Symmetry and Strand Transitivity Properties of Three-way Threefold Fabrics

Final Defense
Symmetry and Strand Transitivity Properties of Three-way Threefold Fabrics

by Kristan B. Liza
PhD Mathematics Candidate

Date: Wednesday, 04 December 2024
Time: 6:00 pm
Venue: Online

Advisers:
Ma. Louise Antonette N. De Las Peñas, PhD
Ateneo de Manila University

Panelists:
Manuel Joseph C. Loquias, PhD (Critic Reader)
University of the Philippines Diliman
Mark L. Loyola, PhD (Critic Reader)
Ateneo de Manila University
Ma. Louise Antonette N. De las Peñas, PhD
Ateneo de Manila University
Job A. Nable, PhD
Ateneo de Manila University
Fidel R. Nemenzo, PhD
University of the Philippines Diliman

A three-way threefold fabric F consists of three layers of congruent strands in the same plane P oriented in three different directions. These strands are woven together with a preferential ordering of the layers at every point of P that does not lie on the boundary of a strand, in such a way that it must hang together. The fabric F is represented geometrically by its design which is a k-coloring of the tiling T by equilateral triangles with 3 ≤ k ≤ 6 . The ordering of the layers on a point in F that lies in the intersection of the strands, corresponds to a color of a tile in T .

This work discusses the symmetry groups of three-way threefold fabrics. Our results point to a total of 27 layer groups that are possible symmetry groups of these fabrics. A method to obtain a design of a three-way threefold fabric with a given symmetry group is also presented, using color symmetry theory.

The strand transitivity properties of three-way threefold fabrics are also discussed. In partic- ular, we determine the number of orbits of strands formed in a given fabric under the action of its symmetry group by investigating the action of the color group on the rows or strips of its corresponding design.

Key Words: three-way threefold fabrics, symmetry, isonemality, color symmetry, layer groups