The Visualization Of 3D Tiling Patterns With The Symmetries Of Planar Crystallographic Groups And The Extended Modular Group

Pattern art has a long history in humankind and is of long standing. From the naissance of humankind, Pattern art has countless relationships with human. We can find it everywhere, from the old pyramid of Egypt to the flourishing development of virtual 3D computer games. With the fast development of computer science, a lot of wonderful and exotic patterns can be generated by the combination of computer technology and mathematics theory, such as the Mandelbrot set and Jualia set, are easily generated by complex dynamical systems with the help of computers. Furthermore, interesting patterns can be generated by other methods, such as, chaotic attractors, L-system, cellular automata, tiling and so on.The above patterns are usually planar patters. Nowadays, more and more methods to produce 3-D patterns can be derived from the above method. The patterns gained by this kind of methods are more interesting and have more use in practice.This thesis mainly concerns the automatic generation of 3D tiling patterns with the help of computer and mathematics theories, it consists of the following two aspects: 1. Considering 12 crystallography groups that have rectangular lattice in the 17 crystallography groups, i.e., we will construct dynamical system functions called 3D-equivariant-function F(x,y,z)=and prove that only with the 8 crystallography groupsP₁,P₂,P₄,P_m,P₄m,P_m,P_mm,C_m,C_mm , every facet of the cube will have the same symmetricalpatterns under the iteration of the 3D-equivariant-function and the invariant function p(x,y,z).As all facets of the cube have the same symmetrical patterns, 3D patterns are very fascinating. We also draw 3D patterns in spheres so that patterns got in this way are more pretty and interesting. 2. Making use of the concept of extended modular group and fundamental region, we also draw patterns in 3D cube so that it has n fold (n is an arbitrary positive integer which can divide 360.) rotations on one facet, while on other facets the patterns own the symmetry of the extended modular group. Furthermore, the artistic symmetrical patterns with the extended modular group in the Poincare sphere are also derived by conformal mappings. This method can be used to create a great variety of aesthetic symmetrical patterns.