The Mathematical Tourist
By Ivars Peterson
As anyone who lives in a tiny apartment or cluttered room has experienced, opening or closing a closet door requires a certain amount of floor space.
Suppose, for example, that a closet has a standard door, which can swing open so that it's at right angles to the closet interior. As it opens (or closes), the door covers a quarter of a circle, which represents the area that must be kept clear of obstacles. If the closet (or door) has width r, the door requires πr2/4 square feet of floor space.
To save floor space, some closets have so-called bifold doors. In this case, the door has hinges at one side and in the middle so that it folds as it opens. The unhinged side typically runs along a track that keeps it aligned with the closet opening. Note that "bifold" is a misnomer. There is only one fold (but two panels).
When Derek Seiple was a high school student, he wondered how much floor space is needed to accommodate the opening and closing of a bifold door. How big is the saving over a standard door? When he got to college (Penn State University at DuBois), Seiple was encouraged to investigate the problem, and the results appear in the April Mathematics Magazine.
"If you have a closet (or any doorway) covered with a bifold door there is an astroid lurking just inside and the only way you can get to it is to coax it carefully with a little bit of calculus," Seiple and his coauthors note. "If your door has more than one fold there are even more interesting objects waiting to be discovered."
Seiple's analysis shows that a bifold door traces out a path that consists of two curves. Given that each panel has a width r/2, a closing bifold door first sweeps out a circular arc of radius r/2. At 45 degrees, however, its path changes. Whereas the first part of the path was convex, the second part is concave. It now traces out part of a type of curve known as an astroid.
In this case, the area swept out by the door is 5πr2/64. That's a saving of nearly 70 percent, compared with a standard door.
"It is clear that adding 2, 3, 4, or n folds will reduce the floor space required even further," Seiple says.
At the same time, "adding more hinges has no effect on the astroidal portion of the curve," he adds. "The very same astroid appears regardless of the number of folds in the door as long as all of the panels are hinged so that they make the same angle with the front of the closet."
Interestingly, the entire path would be an astroid if you happened to have a "door" with no hinges. As one side of the door moved along a track across the closet opening, the other side would move along a track at right angles to the closet opening.
You'd see the same path traced by an initially vertical ladder that slips down and away from a wall.
Comments are welcome. You can reach Ivars Peterson at email@example.com.
Seiple, D., E. Boman, and R. Brazier. 2007. Mom! There's an astroid in my closet! Mathematics Magazine 80(April):104-111.
Last week: Integral Heptagons