
     "quadruple quadrisection constriction":

        1) quadrisect the perimeter of the circle, by points  N,S,E,W;

        2) quadrisect the radius  ON  by the point A;

        3) quadrisect the angle  OAE  by the line  AB;

        4) quadrisect the straight angle BAC by the line AD:

                        N
                    I   |
                        |       J
                     C  |
                        F
                        A
                        |
          W---------G-D-O-B-----H-----E
                        |
                        |
                        |
                        |
                        |
                        |
                        S

         5) draw the semicircle DFE, cutting  ON  in  F;

         6) draw the semicircle GFH, centered at B;

         7) cut the semicircle WNE by the perpendiculars  GI  and  HJ  to WE.

     Then  I  and  J  are points of the regular heptakaidecagon on the
     circle ENWS that has one vertex at  E.

         I first saw this in Hardy and Wright's book on The Theory
         of Numbers, which is where I've just checked up on it.
         H & W confirm my impression of the history.  They say
         that Gauss worked out the general theory in Paragraphs
         335-366 of his Disquisitiones, but that the first explicit
         construction was given by Erchinger, for whom they refer
         to Gauss' Werke, vol II, pp186-187.

       This "Quadruple Quadrisection" construction (my name) is due
       to Richmond, who gave it in the Quarterly Journal of Math, 1893.

       Of the four or five constructions I have seen, it is definitely
       the nicest.  If you intersect the other quadrisectors of that
       straight angle with WE and treat the resulting points similarly,
       you can get more vertices in the same way - but it's easier to
       use your compasses to step around the circle from the ones given,
       for which the constructing points are the most conveniently
       situated.

     John Conway


 -Pat Ballew, for the Teacher2Teacher service

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