Derivation of 3D Braided Geometry Structures from Braided Symmetry Group

ABSTRACT

Considering that there were very a few varieties of 3D braided materials at present, some novel 3D braided geometry structures were derived based on symmetry group theory. Group theory was used for the first time to describe the 3D braided geometry structures was discussed. The whole analyzing procedure from the existing braided geometry structure to the braided symmetry group was described in detail. It is found that because the reflection operation does not exist in some point groups and space groups of braided structure and some odd lattices appeared, the braided symmetry group is not always the same as symmetry groups of crystallographic. The representative volume element of 3D braided geometry structure was deduced from braided space point group, and 3D braided geometry structure was obtained from braided space group. 3D braided geometry structures can not only be classified based on braided group theory, but some of novel 3D braided structures can be deduced through the group's symmetry operators. It is proved that some novel 3D braided processes are feasible. Braided symmetry theory is an effective mathematical method for developing more and rational 3D braided materials.