On one contraction of a discrete group D3

The groups of motions of two-dimensional space-time models 
from a geometric point of view are groups of motions of 
two-dimensional homogeneous spaces of a constant curvature. 
When quantizing these models, discrete subgroups of 
these groups play an important role. The standard model is 
based on the gauge groups U(1), SU(2) and SU(3). Discrete 
subgroups of these groups are now also in the center 
of attention in the elementary particle physics. In this paper 
we present contractions of the discrete dihedron group 
D3 induced by limit transitions in the continuous groupO(2) 
and O(3). The resulting groups are D4 and infinite groups 
D∞, generating discrete lattices of two-dimensional spacetime 
models. We also discuss transitions of discrete symmetry 
groups of the Platonic solids.