Automatic Generation Of 3-D Symmetric Tiling Patterns

In this thesis, we generate some 3-D tiling patterns with three kinds ofsymmetries by means of dynamical system. First, we construct an extendedPicard group P₁. By exploiting this group, we generate 3-D tiling patternswith P₁ symmetry by means of dynamical system equivariant to the extendedPicard group. The derived patterns have the symmetry of the wallpaper groupprom on the cross section plane parallel to the xy-plane, and own the symmetryof the extended modular group on the cross section plane containing the x-axis.Then, by exploiting the P₆ and P₄ groups, we generate two other kinds of 3-Dtiling patterns by invariant method under dynamical systems. These derivedpatterns have the symmetries of the wallpaper group p6m and p4m on the crosssection plane parallel to the xy-plane, and also own some kinds of symmetrieson the cross section plane containing the x-axis. The numerical experimentsshow that the generated patterns are symmetric and beautiful. This methodprovides some novel approaches for devising exotic symmetric tiling patternsfrom a dynamical system's point of view.