  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 L\'evy processes.
