Mr. Hilbert was sitting alone at the dinner table of the Grand Hotel restaurant, waiting for his two friends, Mr. Euler and Mr. Langrange, to show up. It was 5 minutes past six already, but knowing the two of them, it wasn't too much of a surprise to Mr. Hilbert that their arrivals times were a bit variational.
Being slightly bored, Mr. Hilbert started to play with his napkin ring. At first, it looked just like a hollow sphere with a circular hole, but upon closer inspection, the hole seemed more like this:
where the hole is the shaded area, the intersection of four quarter-circles of radius $r$. And this hole was drilled straight through a sphere of diameter $r$:
"Hmm, I wonder how much volume is cut out of the original sphere in order to get this napkin ring?" wondered Mr. Hilbert.
Can you figure it out? Express your answer in terms of $r$.