Honours Project
Symmetry Classifications of
Periodic Tilings
– Escher’s drawings
Name: Kavitha d/o Krishnan
Supervisor: Associate Professor Helmer Aslaksen
Department of Mathematics
National University of Singapore
2002/2003
Acknowledgements
I would like to thank my supervisor Associate Professor Helmer Aslaken for his guidance
and patience. Throughout all the hardships that I had in completing this thesis he was
always very understanding. Without his constant assurance and encouragement this
would not be possible. Also, I would like to thank my family and friends for all their
support and encouragement.
Summary
This thesis serves as an introductory text in understanding the symmetry classifications of
Escher’s periodic tilings in Table 1 (see appendix). Firstly, it aims to provide the readers
an overview of the following classifications: symmetry group, isohedral type, Heesch
type and Escher’s system. Secondly, armed with the knowledge of the classifications it
allows the readers to understand the meaning behind the notations seen in Table 1
(appendix). Therefore, finally, given an Escher’s periodic tiling the reader would be able
to obtain the notation under each classification. Or given an Escher’s periodic tiling with
a notation the readers will be able to give the symmetry properties of the tiling.
Chapter 1 gives a basic understanding of isometry and symmetry. Chapter 2 defines
tilings. In Chapter 3 we study the different types of symmetry groups of tiles and tilings.
The key area in this chapter is symmetry groups of periodic tilings. This is because
Escher’s drawings are periodic tilings. In Chapter 4 we look at isohedral types. In this
chapter we will not study how the classification came about. Instead, we focus on being
able to obtain the isohedral type given a periodic tiling. Chapter 5 discusses the
classification of periodic tilings into Heesch types. Again given a tiling we try to find its
Heesch type. In this chapter we will go a step further and find the relation between
Heesch types and isohedral types. Finally, Chapter 6 touches Escher’s own notations used
for his drawings. In this chapter we will not learn the meaning behind all the notations in
Escher’s system in Table 1 (appendix). We will briefly look at a few notations. The
purpose is to derive symmetry properties of Escher’s tilings from his notations. Also, in
Chapter 6 we went a step further to find the relation between Escher’s system, Heesch
types and isohedral types. This relation allows readers to see the link between the three
different types of classifications.
Author’s Contributions
I have done the following in this thesis:
1. Used Escher’s periodic tilings as examples to illustrate how to obtain a few
symmetry groups. The examples are found on page 19.
2. Used Escher’s periodic tiling to illustrate how to obtain the isohedral type of a
tiling. This is given in Section 4.2.
3. Used Escher’s periodic tiling to illustrate how to obtain the Heesch type, which is
on page 34.
4. Analyzed and found a relation between Heesch types and isohedral types, which
is presented in Table 5.2 on page 35.
5. Analyzed and found a relation between Escher’s systems, Heesch types and
isohedral types. This relation is presented in Table 6.3 on page 43.
Contents
Page
Acknowledgements
i
Summary
ii
Author’s Contributions
iii
1. Introduction
1
1.1 What is an isometry?
1
1.2 What is a symmetry?
4
2. Tilings
5
2.1 Defining tilings
3. Symmetry groups of tiles and tilings
5
7
3.1 Symmetry groups of tiles
7
3.2 Symmetry groups of tilings
9
3.2.1 Symmetry groups of tilings with no translations
10
3.2.2 Symmetry groups of strip tilings
11
3.2.3 Symmetry groups of periodic tilings
15
4. Isohedral types
23
4.1 What is an isohedral tiling?
23
4.2 Identifying isohedral type
27
4.2.1 Topological types
27
4.2.2 Incidence symbols
29
4.2.3 Induced tile group
33
5. Heesch types
35
5.1 Classification of isohedral tilings into Heesch types
35
5.2 Relation between Heesch types and isohedral types
38
6. Escher’s system
40
6.1 Quadrilateral systems
41
6.1.1 Isohedral tilings with a minimum of two colours
41
6.1.2 Relation between Escher’s system, Heesch types
45
and isohedral types
6.2 Triangle systems
46
Appendix
48
Referenc es
53
1. Introduction
In this thesis, we will study periodic tilings, as in many of the prints of M.C.Escher. The
key concept is symmetry. In order to understand symmetry we will start by studying
isometries of the R2 plane.
