If you can divide a circle into 4 even wedges then you're
half way to dividing it into 5.
Let's assume you've divided
a circle into 4 even sections with a horizontal line and a vertical line. Bisect the top
half of the vertical line (the line segment from the center of the circle to the top of
the circle). If you're not sure how to do this, the following is one
method:
Set your compass to the radius of the circle, place
its pivot where the vertical line crosses the top of the circle. Draw an arc that
crosses the circle in two places. Draw a line between the points the arc intersects the
circle - it will bisect the top half of the vertical
line.
Put the pivot of the compass on the bisect point you
just made. Adjust the compass so that it reaches one of the points the horizontal line
intersects the circle and draw an arc from it to the other intersection point
between circle and horizontal line.
Put the pivot of the
compass at the intersection of the arc you just drew and the vertical line it
crossed. Adjust the compass so that it can touch one of the points of intersection
between the horizontal line and the circle. The compass is now at the length of a
secant that will divide the circle into five even sections just like the radius of a
circle is the secant that divides the circle into six even
sections.
So now, just like divide a circle in to six
sections, pick a point as the first of your 5, set the compass pivot there and mark
intersection points clockwise and counterclockwise from it. Using these two new points
to start at, use the compass to make two more and there are your 5 evenly spaced points
on the circle. Draw line segments from these points to the center of the circle for 5
even wedges or extend the segments to the opposite side of the circle for ten
wedges.
There is a drawing at the web site
provided that does the same thing except it bisects one of the horizontal lines – the
results are the same.
You can now get a pretty good
approximation for a dividing a circle into 9 or 11 parts. On one piece of paper draw a
circle and divide it into 10 sections – make sure the line segments that divide the
circle extend past the circle a bit. Now draw two more circles concentric with the
original, divided one; make the first 10% larger and the second 10% smaller than the
original.
On a separate piece of paper draw a circle the
same size as the one you divided into 10 parts. The length between adjacent division
lines where they intersect with the smaller concentric circle you drew on the other
piece of paper are pretty close to the length of a secant that can be used to divide
your fresh circle into 11 sections. Conversely the adjacent intersections from the
larger circle will give you a good first try at dividing a circle of the original size
into 9 sections.
There’s also a geometric method (done
with compass and straightedge only) for approximating dividing a circle into seven even
wedges, but it has slipped my mind and I haven’t been able to find my notes on it
either.
No comments:
Post a Comment