Classification of tilings in general
- and M.C. Escher’s tilings in particular
by
Erik Kjellqvist
TABLE OF CONTENTS
1 ABSTRACT
3
2 INTRODUCTION
3
3 LITERATURE STUDY
4
3.1 Heesch types
4
3.2 Symmetry groups
7
3.3 Examples of Classification of Escher-tilings
9
Escher drawing no 105 – Symmetry group p1
10
Escher drawing no. 63 – Symmetry group pg
11
Escher drawing no 20 – Symmetry group p4
12
Escher drawing no 103 – Symmetry group p31m
13
Escher drawing no. 104 – Symmetry group p4
14
4 CONCLUSIONS
15
5 REFERENCES
16
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Classification of tilings in general – and M.C. Escher's tilings in particular
1 Abstract
This report explains two ways of classifying plane-filling tilings, through Heesch
types and through the 17 planar symmetry groups. Furthermore some of the works of
M.C. Escher’s tilings will be analysed and classified according to the 17 planar
symmetry groups.
2 Introduction
One can study M.C. Escher’s tilings from many points of views. Strictly aesthetically,
as a form of art, mathematically, by finding the underlying symmetrics, through
Computer Science, by generation, optimisation and so on.
This variety of viewpoints is surely one of the great qualities of Escher’s works and
this report will try to grasp the basic structures that bind the mathematics, the
computer science and the art of tilings all together.
The report consists of three parts. The first part begins by explaining how to
categorize tilings in accordance to the nine Heesch types. The second part explains
how to categorize tilings in accordance to the 17 plane symmetry groups and finally
the last part consists of Escher drawings being categorized according to these 17 plane
symmetry groups.
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Classification of tilings in general – and M.C. Escher's tilings in particular
3 Literature Study
3.1 Heesch types
When trying to generalize the classification of tilings, one may find it logical to begin
with the fundamentals of how the tilings generally are created. A tiling should not
consist of holes or gaps in between the tilings. Clearly an arbitrary quadrangle with
straight sides fit together under these properties. Thus, one might try to backtrack the
creation of an arbitrary tiling to its “quadrangular ancestor” stepwise, and in each step
make sure that these properties are preserved. The operations done in each step are
called moves. The collection of moves needed to be done is called the tilings Heesch
type. The moves can easily be explained in an informal way by visualizing scissorwork on a given quadrangular piece of paper as follows:
1. T (Translation)
Cut out an arbitrary piece of paper along one of the sides of the quadrangle and move
that piece across the quadrangle to the opposite side and tape the piece back onto the
quadrangle.1
Figure 1: Move T
Clearly, two (and more) copies of this modified quadrangle fit together by a simple
translation.
2. G (Glide reflection)
Repeat the same procedure as in T but this time flip the cut-out piece before taping it
back onto the opposite side of the quadrangle. 1
Figure 2: Move G
Clearly two or more copies of the above figure fit together using a translation and a
reflection (a so-called glide reflection).
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Classification of tilings in general – and M.C. Escher's tilings in particular
3. C (Centre point rotation)
This move involves only one side, but begins as usual by cutting out an arbitrary piece
of paper along one chosen side of the quadrangle. Now rotate the piece along the midpoint of the chosen side and tape it back onto the quadrangle. 1
Figure 3: Move C
Here we can understand that two copies will fit together using a rotation around the
modified side of the original.
4. C4 (Corner Rotation)
This is more or less the same procedure as in C, but this time you rotate the piece
around one corner of the quadrangle and tape it back onto one adjacent side of the
originally selected one. 1
Figure 4: Move T
Copies of this tiling fit together by a rotation around the selected corner.
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Classification of tilings in general – and M.C. Escher's tilings in particular
5. G’ (Glide reflection, adjacent sides)
The final move involves, like the previously defined, a side and a touching or adjacent
side. After an arbitrary piece has been cut out from a side, that piece “glide-reflects”
onto an adjacent side along a reflection line going through the midpoints of the two
mentioned sides. 1
Figure 5: Move G´
Copies of this tiling fit togheter by a reflection and a corner rotation of the original.
The tile is then created from these moves by combining them according to the specific
needs. Some combinations of moves are illegal since they create more than one move
on each side, thus making them interfere with each other. Try out for yourself and you
will soon convince yourself that there are only nine different possible combinations of
these edge modifications, which give rise to the nine Heesch types: 1
Figure 6: The Nine Heesch Types
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Classification of tilings in general – and M.C. Escher's tilings in particular
3.2 Symmetry groups
In order to create a symmetric tiling with some fundamental tile (or prototile) we must
find ways to cover the plane with it. If we want to obtain a monohedral tiling (a tiling
consisting of a single prototile) we cannot transform the shape of the original
prototile.
