MIRROR
DESIGNS
AND
MIRROR
CURVES
Figure 1: Examples  (a) Tchokwe sand drawing,
(b) Tamil threshold design.
When analysing sand drawings from the Tchokwe (Angola)
[Gerdes, 199394] and threshold designs from the Tamil
(South India) [Gerdes, 1989; 199394,
chap. 11; 1995], I found
that several of them (see the two examples in Figure 1)
may be generated in the following way.
Figure 2: RG[6,5] and RG[5,5].
Consider a rectangular grid RG[m,n]
with vertices (0,0),(2m,0), (2m,2n), and (0,2n)
and having as points (2s1, 2t1), where s = 1,...,m,
and t = 1,...,n, and m and n two arbitrary
natural numbers. Figure 2 displays RG[6,5] and RG[5,5]. A curve like
that shown in Figure 1 is the smooth version of a closed polygonal
path described by a lightray emitted from the point (1,0), making an
angle of 45 degrees with the sides of the rectangular grid RG[m,n]
(see the example in Figure 3).
Figure 3: Emission of a light ray from the point (0,1).
The ray is reflected on the sides of the rectangle and on
its way through the grid it encounters doublesided mirrors
which are placed horizontally or vertically in the
centre between two neighbouring grid points (see Figure 4).
Figure 4: Possible positions of mirrors.
Figure 5 shows the position of the
mirrors in order to generate
the two curves of Figure 1.
Figure 5: Mirror designs that generate
the curves of Figure 1.
Both curves are rectanglefilling in the sense
that they 'embrace' all grid points. Such curves we will call
(rectanglefilling) mirror curves. The rectangular
grids together with the mirrors which generate the curves
will be called mirror designs.
Figure 6a displays the mirror design that leads
to the Celtic knot design in Figure 6b
(Gerdes, 199394, chap. 12).
Figure 6: Example of a Celtic knot design as a mirror curve.
Properties and classes of mirror curves are analysed
by Gerdes (1990,
199394, chap. 48)
and by Jablan (1995),
who also establishes links with the theory of cellular automata,
Polya's enumeration theory, combinatorial geometry, topology, and knot theory.
