English: The boundary of the boundary of a 2-simplex (left) and the boundary of a 1-chain (right) are taken. The boundary of a boundary is always trivial, always sums to 0. A nontrivial cycle is something that closes up like the boundary of a simplex, in that its boundary is 0, but which isn't actually the boundary of a simplex or chain. Two cycles are homologous when their difference is a boundary, and in that sense two closed paths are equivalent if one can turn one into the other by passing through a solid region.