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Homogeneous (i.e. spatially stationary) random tessellations of the Euclidean plane are constructed which have the characteristic property to be stable under the operation of iteration (or nesting), STIT for short. It is based on a Poisson point process on the space of lines that are endowed with a time of birth. A new approach is presented that describes the tessellation in the whole plane. So far, an explicit geometrical construction for those tessellations was only known within bounded windows.