  We use numerical simulations and linear stability analysis to study an active
nematic layer where the director is allowed to point out of the plane. Our
results highlight the difference between extensile and contractile systems.
Contractile stress suppresses the flows perpendicular to the layer and favours
in-plane orientations of the director. By contrast, extensile stress promotes
instabilities that can turn the director out of the plane, leaving behind a
population of distinct, in-plane regions that continually elongate and divide.
This supports extensile forces as a mechanism for the initial stages of layer
formation in living systems, and we show that a planar drop with extensile
(contractile) activity grows into three dimensions (remains in two dimensions).
The results also explain the propensity of disclination lines in three
dimensional active nematics to be of twist-type in extensile or wedge-type in
contractile materials.
