Take a square piece of paper oriented with its top horizontal. Fold it along any line that passes through the center and forms an angle of between 0 and 45 degrees with the horizontal.
The outline of the resulting shape is a nonagon.

What angle will maximize the perimeter of this nonagon?

What angle will maximize the area of this nonagon?

Is there any other single fold (not through the center) that can do better for either of these?

(In reply to refining the answer-computer solution by Charlie)

But it seems like we still need to end up with a nonagon after our non-central fold. Just folding post-its looks like I can achieve a nonagon with area 5/8 or so.

In fact, if I fold the unit square just above the line y=(1/2)x + 1/8 I get a nonagon with area 5/8 + 1/96

Posted by Eric on 2006-03-08 02:13:16