  Causal Dynamical Triangulations is a non-perturbative quantum gravity model,
defined with a lattice cut-off. The model can be viewed as defined with a
proper time but with no reference to any three-dimensional spatial background
geometry. It has four phases, depending on the parameters (the coupling
constants) of the model. The particularly interesting behavior is observed in
the so-called de Sitter phase, where the spatial three-volume distribution as a
function of proper time has a semi-classical behavior which can be obtained
from an effective mini-superspace action. In the case of the three-sphere
spatial topology, it has been difficult to extend the effective semi-classical
description in terms of proper time and spatial three-volume to include genuine
spatial coordinates, partially because of the background independence inherent
in the model. However, if the spatial topology is that of a three-torus, it is
possible to define a number of new observables that might serve as spatial
coordinates as well as new observables related to the winding numbers of the
three-dimensional torus. The present paper outlines how to define the
observables, and how they can be used in numerical simulations of the model.
