When the lithosphere subducts it sinks into the planetary interior, driving tectonic processes that determine where minerals and hydrocarbons form, and where earthquakes take place. It stands to reason then that we should get to know as much as possible about what happens to the lithosphere as it founders, and the tectonic processes that such a process engenders. Here the 4D (i.e. 3D + time) geometrical evolution of the planetary lithosphere is our major concern, and we are discovering that astonishingly little is known about what might seem at first such a simple thing. A summer scholar, or a PhB student, would gain the benefit of being immersed in a hard-hitting dynamic team environment, at the very forefront of tectonics research.