In section 2,
we will make precise the notion of cutting a manifold into tubes,
and show how a cutting into tubes gives
a lower bound for the bass note.
As the subject is physics and not geometry,
the tone will be formal.
In section 3,
we will show how to cut 
into tubes,
so as to prove the existence of
a universal lower bound for 
.
Here we are back on solid ground,
and the tone will be more relaxed.