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Re: Consturction of 17sided regular polygon
Posted:
Feb 8, 1999 2:23 PM


On Sun, 31 Jan 1999, John Conway wrote: (in geometryprecollege NG/ML)
>On Sun, 31 Jan 1999, Peter Hung wrote: > >> I have trying to find out the procedure for constructing a regular 17sided >> polygon. > >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 >  > WGDOBHE >  >  >  >  >  >  > S > > 5) draw the semicircle DFE, cutting ON in F; > > 6) draw the semicircle GFH, centred 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. > > John Conway
John, I discovered another nice (IMHO) construction by Henri Lebesgue.
Let me tell the story: Henri Lebesgue published in 1937 the following paper: Lebesgue, Henri: Sur une construction du polygone regulier de 17 cotes, due a AndreMarie Ampere, d'apres des documents conserves dans les archives de l'academie des sciences. C. R. Acad. Sci., Paris 204(1937) 925928. [Republished in: Enseign. Math., II. Ser. 3(1957) 3134]
Also, he is the author of the book: Henri Lebesgue: Lecons sur les constructions geometriques au college de France en 19401941. Paris : Gauthier  Villars, 1950 In pp. 148  149.he describes the construction of the r. heptakaidecagon (and in p. 145 of the r. pentagon).
I haven't seen HL's paper/book. Only brief descriptions of his constructions (in the book) published in a Greek periodical.
Here is the construction of the r. heptakaidecagon: Let (O) be a circle of center O.
Y A_6  A_4  K     XF*D*HE*COFDGEA  B       Z
1. Draw the diameters XA _L YZ
2. Quadrisect the radius OZ by B.
3. Draw CB _L BA (C lies on OX)
4. Draw the semicircle (C, CB) intersecting XA at D, D*
5. Draw the semicircle (D, DB) intersecting XA at E,E*
6. Draw the semicircle (D*, D*B) intersecting XA at F,F*
7. Draw the semicircle of diameter AE*, intersecting OY at K
8. Draw the semicircle (F, FK) intersecting XA at G, H.
9. Draw the _L _L from G, H, intersecting the (O) at A_4, A_6.
Now, A:=A_1, A_4, A_6 are vertices of the r. 17gon. (We have arc(A_4A_6) = 4Pi/17. We bissect it to find A_5, and therefore the r.17gon's side).
It is a memorizable construction: 5 semicircles and 4 _L _L (:peprpendiculars).
The question is: Is it a construction of AndreMarie Ampere (see paper above) or H. Lebesque's himself ? (My source calls it as Lebesgue's)
Antreas



