Kakeya Needle
You have a needle of length 1 and you are spinning it around. If it turns around its own centre then it will trace out a circle with radius 1/2, and therefore an area of π/4 = 0.785... But could we turn it in such a way that it traces out a smaller area?
Well if you draw out an equilateral triangle of height 1 like the deathly hallows below, then the needle (elder wand) can manoeuvre itself around by swinging around one end into a corner of the triangle and then repeating the process.
It was easier to nick this than load up paint. Plus everyone loves Harry Potter. Ignore the circle.
This has a rather reduced area of 1/√3 = 0.577... and it turns out to be the best solution for a convex shape. However if we allow a concave shape instead we could reduce the area slightly by buckling the sides of this triangle.
A 3-pointed deltoid
Glorious! It is difficult to imagine how we could possibly improve on this shape in terms of area. However by adding in more "points" which we can park one end of our needle in, so that we can turn the opposite end, we can create something rather fractaly. Here is one which has been formed by an iterative process which has a lower bound on its area of (π/24)(5-2√2) = 0.284...
If I ever start a Scientology style cult then I think I could run with this logo.
If you drop the restriction that the shape should be simply connected then there becomes no lower bound for the area and you can turn the needle in a space with measure 0; which you can read as it tends to zero as you get deeper into the iterations of your fractal.
