  Discretization of general relativity is a promising route towards quantum
gravity. Discrete geometries have a finite number of degrees of freedom and can
mimic aspects of quantum geometry. However, selection of the correct discrete
freedoms and description of their dynamics has remained a challenging problem.
We explore classical area Regge calculus, an alternative to standard Regge
calculus where instead of lengths, the areas of a simplicial discretization are
fundamental. There are a number of surprises: though the equations of motion
impose flatness we show that diffeomorphism symmetry is broken for a large
class of area Regge geometries. This is due to degrees of freedom not available
in the length calculus. In particular, an area discretization only imposes that
the areas of glued simplicial faces agrees; their shapes need not be the same.
We enumerate and characterize these non-metric, or `twisted', degrees of
freedom and provide tools for understanding their dynamics. The non-metric
degrees of freedom also lead to fewer invariances of the area Regge action---in
comparison to the length action---under local changes of the triangulation
(Pachner moves). This means that invariance properties can be used to classify
the dynamics of spin foam models. Our results lay a promising foundation for
understanding the dynamics of the non-metric degrees of freedom in loop quantum
gravity and spin foams.