1.1 What is an isometry?
An isometry is a distance-preserving transformation of the real plane R² onto itself. More
precisely, let P and Q be any two points in R² and let P’ and Q’ in R² be their respective
images under a transformation. If distance PQ = distance P’Q’ then the transformation is
an isometry. A distance-preserving transformation is called an ‘isometry’ because in
Greek, isos means equal and metron means measure. Another name given to a distancepreserving transformation is rigid motion. There are only four types of isometries of the
plane. The four isometries are listed below. A proof that there are only four isometries
can be found in Martin [1].
1.
2.
3.
4.
Reflection
Translation
Glide reflection
Rotation
Reflection
A reflectio n maps all points of a figure on one side of any straight line L, to the opposite
side of L such that the perpendicular distance between any point P, on one side of L, to
the line L is equal to the perpendicular distance of P’, the image of P on the opposite side
of L, to the line L. The line L is called the mirror or the axis of reflection. Figure 1.1
shows the image of the point P and the image of a triangle after reflecting about different
axis of reflection L.
P
P
P’
P’
L
P
L
L
P’
Figure 1.1. Reflection of the point P and a triangle about different axis of reflection L.
Translation
Translation shifts all points of a figure in a given direction through a given distance. We
will use an arrow to represent a translation. The magnitude and directio n of the arrow
denote the distance and direction of translation, respectively. Similar to reflection,
translation can also take place in any direction. Figure 1.2 shows translation of the point
Q and translation of a triangle in different directions and distances as indicated by the
arrows.
Q’
Q’
Q
Q’
Q
Q
Figure 1.2. Translation of the point Q and a triangle in different directions and
distances.
Glide reflection
Glide reflection, as the name suggests, is a combination of the two isometries mentioned
earlier, namely, translation (glide) and reflection. Glide reflection is carried out by
following either of the two methods given:
Translation followed by reflection
Translation of all points of a figure through a given distance d parallel to a line L,
followed by reflection of the image under translation in the same line L.
Reflection followed by translation
Reflection of all points of a figure about a line L, followed by translation of the image
under reflection through a given distance d parallel to the same line L.
Both methods give the same final result, meaning, the position of the final image under
both methods is the same. We will illustrate this concept with triangles. Figure 1.3 shows
the position of the image of a triangle P, P’, under glide reflection. In Figure 1.4 we
perform both method one and two, separately, on triangle P. The dashed triangle in
method one represents the intermediate image under translation. The dashed triangle in
method two represents the intermediate image under reflection. Notice that the position
of the final image in both method one and two in Figure 1.4 is the same as that in Figure
1.3.
P
glide reflection
P
L
P’
Figure 1.3. Glide reflection of triangle P and its image P’.
Method 1: Translation followed by reflection
translation
reflection
P
P
L
P’
L
Method 2: Reflection followed by translation
reflection
P
translation
P
P
L
P’
L
Figure1.4. The two methods of glide reflection of a triangle P.
The line L in glide reflection is called the axis of glide reflection. Similar to reflection
and translation, glide reflection can also take place in any direction as seen in Figure 1.5.
L
L
L
Figure 1.5. Glide reflection of the same triangle in different axis of glide reflection L.
Rotation
Rotation maps all points of a figure through a given angle θ about a given point O. The
point O is called the center of rotation. Let θ be the angle of rotation about point O that
maps a figure onto itself. If 360°/θ = n (a natural number) then the order of rotation of
the figure is equal to n. The point O is then referred to as the center of n-fold rotational
symmetry. Therefore, if a figure admits rotations through 60°, 120°, and 180° then its
order of rotation is six, three and two, respectively. The highest order of rotation of the
figure is six.
We will use an equilateral triangle as an example to illustrate the way to obtain the order
of rotation. The angles of rotation of the equilateral triangle about its center O that maps
the triangle onto itself are 120° and 240°. 360°/120° is equal to three, a natural number,
while 360°/240° is not a natural number. Thus, the order of rotation is only three. Thus,
the center O of the equilateral triangle is then called the center of 3- fold rotational
symmetry.
A point that remains at the same position under an isometry is called a fixed point. Only
reflection and rotation have fixed points. Under reflection, points that lie on the axis of
reflection are the fixed points. Under rotation, the centre of rotation is the fixed point.
Fixed points do not exist in translation and glide reflection.
