MIRROR
CURVES
by
Symmetry: Culture and Science
1, 2 (1990), 154-170.
Symmetry: Culture and Science
1, 2 (1990), 154-170.
Mirror curves are
present in ethnical art, as
Tamil threshold designs or Tchokwe sand drawings. Historically, they are to be
found in the art of most peoples surrounding the Mediterranean, the Black and
Caspian Seas, in the art of Egyptians, Greeks, Romans, Byzantines, Moors,
Persians, Turks, Arabs, Syrians, Hebrews and African tribes. Highlights are
Celtic interlacing knotworks, Islamic layered patterns and Moorish floor and
wall decorations. In this paper mirror
curves are considered from the point of view of geometry, tiling theory, graph
theory and knot theory. After the enumeration of mirror curves in a rectangular
square grid, and a discussion of mirror curves in polyominoes and uniform
tessellations, the construction of mirror curves is generalized to any surface.
Let a connected edge-to-edge
tiling of some part of a plane by polygons be given. Connecting the midpoints of
adjacent edges we obtain a 4- . A steps in that graph is a connected
series of steps, where each step appears only once. Every closed path in
that graph is called a path. The set of all components of such graph
is called a component. In every vertex we have three possibilities to
continue our path: to choose left, middle, right edge. If the middle edge
is chosen such vertex will be called a mirror curve. By introducing in every
crossing the relation over-under, every mirror curve can be converted into
a crossing knotwork design.The term “mirror curve” could be
simply justified if we take a a,b] with the
sides a and b, where that sides are mirrors, and the additional internal
two-sided mirrors are placed between the square cells, coinciding with
an edge, or perpendicular to it in its midpoint. In this case, a ray of
light, emitted from one edge-midpoint, making with that edge a 45°
angle, after the series of reflections will close a component. Beginning
from different edge-midpoints, till exhausting the complete step graph,
we obtain a mirror curve. It is easy to conclude that the preceding description
could be extended to any connected part of a regular triangular, square
or hexagonal tessellation, this means to any , polyamond or
polyomino,
respectively.
polyhexeAfter the historical remarks about
Tamil and Tchokwe mirror curves and knotwork designs created by Leonardo
and Dürer, the rules for the construction of one-component mirror
curves are given and mirror curves are generalized to any surface. Those algorithm rules are used for the combinatorial enumeration
of mirror curves obtained from RG[ NEXT
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