A Besicovitch (Kakeya) set contains a unit line segment in every direction, yet can have arbitrarily small area. This shows the classic Perron-tree construction: a triangle is repeatedly split and its halves slid over each other, so the 2n pieces still point in all directions but overlap more and more. Increase n and push α toward 1 to watch the area shrink.
A modern JavaScript + Canvas port of a 1998 Java applet by Terence Tao
(originally Besicovitch.java / Kakeya.java).