Compass and straightedge - the regular Heptadecagon

One of the oldest and best-known topics in the history of mathematics are compass and straightedge (or compass and ruler) constructions. I will not illustrate them here, and neither will I explain why it's so remarkable that the regular heptadecagon, the 17 equal-sided polygon, actually CAN be constructed that way (you can find all you need just asking Saint Google about "compass and straightedge" and "regular heptadecagon"). Instead I prefer letting you get astonished by the amazing complexity of the procedure (which you can find described step by step here below). Enjoy!

1. Draw a line and choose on it points O and A. Draw a circle centered at O with radius OA, it will be the heptadecagon's circumcircle.

2. Draw the line through O that is perpendicular to OA and call B one of its two intersections with the circle.

3. Find the midpoint C of segment OB.

4. Find the midpoint P of segment OC.

5. Find the point V where segment PA cuts the circle centered at C with radius CO.

6. Draw the circle centered at P with radius AV locating F on PA and I on BO (there are two intersection points, choose the one nearer to O than to B).

7. Find H, the midpoint of arc FI.

8. Find J, the midpoint of arc HI, and K, symmetric to J with respect to P.

9. Locate G, the midpoint of the semicircle KJ (choose the one containing I).

10. Draw the line bisecting the right angle JPG, and let L denote its interception with OA.

11. Find the midpoint Z of segment AL.

12. Locate N where the circle centered at Z with radius ZL cuts segment CP.

13. Draw the circle centered at R, intersection point between PJ and OA, with radius RN. It strikes OA at point U, which is situated inside segment AZ very near to Z.

14. Draw the perpendicular line to OA through U. Call S its intersection point (choose one of the two possible ones) with the circle centered at O with radius OA.

15. A and S are vertices of the heptadecagon, use the arc AS to find the others. If you choose and keep a direction (clockwise for example) you should encounter exactly your starting point after three complete turns.

16. Draw the heptadecagon's sides.

17. If desired, erase construction lines.

Category:

Howto & Style

Tags:

• compass
• straightedge
• ruler
• construction
• regular
• polygon
• 17
• heptadecagon
• mathematics
• maths
• math

[MrLuis5581]

hi do u know how to draw a squre inside a circle with a diameter of 1 1/2?

[lesath82]

@MrLuis5581 I don't understand exactly what you ask: what do you mean a circle with a diameter of 1 1/2? With respect to what?

[VictoryAtNight]

In step 6, does the radius of the arc centered on P going through F and I matter? I can't see where the radius it came from, it just seems to be an arbitrary radius arc used in order to bisect several angles in steps 7 through 10, and I'm not sure if it's ever used again later in the construction.

[lesath82]

@VictoryAtNight It is all explained step by step in the description! Indeed, what you ask was the hardest point to figure out!

[lesath82]

@VictoryAtNight You might be right. To be honest, I won't probably check it in the near future, sorry! :-(

[OhSnapSpaceCodet]

on step 2 whats with the extra lines?? i dont get it

[lesath82]

Which extra lines??

[FlyDan11]

Its amazing to think that Carl Friedrich Gauss discovered this when he was a teenager. Damn.

[Hewpie]

Wahey! Did it using GeoGebra software - worked out perfectly with no gaps on full zoom...

[VictoryAtNight]

@lesath82 I've watched the construction over again, and I'm still curious why the radius of the arc matters, as it seems to me that in steps 7 through 10 you're just using the arc to identify points on segments radiating out of P which you can then use for bisecting several angles at P and creating new segments raiding out of P at certain angles. It seems to me that using AV as the radius might have been a choice of convenience.