SUBGROUP LATTICES FORCRYSTALLOGRAPHIC GROUPS
Introduction
An isometry is any mapping, The discrete symmetry groups of the plane, which contain translations in nonparallel directions, are called 2-dimensional crystallographic groups or simply wallpaper groups. Patterns of these seventeen groups are shown in Table 1.
The
elements of a symmetry group This
means the symmetry
(n,j) maps
the point Group
multiplication in ((n,j) or (n,j) With
this definition of multiplication in the symmetry group,
Let
With
the multiplication, (n,j)
If (j Therefore,
(j
Here is a concrete way to view these ideas
in terms of symmetrical patterns in the plane. The Islamic pattern (Figure
1) consists of two types of stars, octagons and hexagons along with
some overlapping knotwork. The figures below show a sequence of steps for
determining a fundamental region, which may be used to generate the full
tiling.
First, the tiling is rendered in black,
white, and gray (Figure 2). By placing a point at the
center of each eight-pointed star (Figure 3), the lattice
Connecting the lattice points with lines
of translational symmetry in two linearly independent directions (Figure
4) forms a lattice grid for the pattern. For this tiling, a square
formed by this lattice grid (Figure 5) may be used to
generate the pattern by translations alone. By determining the symmetry
of this square region the point group can be determined. In this example,
the region has rotational symmetry of order four. Also, the region cannot
be reflected because of the interlocking knotwork so the point group for
this pattern is With these symmetries in mind, diagonal
lines may be added to the lattice grid (Figure 6) to
form a system of congruent isosceles right triangles. One of these triangles
serves as a fundamental region (Figure 7) and will generate
the entire pattern by rotations. The point group,
If the overlapping of the knotwork was
ignored, the pattern would have reflective symmetry, the point group would
be
This
method may be generalized to include all the crystallographic groups. In
the crystallographic group
By viewing the seventeen two-dimensional crystallographic groups and patterns all together, connections among the groups may be made. Questions may be answered concerning how the groups sit inside one another as subgroups and how the groups relate in terms of the invariants described above. The subgroup lattices that follow are designed as a tool to illuminate some of these connections. The last three subgroup lattices show techniques involving the removal of symmetry from one crystallographic pattern to produce another. Subgroup
Lattice
1 details a subgroup relationship of the crystallographic groups and
gives the minimal index in which each group lies as a subgroup of those
groups directly above it. Horizontal arrows between two groups denotes
the minimal index in which the group on the left lies as a proper subgroup
in the group on the right and vice versa. The groups
Each of the crystallographic groups is
a subgroup of either
Subgroup Lattice 4 gives
a periodic pattern for each of the seventeen groups. Subgroup Lattices
5 and 6 look at crystallographic lattices and point
groups, respectively. Subgroup Lattice 7 categorizes
the crystallographic groups in terms of rotations, reflections and glide
reflections. Subgroup Lattices 8, 9,
and 10 show possible paths to create groups from one
another by deleting symmetry.
Coxeter, H.S.M (1965)., W. O.Moser, Generators and Relations For Discrete Groups, Springer-Verlag, Berlin, New York, 1965. Jablan, S. (1995), Theory of Symmetry and Ornament, Retieved June 1, 1999 from http://www.emis.de/monographs/jablan/index.html Jablan, S.V. (2002), Symmetry, Ornament and Modularity, World Scientific, Singapore. Mathematics Museum (Japan) (1998), Retrieved
August 15, 2002 from
Schattschneider, D. (1978), |