This leads us to the question: in how many ways can we copy our prototile to another
location in the plane and still preserve its original shape? The transformations needed
for this are the so-called isometric transformations.
Definition A similarity transformation T of the plane is called an isometry or (rigid) motion if it preserves distances.2
There are four types of isometric motions in the plane, called the Euclidean
isometries, namely reflection, translation, rotation and glide reflection. Actually the
three last mentioned isometries are all composites of reflections.3
At a first glance one may think it is not enough with these isometries, that a composite
of these should lead to a new isometric motion, and a composite of that should in its
turn lead to another isometric motion and so on. This is, however, not the case. Each
and every isometric motion in the plane can be described by applying one of these
four groups.
Reflection
Translation
Rotation
Glide Reflection
Figure 7: The four isometries in the Euclidean plane
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Classification of tilings in general – and M.C. Escher's tilings in particular
These isometries can be used in order to classify the symmetry of tilings. The
symmetry group of a tiling is the collection of all the symmetries of the tiling. Such a
symmetry group is described by a finite number of basic transformations from which
all remaining symmetries can be obtained by composition. Russian mathematician
Fedorov proved that there are only 17 symmetry groups of a symmetric tiling in the
plane. Each one of the symmety groups is based on collections of the four isometric
motions.4
Escher made his own system of grouping the tilings, which corresponds to five of the
17 symmetry groups originally discovered by Fedorov5, which I hereby will present
and explain further. For a more detailed presentation of all the 17 groups one might
try to search the topic crystallography in which they all are frequently used in order to
model compositions of chemical molecules or have a look at the book Tilings and
Patterns by Grünbaum and Shephard.
The first symmetry group by Escher is symmetry with respect to translation. Think of
the prototile contained in a parallelogram. Take a copy of the prototile and shift the
copy one “prototile-unit” along any of the different edges of the parallelogram. This
symmetry group corresponds to group p1 in crystallographic notation, often referred
to as a periodic tiling in the vocabulary of tilings (see also Picture Generation- A
Tree-Based Approach by Frank Drewes).
Figure 8: A p1 prototile
The second symmetry group, p2, tiles the plane by successively rotating the prototile
180 degrees and then translate it horizontally, so that the rotation covers the plane
vertically and the horizontal translation covers the plane horizontally.
Figure 9: Generation of a tiling from a p2 prototile
The third symmetry group, denoted pg, involves glide reflections. In this group the
prototile is translated horizontally half the width of the prototile and then reflected
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Classification of tilings in general – and M.C. Escher's tilings in particular
upwards and downwards, so that the reflections cover the plane vertically and the
translations covers the plane horizontally.
Figure 10: Generation of a tiling from a pg prototile
The fourth symmetry group involves a 180-degree rotation and a glide reflection in
such a way that the rotation covers the plane vertically and the glide reflection
horizontally. The rotations are made around the midpoints of the upper and lower
edge of the prototile, making the two half squares of the prototile becoming whole
squares. The glide-reflection creates the chessboard effect by translating the prototile
horizontally and then reflecting it so that the upper part of the prototile becomes the
lower and vice versa in each horizontal translation.
Figure 11: Generation of a tiling from a pgg prototile
The last symmetry group by Escher, p4, uses rotations of 90 degrees clockwise three
times and then rotation by 180 degrees counter clockwise one time to tile the plane.
An easier way to understand this symmetry group is to consider the collection of
prototiles after the initial three rotations. A big square containing a swastika-like
shape appears. This square can obviously tile the plane by translation; therefore one
may say that p4 tilings are symmetric with respect to translations.
Figure 12: Generation of a tiling from a p4 prototile
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Classification of tilings in general – and M.C. Escher's tilings in particular
3.3 Examples of Classification of Escher-tilings
Escher drawing no. 105 – Symmetry group p1
Figure 13: Escher drawing no. 105
The colouring of this drawing can make one feel that a relatively large prototile is
needed, however if one disregards the colouring one finds that the following prototile
will do:
Figure 14: Periodic prototile for Escher drawing no. 105
The square then tiles the plane simply by translations perpendicular to the squares
edges7, in other words: this drawing has symmetry group p1.5
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Classification of tilings in general – and M.C. Escher's tilings in particular
Escher drawing no. 63 – Symmetry group pg
Figure 15: Escher drawing no. 63
One can observe the prototile by understanding that the following two parallelograms
tiles the plane by translation:
Figure 16: Periodic prototile for Escher drawing no. 63
One can see that the lower parallelogram is made up from the upper by a glidereflection7. Just reflect a copy of the upper parallelogram around its rightmost side
and then translate this copy underneath the original parallelogram. Clearly this tiling
has symmetry group pg.5
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Classification of tilings in general – and M.C. Escher's tilings in particular
Escher drawing no 20 – Symmetry group p4
Figure 17: Escher drawing no 20
If one observes this drawing carefully one might find that the authors grid lines are
still visible as a net of squares.
