When I was younger my brother taught me how to use straight lines on graph paper to draw the things that comprise the Pittsburgh Steelers logo (which I learned much later are called hypercycloids). I thought it was awesome and I did it all the time. It was my version of that dumb angled S thing that everyone else drew when they were 10 years old.
I have often wondered if the circular shape you get out of such a drawing is, in fact, a circle. So I decided to figure it out.
So it looks like the shape would approach a circle as the number of line segments used increases. However, when an odd number of line segments are used, the middle one is a forty-five degree line. On the 3x3 figure, the line segment is two-thirds of the way between the center and the corner. On the 5x5 figure, it is seven-tenths of the way between the center and the corner. This is weird to me because for an nxn figure it will always be connecting the points (n+1)/2 units from the corner, and as n limits to infinity (n+1)/2 will limit to the midpoint. But the line segment actually connecting the midpoints would lie 1.06 radii away from the center, so it will not be on a circle.
We have two options here:
No comments:
Post a Comment