Let ABC be the vertices of a triangle named in a clockwise (anticlockwise) direction. Let
A’B’C’ be the respective images of the vertices ABC under an isometry. If A’B’C’ forms
the vertices of the image of the triangle named in a clockwise (anticlockwise) direction
then the isometry is called direct. On the other hand, if A’B’C’ forms the vertices of the
image of the triangle named in an anticlockwise (clockwise) direction then the isometry
is called indirect. Only rotation and translation
are direct. Reflection and glide reflection are indirect. Table 1.1 shows us whether each
of the four isometries has fixed points and also whether it is direct.
Direct
Indirect
Have fixed points
Rotation
Reflection
No fixed points
Translation
Glide reflection
Tabl
e
1.1.
Prop
erties of the four isometries.
1.2 What is a symmetry?
A symmetry of a figure is an isometry that maps the figure onto itself. Besides the four
isometries mentioned, there is a trivial isometry called the identity isometry. The identity
isometry maps every point of a figure onto itself and it is a symmetry of every figure. The
rotation of a figure through 360° is an identity isometry of the figure. For example, in
Figure 1.6 the symmetries of the square are reflections about lines L1 , L2 , L3 and L4 and
rotations through angles π/2, π and 3π/2 about its center O. The 360° turn of the square
about point O is the identity isometry. Therefore, the square has eight distinct
symmetries: four reflections, three rotations and the identity isometry.
L1
O
L2
L3
L4
Figure 1.6. Reflections in the lines L1, L2 , L3 and L4 are symmetries of the square.
Other symmetries of the square are the identity symmetry and rotations through angle
π/2, π and 3π/2 about the center O.
Note that a figure has a proper glide reflection if and only if the translation component of
the glide reflection and the reflection component of the glide reflection are not
symmetries of the figure. For example, the figure consisting of two rows of triangles in
Figure 1.7 seems to map onto itself under glide reflection about line L, which is
performed by reflection about line L and translation parallel to line L. However,
reflection about line L and translation parallel to line L are symmetries of the figure.
Thus, the figure has no proper glide reflection. In this thesis, we will only consider
figures with proper glide reflection as having glide reflection as a symmetry.
L
Figure 1.7. Figure that has no proper glide reflection.
2. Tilings
Many of M.C.Escher’s prints that we will be studying are considered tilings in the plane.
Therefore it is necessary to know what are tilings before we discuss the symmetry in
tilings.
2.1 Defining tilings
Following is the definition of tilings given by Gru/nbaum and G.C.Shephard [2]. Let Ti
(i=1,2,…) denote the tiles of a tiling T in the plane. A plane tiling T satisfies the
following conditions:
1. The tiles Ti are closed sets. Meaning, each tile contains all its boundary points.
2. T consists of a countable family of sets T ={ T1 , T2 …}. Meaning the number of
tiles in the tiling can be counted.
3. The tiles Ti cover the plane without gaps. Meaning, the union of the sets T1 , T2 …
is to be the whole plane
4. The tiles Ti do not overlap. Meaning, the interiors of the sets T1 , T2 … are to be
pairwise disjoint.
Condition two excludes tilings in which every tile has zero area (such as points or line
segments). This is because tiles with zero area are uncountable when condition three is
satisfied. In other words, tiles with zero area are uncountable when they are arranged
without any gaps between them. However, the above definition admits tilings in which
some tiles have bizarre shapes and properties. Figure 2.1 gives a few examples of some
of the tiles with bizarre shapes and properties. Note that the few examples given in Figure
2.1 are not exhaustive. There are many other bizarre tiles. The following explains the
properties of the bizarre tiles given in Figure 2.1:
.
(a) A tile which is not connected. Meaning the tile consists of two or more
separate pieces.
(b) A tile which is not simply connected. Meaning the tile encloses at least one
hole.
(c), (d) Tiles which become disconnected when a suitable finite set of points are
are deleted.
(e), (f) Tiles which are made up partly of figures of zero area such as line
segments and arcs of curves.
(g) A tile which is unbounded. Meaning the tile cannot be enclosed by a finite
circle, however large. In this particular example (g) the tile is part of an
infinite strip. Note that tile (g) is a closed set since it contains all its
boundary points. Therefore, (g) satisfies condition one.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Figure 2.1. Gives a few examples of tiles with bizarre shapes and properties.
In this thesis, we only want to consider tilings consisting of tiles that are closed
topological disks. A disk is called a topological disk if it satisfies the following:
1. It is bounded and connected.
2. It is a simply connected set. This means that the boundary of a topological disk is a
single simple closed curve. A simple closed curve is a curve whose ends join up to
form a loop and has no crossings or branches.
