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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