Grab a pen and draw a figure. Follow a few rules: keep your lines straight, don’t pick up your pen, don’t cross the lines, and finish at the spot where you started. You’ll have a polygon.
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2007/05/6790.jpg?resize=150%2C145&ssl=1)
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2007/05/6791.jpg?resize=300%2C119&ssl=1)
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2007/05/6792.jpg?resize=300%2C145&ssl=1)
Polygons are among the simplest mathematical objects in existence. Even so, they hold mysteries. Here’s one: What is the polygon with the largest area that has n sides and fixed diameter?