A mapping of the plane onto itself is called a homeomorphism if it is one-to-one and
bicontinuous. Bicontinuous means both the mapping and its inverse are continuous. We
describe a closed or open topological disk as any plane set which is the image of a closed
or open circular disk, respectively, under a homeomorphism.
Therefore, we include the following condition five in the definition of tilings given
earlier: 5. Each tile Ti in the tiling is a closed topological disk. We will deal only with
tiles that are closed topological disks. For brevity, we will call them simply “tiles”.
The additional condition five removes tilings with bizarre shapes and properties such as
those given in Figure 2.1. This is because in Figure 2.1 (a) the tile is not connected and
the boundary is not a single curve. Similarly, in Figure 2.1(b) the boundary of the tile is
also not a single curve. In Figures 2.1 (c), (d), (e) and (f) the boundary of the tile is not a
simple curve because it has branches or crossing points. In Figure 2.1 (g) the boundary of
the tile is not a closed curve.
3. Symmetry groups of tiles and tilings
A symmetry of a tile or a tiling is an isometry that maps the tile or the tiling onto itself,
respectively. The symmetry group of a tile or a tiling is the set consisting of all the
symmetries of the tile or the tiling, respectively. We will denote the symmetry group of a
single tile T by S(T) and the symmetry group of a tiling T by S(T).
3.1 Symmetry groups of tiles
Before we study the symmetry groups of tiles we have to first know the following: the
number of symmetries in a symmetry group is called the order of the symmetry group.
The symmetry groups of tiles are classified into two types: cyclic groups and dihedral
groups.
Cyclic groups
The first type of symmetry groups is called cyclic groups. A cyclic group consists of
rotations through angle 360°j/n (j = 0,1,…,n-1) about a fixed point. Therefore, order of a
cyclic group is n because the cyclic group consists of only n rotations. The notation for a
cyclic group of order n is cn (n ≥1). c1 is the group consisting of only the identity
symmetry. In the later chapters we will come across the term “asymmetric tiles”. Thus, it
is important to define asymmetric tiles. Asymmetric tiles are tiles that admit only the
identity symmetry. This means that the symmetry group of an asymmetric tile T is c1
(S(T) = c1). Figure 3.1. gives examples of a tile with cyclic group c2 and an asymmetric
tile.
c2
c1 (asymmetric tile)
Figure 3.1. Examples of a tile with cyclic group c2 and an asymmetric tile.
Dihedral groups
The second type of symmetry groups is called dihedral groups. A dihedral group includes
all the isometries in cn and reflections in n lines. The n lines of reflection must be equally
inclined to one another. This means that the angle between any two lines of reflection is
the same. The order of a dihedral group is 2n. This is because a dihedral group consists of
n rotations and n reflections. Thus, the total number of symmetries in a dihedral group is
2n. The notation for a dihedral group of order 2n is dn (n ≥1). d1 is the group consisting
of the identity symmetry and reflection about a line. Figure 3.2 gives a few examples of
tiles with dihedral groups. Table 3.1 gives a summary of cyclic groups and dihedral
groups.
d1
d3
(equilateral triangle)
d4
(square)
d8
(regular octagon)
Figure 3.2 A few examples of tiles with dihedral groups.
Table 3.1. Summary of the description of the two types of symmetry groups of tiles.
A marked tile is simply a tile with a marking on it. The symmetry of a marked tile is an
isometry that not only maps the tile onto itself but also maps the marking on the tile onto
itself. Therefore, the symmetry group of a marked tile consists of all the symmetries that
map the tile and its marking onto themselves. The symmetry groups of marked tiles are
1.
Type of S(T)
Symmetries in S(T)
Notation
Cyclic group
rotations through
angle cn (n ≥ 1)
360°j/n (j = 0,1,…,n-1) about
a fixed point
Order of S(T)
n
no reflection
2.
Dihedral group
rotations through
angle dn (n ≥ 1)
360°j/n (j = 0,1,…,n-1) about
a fixed point
2n
reflections about n lines
equally inclined to one
another.
also classified into cyclic groups and dihedral groups. Figure 3.3 gives a few examples of
marked square tiles and their symmetry groups. The center of rotation is the center of
each marked square and the axis of reflection is denoted by a red dotted line.
c1
c2
d1
d1
d2
Figure 3.3 A few examples of marked square tiles and their symmetry groups.
