We provide a new geometric representation of a family of fragmentation processes by nested laminations, which are compact subsets of the unit disk made of noncrossing chords. We specifically consider a fragmentation obtained by cutting a random stable tree at random points, which split the tree into smaller subtrees. When coding each of these cutpoints by a chord in the unit disk, we separate the disk into smaller connected components, corresponding to the smaller subtrees of the initial tree. This geometric point of view allows us in particular to highlight a new relation between the Aldous-Pitman fragmentation of the Brownian continuum random tree and minimal factorizations of the $n$-cycle, i.e. factorizations of the permutation $(1 \, 2 \, \cdots \, n)$ into a product of $(n-1)$ transpositions. We discuss various properties of these new lamination-valued processes, and we notably show that they can be coded by explicit Lévy processes.