These are the instuctions for the ruler and compass constution of a
17-sided polygon shown on YouTube..
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.