The iteration of random tessellations and a construction of a homogeneous process of cell divisions

A random tessellation of .d is said to be homogeneous if its distribution is invariant under all shifts of .d. The iteration of homogeneous random tessellations is described in a new manner that makes it evident that the resulting random tessellation is homogeneous again. Furthermore, a tessellation-valued process is constructed, the random states of which are homogeneous random tessellations stable under iteration (STIT). It can be interpreted as a process of subsequent division of cells.