  We present the distributions of geometrical covering factors of active
galactic nuclei (AGNs) dusty tori (f2) using an X-ray selected complete sample
of 227 AGN drawn from the Bright Ultra-hard XMM-Newton Survey. The AGN have z
from 0.05 to 1.7, 2-10 keV luminosities between 10^42 and 10^46 erg/s and
Compton-thin X-ray absorption. Employing data from UKIDSS, 2MASS and the
Wide-field Infrared Survey Explorer in a previous work we determined the
rest-frame 1-20 microns continuum emission from the torus which we model here
with the clumpy torus models of Nenkova et al. Optically classified type 1 and
type 2 AGN are intrinsically different, with type 2 AGN having on average tori
with higher f2 than type 1 AGN. Nevertheless, ~20 per cent of type 1 AGN have
tori with large covering factors while ~23-28 per cent of type 2 AGN have tori
with small covering factors. Low f2 are preferred at high AGN luminosities, as
postulated by simple receding torus models, although for type 2 AGN the effect
is certainly small. f2 increases with the X-ray column density, which implies
that dust extinction an X-ray absorption takes place in material that shares an
overall geometry and most likely belongs to the same structure, the putative
torus. Based on our results, the viewing angle, AGN luminosity and also f2
determine the optical appearance of an AGN and control the shape of the
rest-frame ~1-20 microns nuclear continuum emission. Thus, the torus
geometrical covering factor is a key ingredient of unification schemes.
