Wednesday, 10 February 2010

A Little Geometry Music Please

Just for a kick, I thought I would pose a math problem.

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.
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