  The twisted geometries of spin network states are described by simple
twistors, isomorphic to null twistors with a time-like direction singled out.
The isomorphism depends on the Immirzi parameter, and reduces to the identity
when the parameter goes to infinity. Using this twistorial representation we
study the action of the conformal group SU(2,2) on the classical phase space of
loop quantum gravity, described by twisted geometry. The generators of
translations and conformal boosts do not preserve the geometric structure,
whereas the dilatation generator does. It corresponds to a 1-parameter family
of embeddings of T*SL(2,C) in twistor space, and its action preserves the
intrinsic geometry while changing the extrinsic one - that is the boosts among
polyhedra. We discuss the implication of this action from a dynamical point of
view, and compare it with a discretisation of the dilatation generator of the
continuum phase space, given by the Lie derivative of the group character. At
leading order in the continuum limit, the latter reproduces the same
transformation of the extrinsic geometry, while also rescaling the areas and
volumes and preserving the angles associated with the intrinsic geometry. Away
from the continuum limit its action has an interesting non-linear structure,
but is in general incompatible with the closure constraint needed for the
geometric interpretation. As a side result, we compute the precise relation
between the extrinsic geometry used in twisted geometries and the one defined
in the gauge-invariant parametrization by Dittrich and Ryan, and show that the
secondary simplicity constraints they posited coincide with those dynamically
derived in the toy model of [1409.0836].
