English for Math III
M.C. Escher - Interlocking Tiling
Many of M.C.Escher's drawings contain mathematical ideas never seen elsewhere.
The best way to understand Escher's interlocking shapes is to dissect the figures.
Take a piece of tracing paper and copy the outline of a shape (in the case of the
horseman, the combined outline of the horse and the rider). This silhouette allows
you to focus on the line properties between the shapes. Look in the original periodic
drawing and locate the vertices: points where 3 (or more) lines joint together. The
vertex is also a place where 3 or more neighbouring shapes touch. There are at least
3 vertices and up to 6 vertices on the silhouette on the tracing paper. The line
segments between the vertices hold the secrets of mathematics. Look for rotation
symmetry at these vertices. Let's discuss two famous Escher drawings to illustrate
the structure: "Reptiles" where the lizards claw out of a book, and "Cycle" where a
short person runs down the stairs and tumble into a hexagon form. In these figures,
there are 6 vertices and 6 line segments. Upon closer examination, there are only 3
distinct line segments and 2 copies of each of these three lines. Such simplicity! Now,
the secret becomes lucent, half of the silhouettes are made out of copies from the
other half. Three of the 6 vertices have rotation symmetry C3, which means the line
segments attached to these vertices can pivot about this point and radiate out 3
copies like the propeller of an electric fan. The two copies of identical line segments
are connected at the rotation vertices C3. The other 3 vertices are the junctions of
the dissimilar line segments. The line segments repeat across the page periodically
and hook up with other line segments at the vertices to create the repetitive pattern.
In all the drawings made by Escher there are 6 different kinds of repetition method
for the line segments: translation T, half-turn C2, one-third turn C3, quarter turn C4,
one-sixth turn C6 and gliding mirror reflection G. Only 6 different congruent lines
and you can do everything! The lines are equivalent to the sub-atomic particles in
physics. The atoms are recognized forms which have an interesting meaning to our
eyeballs: people, birds, fish, horse, animals. These atomic units may or may not nest
on a surface. The molecules are the smallest unit that can interlock - what I call
interlocking shapes. The molecules may be a silhouette of a horseman, which
interlocks perfectly, and consists of two atoms: the horse and the rider. The atom
has meaning to our eyeballs, but no mathematical interpretation. The silhouette is
formed by groups of congruent lines just like a molecule is built with shared orbits of
electrons. The method of repetition of the interlocking shapes is rooted in the
congruent movements of the lines. Thus, a silhouette that is build with C6 lines also
repeat in 60 degree rotations all over the flat surface.
1. A silhouette is __________________________________
The internal lines in a tessellation are not important.
The silhouette may contain 2 things, a horse with rider,
a devil/angel, etc.
2. A vertex is _______________________________________.
A vertex is also a place where 3 adjacent shapes meet.
3. A line segment is _________________________________________________
If the silhouette has 4 vertices, there are 4 line segments.
4. T line is a translation line.
C line is _________________; this line can be rotated at its mid-point.
C3 line can rotate at one of the end points by _________________.
C4 line can rotate at one of the end points by __________________.
C6 line can rotate at one of the end points by __________________.
G line has a mirror clone which may also translate and rotate.
5. With the exception of the C line, each of the above line segments must
appear in pairs on the silhouette. The C line is considered a pair by itself.
Principles of Plane Tessellations
In the next figure, there is a simple design. The whole surface is covered with
equilateral triangles. If we shift the whole plane over the distance AB, it will cover
the underlying pattern once again. This is a translation of the plane. We can also
turn the duplicate through 60 degrees about the point C, and we notice that again it
covers the original pattern exactly. This is a rotation. Also if we do a reflection about
the line PQ, the pattern remains the same.
A pattern can be made to map to itself by means of translation, rotation, reflection
and glide reflection. There are 17 different groups of patters. Each groups admits
only some kinds of shifts whereby they map onto themselves (some admit only
translation, others translation and reflection etc.). Escher discovered all these
possibilities without any previous mathematical knowledge. A particular
characteristic of Escher's tessellation is that he chooses motifs that represent
concrete objects or beings.
REPTILES
CYCLE