Suppose you pick n points equally spaced around a circle of Radius R. Now pick a number relatively prime to n, call it p, and starting at some point on the circle, draw the chord connecting that point to the point p steps away around the perimeter. Start at that point and repeat the process with the point p steps away again.. keep going and eventually you come back to the point where you started. In the image below, n=12 and p = 5. Now draw the new circle that is tangent to all the chords. Call the radius of this circle r. Now, what is the ratio of the two radii, r/R?
You can download a file here that will draw this for up to n=19 (of course you can adjust the slider to make more). You have to have geogebra downloaded first (its free). First solution wins a shiny "attaboy" (or atta-person for the gender sensitive).
When I took those old Woodrow Wilson summer programs these were called "star polygons". The one in the picture is called the {12/5} star polygon.. This notation is often called a Schläfli symbol after the 19th-century mathematician Ludwig Schläfli. For example, you can indicate an equilateral triangle by {3} or a square by {4}. The classic pentagram (the star you draw without picking up the pencil, is the {5/2} star polygon. Of course {5,3} would be the same figure. An interesting challenge for students is to find the angle at any vertex of {n/p} as a function of n and p.
kind of thinking out loud...
ReplyDeleteI look at the angle between the chord and the x-axis (the chord that has A as the endpoint). Using inscribed angles of circles, this angle is pi/12 radians (cuts off an arc of pi/6). Since the radius of the larger circle is 1, I consider a perpendicular from the origin to this chord. The radius of the circle tangent to each of these chords must be sin(pi/12). That would need to be the desired ratio.
I will need to think about that some more, but I think that is correct.
It really is that easy. I often lament my students inability/unwillingness to think geometrically and used a very simple version of this as an example. I drew a square but rotated it 30 degrees so it was not aligned with the 12 o'clock and ask them to find the radius of the inscribed circle... offered them the problem for homework in place of that night, or any night they had missed home work credit... NO takers, not one.. This is a Pre-calc class... Some days, they make me feel old.
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