3.2 Symmetry groups of tilings
Symmetry groups S (T) of tilings is classified into three cases as shown in Table 3.2
Figure 3.4 gives an example of a tiling from each of the three cases of S(T).
Table 3.2. The three cases of S(T) for tilings.
Properties of S(T)
S(T) contains no translations.
Name of S(T)
Cyclic groups and
dihedral groups
Name of tiling
S(T) contains translations, all of Strip groups or
which are parallel to a given frieze groups
direction L.
Strip tiling or
frieze tiling
S(T) contains translations in non- Periodic groups
parallel directions.
wallpaper groups or
crystallographic groups
Periodic tiling or
wallpaper tiling
Tiling with no
translation
Strip tiling
Periodic tiling
Figure 3.4. Examples of a tiling with no translations, a strip tiling and a periodic tiling.
3.2.1 Symmetry groups of tilings with no translations
Similar to the symmetry groups of tiles, the symmet ry groups of tilings with no
translations are also classified into cyclic groups and dihedral groups. The description of
the cyclic groups and the dihedral groups for tilings with no translations is similar to that
given in Section 3.1.
A tiling with no translations that contains rotations only has cyclic group cn (n ≥1).
Figure 3.5 gives an example of a tiling with no translations. The symmetries of the tiling
are rotations through angle 60°, 120°, 180°, 240°, 300° about the center of the tiling.
Since the tiling has only rotations it has a cyclic group. Also, the order of cyclic group of
the tiling consist of five rotations plus the identity symmetry. Thus, the order of the cyclic
group of the tiling is six. Hence, in Figure 3.5 the symmetry group of the tiling with no
translations is cyclic group c6.
Figure 3.5. An example of a tiling with no translations and with cyclic group c6.
Tilings with no translations that admit only rotations and reflections have dihedral group
dn (n ≥1) of order 2n. If the tiling admits more than one reflection then the lines of
reflection are not parallel but meet at a point. This is because if the lines of reflection are
parallel then the tiling will admit translation but we are considering tilings with no
translations. Therefore, in a tiling with no translations and with more than one reflection
the center of rotation is the point at which the lines of reflection meet. Also, in dihedral
groups of tilings with more than one reflection, the smallest angle of rotation is twice the
angle between any two lines of reflection. All other angles of rotation (< 360°) are a
natural multiple of the smallest angle.
For example Figure 3.6 gives a tiling with no translatio ns. The tiling admits reflections
about the three lines of reflection denoted by red lines. The angle between any two lines
of reflection is 60°. Therefore, the smallest angle of rotation of the tiling is 120°. The
other angle of rotation is 240° as it is the only natural multiple of 120° that is less than
360°. Thus, the tiling admits two rotations, three reflections and the identity symmetry.
Therefore, the order of the dihedral group of the tiling is six and the symmetry group of
the tiling is d3.
Figure 3.6. An example of a tiling with no translations and with dihedral group d3.
3.2.2 Symmetry groups of strip tilings
There are only seven types of symmetry groups for strip tilings. Unlike tilings with no
translations, which have infinite number of symmetry groups in both cyclic groups and
dihedral groups, strip tilings have a finite number of symmetry groups. Readers can refer
to Martin [1] for a proof that there only seven strip groups. In this thesis, we are
interested in understanding the notations for each strip group so that given a tiling we
know the type of strip group it belongs to.
The restrictions for the symmetries present in strip tilings are as follows. A strip tiling
contains translations parallel to a given direction L. Therefore, a strip tiling can only
admit reflection in lines perpendicular to L and at most one reflection in a line parallel to
L. This is because if a strip tiling admits reflections in lines neither perpendicular nor
parallel to L then consequently it will not admit translations parallel to L. If a strip tiling
admits more than one reflection in lines parallel to L then consequently it will admit
translations in the direction perpendicular and parallel to L and thus it will be a periodic
tiling. Also, a strip tiling can only admit rotation of order two or 180° rotation and the
centers of rotation must lie at equal distances on a line parallel to L. The restrictions give
the possible symmetries of a strip tiling that are denoted in its notations.
Each symmetry group of a strip tiling is named in the form of a four-symbol notation
pxyz. The name of each symmetry group begins with the letter p. The rest of the four symbol notation is described below. Note that we assume strip tiling to admit translation
in a given direction L. Reflection in lines perpendicular to L are called vertical
reflections. Reflections in lines parallel to L are called horizontal reflections.