Figure 18: Periodic prototile for Escher drawing no. 20
Begin by recognizing the translating square. If you look carefully at the outline of the
fishes inside the square you see the swastika recognizable from symmetry group p4. If
we take the uppermost fish in the square and combine it with the fish head to the right
of that fish and rotate it counter clockwise three times around the centre of the
swastika we get the indicated tiling square, exactly as in p4. Thus this tiling belongs
to symmetry group p4.5
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Classification of tilings in general – and M.C. Escher's tilings in particular
Escher drawing no 103 – Symmetry group p31m
Figure 19: Escher drawing no 103
This drawing uses a symmetry group called p31m5, which is made up from an initial
kite-shaped prototile, which is rotated 120 degrees twice to create a triangle. This
triangle is then reflected over its three edges to tile the plane. Escher never formally
classified p31m in his own classification system, even though he used it in his
artwork.
To illustrate the group p31m in accordance with drawing no. 103 we may think of the
initial kite to be one of the three curved lines below. When this curved line is rotated
twice around its “starting point” a triangle is created, as shown below. This triangle
can then be used to tile the plane by continuously rotating the triangle along its three
edges.
Figure 20: p31m in accordance with drawing no. 103
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Classification of tilings in general – and M.C. Escher's tilings in particular
Escher drawing no. 104 – Symmetry group p4
Figure 21: Escher drawing no 104
If one looks carefully on this one, one might find the characteristic swastika of p4 (at
the lower left leg of the white lizard). One might also notice that a clockwise rotation
of the lizard around the centre of that swastika crates a “prototile” of four lizards that
tiles the plane simply by translation, exactly like p4.
A little experimentation can lead us to outline the lizard with its origin in the centre of
the swastika like this:
Figure 22: Initial outline of lizard in Escher drawing no 104
The two curved lines made from the upper and lower edges of the square can then be
rotated 90 degrees counter clockwise around the upper right corner and the lower left
corner respectively to create the entire white lizard, which is then rotated around the
centre of the swastika 90 degrees clockwise three times. The resulting composition of
four lizards can then tile the plane simply by translation7. Thus this drawing belongs
to symmetry group p4.5
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Classification of tilings in general – and M.C. Escher's tilings in particular
4 Conclusions
This report became a rather hands-on description of the different ways of categorizing
tilings. Finding in-depth information about the subject is actually easier than finding
information about the basics, according to my own experiences, especially if one is
interested in mathematical properties of the tilings.
This report focused on simple structures from which more complicated tilings were
made. If one would like to go the other way around, to tile the plane with a closed
picture not made from the different Heesch types I would recommend reading Kaplan
and Salesin’s article Escherization in which they present their solution to that
problem, at WWW:
http://grail.cs.washington.edu/pub/papers/kaplan_siggraph2000.pdf
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Classification of tilings in general – and M.C. Escher's tilings in particular
5 References
1. Wolfe, John , Escher Style Tesselations (from Math 3403: Geometric
Structures), 2000
2. Bulman-Fleming, Sydney, Transformations of the Eucledian plane, (verified
030306), http://www.wlu.ca/~wwwmath/faculty/b-fleming/01318ch4.pdf
3. Mirror, Mirror…: Reflections and Congruence, (verified 030306),
http://www.andrews.edu/~calkins/math/webtexts/geom04.htm#TRNL
4. Akleman, Ergun et. al., Intuitive and Effective Design of Symmetric Tiles
5. Schattschneider, Doris, Visions Of Symmetry, New York, 1999
6. Edwards, Steve, Identifying the 17 Plane Symmetry Groups, (verified 030306),
http://www2.spsu.edu/math/tile/symm/ident17.htm
7. Totally Tessellated, (verified 030306), http://www.abc.lv/thinkquest/tqentries/16661/index2.html
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Classification of tilings in general – and M.C. Escher's tilings in particular