I am beginning to learn algebraic topology. For a 2-simplex $\{v_0v_1v_2\}$, the boundary is the chain 1-simplexes: $\partial(v_0v_1v_2)=v_0v_1+v_1v_2+v_2v_0$, which naively and intuitively, I see as the triangle that bounds the 2-simplex, that triangle being made up of the edges (1-simplexes) of the little triangular surface element. So far, so good. Then, $\partial\partial(v_0v_1v_2)=v_0-v_1+v_1-v_2+v_2-v_0=0$, which is algebraically convincing, and means that "the boundary of the boundary is empty".
Is this intuitively (and naively) because the boundary has to "loop around" the initial simplex, that characteristic of being a "loop" necessarily implying that following the boundary gets you to where you started from, which is as if you had not moved at all? Or is there a better way to intuitively grasp what is going on with that $\partial\partial = 0$?