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.