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Re: Consturction of 17-sided regular polygon
Posted:
Feb 8, 1999 2:23 PM
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On Sun, 31 Jan 1999, John Conway wrote:
(in geometry-pre-college NG/ML)
>On Sun, 31 Jan 1999, Peter Hung wrote:
>
>> I have trying to find out the procedure for constructing a regular 17-sided
>> 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
> |
> W---------G-D-O-B-----H-----E
> |
> |
> |
> |
> |
> |
> 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 Andre-Marie Ampere, d'apres des documents conserves dans les
archives de l'academie des sciences.
C. R. Acad. Sci., Paris 204(1937) 925-928.
[Republished in: Enseign. Math., II. Ser. 3(1957) 31-34]
Also, he is the author of the book:
Henri Lebesgue: Lecons sur les constructions geometriques au college de
France en 1940-1941.
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
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K
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X--F*----D*-H--E*-C--O-F-D-----G-E----A
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B
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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. 17-gon.
(We have arc(A_4A_6) = 4Pi/17. We bissect it to find A_5, and therefore
the r.17-gon's side).
It is a memorizable construction: 5 semicircles and 4 _L _L (:peprpendiculars).
The question is: Is it a construction of Andre-Marie Ampere (see paper above)
or H. Lebesque's himself ? (My source calls it as Lebesgue's)
Antreas
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