Strip tiling: four-symbol notation pxyz
p
x=
y=
z=
m if there is vertical reflection
1 otherwise
m if there is horizontal reflection
a if there is glide reflection but no horizontal reflection
1 otherwise
2 if there is rotation of order 2 (half- turn)
1 otherwise
Note that under symbol y in the four-symbol notation, we did not consider the case where
both horizontal reflection and glide reflection occur in a strip tiling. This is because if
both occur then the strip tiling has no proper glide reflection (see Section 1.2). Thus, the
strip tiling is said to not have a proper glide reflection but only horizontal reflection.
Hence, a strip tiling will not have horizontal reflection and proper glide reflection
occurring simultaneously.
In the four-symbol notation pxyz, x has two possible symbols (m or 1), y has three
possible symbols (m, a or 1) and z has two possible symbols (2 or 1). Therefore,
altogether 12 (=2x3x2) possible four-symbol notations can be obtained. However, only
seven of these possibilities can occur as symmetry groups of strip tilings. Strip tilings can
have only five types of symmetry. They are translation, vertical reflection, horizontal
reflection, glide reflection and rotation of order 2. Translation is present in all the
symmetry groups of strip tilings. The seven four -symbol notations, as seen in Table 3.3,
consist of all possible combinations of one type of symmetry besides translation (p1m1,
pm11, p112 and p1a1), all possible combinations of three types of symmetry besides
translation (pmm2 and pma2) and the symmetry group with only translation (p111).
The remaining five four -symbol notations that are missing are pmm1, p1m2, pma1, p1a2
and pm12. These five notations consist of all possible combinations of two types of
symmetry besides translation. When two types of symmetry are present in a strip tiling it
will result in a third type of symmetry. For example, strip tiling with symmetry group
pmm1 consists of vertical reflection and horizontal reflection. When vertical and
horizontal reflections are present in a strip tiling this will result in rotation of order 2 to
also be a symmetry of the strip tiling. Thus, pmm1 corresponds to pmm2. In this way,
each of the five missing notations corresponds to either pmm2 or pma2. This is because
pmm2 and pma2 are the only two symmetry groups with three types of symmetry besides
translation. Therefore, the five missing notations do not occur as symmetry groups of
strip tilings as they correspond to symmetry groups with translation and three types of
symmetry.
Let
VR – vertical reflection
HR – horizontal reflection
G - glide reflection
2 - rotation of order 2
Each
symm
etry
group
pmm2
VR + HR → 2
of a
HR + 2 → VR
strip
pma2
VR + G → 2
tiling
G + 2 → VR
can
pmm2
also
VR + 2 → HR
be
The centers of rotation of order 2
named
lie on the axis of reflections.
in the
pm12
form
pma2
VR + 2 → G
of
an
The centers of rotation of order 2
abbre
lie between the axis of reflections.
viated
two-symbol notation xy. The abbreviation is obtained from the four-symbol notation pxyz
by deleting the first (p) and fourth (z) symbol and replacing symbol a by symbol g. The
two-symbol notation is derived in the following way:
The five missing
four-symbol
notations
pmm1
p1m2
pma1
p1a2
Corresponding
four-symbol notation
Reason
Strip tiling: two-symbol notation xy
x=
y=
m if there is vertical reflection
1 otherwise
m if there is horizontal reflection
g if there is glide reflection but no horizontal reflection
2 if there is rotation of order 2 (half-turn)
but no horizontal reflection and no glide reflection
1 otherwise
In the two-symbol notation xy, x has two possible symbols (m or 1), y has four possible
symbols (m, g, 2 or 1). Therefore, altogether 8 (=2x4) possible two-symbol notations can
be obtained. However, only seven of these possibilities can occur as symmetry groups of
strip tilings. As seen in Table 3.3, each of the seven two-symbol notations corresponds to
a distinct four -symbol notation that occurs as a symmetry group of strip tilings.
Clearly, 11 correspond to p111 and the two-symbol notation with one type symmetry
besides translation (1m, m1, 12, 1a) corresponds to a four-symbol notation with the same
type of symmetry. The two- symbol notation mm corresponds to pmm2. This is because
strip tilings with symmetry group mm consist of vertical and horizontal reflections. As
seen earlier, vertical and horizontal reflections result in rotation of order 2. Thus, when a
strip tiling has vertical and horizontal reflection it will also have rotation of order 2.
