Theoretical modeling demonstrates an orbifold notation for spatial layer groups in doubly periodic structures, highlighting applications in classifying knitted fabric symmetries.
Key Points
To develop a three-dimensional orbifold framework and Conway-type notation for classifying crystallographic layer groups in doubly periodic materials.
Extended planar orbifold theory into three dimensions to account for finite transverse thickness.
Constructed a complete set of Conway-type topological symbols representing all spatial layer groups.
Applied the framework to analyze and describe the repeating symmetry motifs of several knitted fabric structures.
Established a unified topological notation that fully classifies all 80 crystallographic layer groups.
Demonstrated that complex spatial symmetries in common knitted fabric patterns map directly into the new orbifold symbols.