Computer Algorithms to Determine Colorings that arise from Plane Crystallographic Groups
Ma. Louise Antonette N. De Las Penas
Department of Mathematics
Ateneo De Manila University
Loyola Heights, Quezon City, Philippines
Color symmetry has generated research in many areas of symmetry theory and many problems continue to be of interest today, particularly the classification and analysis of colored symmetrical patterns.
To analyze a colored symmetrical
pattern, we not only consider the
symmetry group G of the (uncolored)
pattern but also the symmetry group
K of the pattern when it
is colored. Since not all elements
of G permute colors, we also
consider the subgroup H of
elements of G which effect
color permutations. This subgroup
H contains K as a
normal subgroup of elements of H
which fixes the colors.
In this paper, we develop a computer program to assist us in our study of colored symmetrical patterns. We determine all colorings of a symmetrical pattern for which the elements of a given subgroup H of the symmetry group G of the uncolored pattern permute the colors and the elements of a given subgroup K of G fix the colors. It is often a difficult task to generate such colorings by hand especially when dealing with a coloring with a large number of colors. The computer program was conceptualized to make the enormous task of listing such colorings manageable. In this paper, we look at colored patterns whose symmetry groups are plane crystallographic groups.
Consider the colored patterns below which are assumed
to repeat over the entire plane.
For both the colored patterns, the
symmetry group G of the patterns
with the colors disregarded is a
hexagonal plane crystallographic
group of type p6m generated
by a,b,x,y where a
is a 60 degree counterclockwise
rotation about the indicated point
P, b is a reflection
in a horizontal line through P,
and x, y are translations
as indicated. These colored patterns
have been generated using the program
by choosing the subgroups H=<a,x,y>
of G. H and K
are hexagonal plane crystallographic
groups of types p6 and p2
respectively. Moreover, the program
identifies the first coloring as
perfect (all elements of G
permute the colors) and the second
coloring as non-perfect (there exist
at least one element of G
that does not permute the colors).
Note that for the second coloring,
the reflectionb does not
permute the colors.