A line of unit length is enclosed within a two-dimensional area.

geometry/rotating.line New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A line of unit length is enclosed within a two-dimensional area.  What
is the smallest area that will allow the line to be rotated through 360
degrees in the two dimensions, remaining enclosed at all times?
Describe the shape of the area.

Show solution

You can actually make the enclosed area as small as you want.

Create a "star-shaped" area, with spikes (triangles?) radiating from a
small central part.  Each spike should be as long as the line.  You can
then rotate the line through 360 degrees by moving it from one spike to
the one nearly opposite, repeatedly, changing its angle by a small
amount at each step.  With a large enough number of spikes, you can
make the enclosed area as small as you like.

-- Jeff Kenton <jkenton@world.std.com>, March 1998