Moreover, all strip tilings have translation. Therefore, we drop the symbols p and 2 in
pmm2 to give a short form mm. Similarly, mg corresponds to pma2. This is because strip
tilings with symmetry group mg consist of vertical and glide reflections. As seen earlier,
vertical and glide reflections result in rotation of order 2. Thus, when a strip tiling has
vertical and glide reflections it will also have rotation of order 2. Therefore, we drop the
symbols p and 2 in pma2 and replace symbol a by g to give a short form mg.
The two-symbol notation that is missing is m2. This is because m2 corresponds to pm12.
And as seen earlier, pm12 corresponds to pmm2 or pma2 depending on the position of
centers of rotation of order 2 with respect to the axis of reflection. From Table 3.3 we
know pmm2 and pma2 correspond to mm and mg, respectively. Therefore, m2 does not
occur as a symmetry group because it corresponds to mm or mg. The case where a strip
tiling admits rotation of order 2, horizontal reflection and glide reflection is not possible.
This is because as we saw earlier under the four-symbol notation, a strip tiling with
Symmetry groups of strip tilings
Four-symbol
notation
p111
p1m1
pm11
p112
p1a1
Two-symbol
notation
11
1m
m1
12
1g
pmm2
mm
Symmetries present
(besides translation)
None
Horizontal reflection
Vertical reflection
Rotation of order 2
Glide reflection
Vertical reflection,
horizontal reflection,
rotation of order 2
pma2
mg
Vertical reflection,
glide reflection,
rotation of order 2
horizontal reflection and glide reflection has no proper glide reflection.
Table 3.3. The four-symbol notation, the two-symbol notation, and the symmetries
present for each of the seven symmetry groups of strip tilings.
An example of a strip tiling for each of the seven symmetry groups is given in Figure 3.7.
The elements of symmetry in Figure 3.7 are denoted in the following way. The lines L
denote the axis of reflection. The diamond ◊ denotes the center of 2-fold rotational
symmetry and the dotted line denotes the axis of glide reflection.
L
p111
p1m1
p112
L
L
L L
pmm2
L
p1a1
L
L
L
L
pm11
L
L
L
L
pma2
Figure 3.7. An example of a strip tiling for each of the seven symmetry groups.
3.2.3 Symmetry groups of periodic tilings
There are only 17 types of symmetry groups for periodic tilings. Similar to strip tilings,
periodic tilings also have a finite number of symmetry groups. Readers can refer to
Martin [1] for a proof that there are only seventeen periodic groups. In this thesis, we are
interested in understanding the notations for each periodic group so that given a tiling,
especially an Escher’s periodic tiling, we know the type of periodic group it belongs to.
Thus, enabling us to understand how the symmetry groups in Table 1 in the appendix was
derived for Escher’s periodic tilings.
In order to understand the notations used for periodic groups we need to learn about
lattices and primitive cells (also called lattice units) of periodic tilings. A lattice of a
periodic tiling is obtained by the following method:
1. Choose any arbitrary point P on the periodic tiling. However, if the periodic
tiling admits rotations then usually any center of rotation of the highest order is
chosen to be point P.
2. Apply translations of the periodic tiling on the point P. The set of all images of P
under the translations forms the lattice of the periodic tiling.
A primitive cell or a lattice unit is a parallelogram of a periodic tiling with the following
properties:
1. The vertices of the primitive cell are the lattice points of the periodic tiling.
2. The lattice points of the periodic tiling are not found on the interior or on the
edges of the primitive cell.
3. The vectors which form the sides of the primitive cell generate the translation
group of the periodic tiling. Meaning the sides of the primitive cell give the
directions of translations of the periodic tiling.
The notation of the symmetry group of a periodic tiling requires us to identify a primitive
cell of the tiling. To find a primitive cell of a periodic tiling we need to first find a lattice
of points of the periodic tiling. Secondly, we need to form a parallelogram whose vertices
are lattice points but whose interior and edges have no lattice points. For some tilings we
will get more than one type of primitive cell. Use the following method to decide which
type of primitive cell to choose when a tiling admits two or more types of primitive cells.
Note that a periodic tiling can only admit rotations of order 2, 3, 4 or 6 (see
Schattschneider [3] for a proof).
Method: 1. For a periodic tiling with order of rotation 3 or 6 choose a hexagonal cell.
2. For a periodic tiling with order of rotation 4 choose a square.
3. For a periodic tiling with order of rotation one or two we consider the
following two cases:
(a) if there is no reflection and no glide reflection choose a parallelogram
(b) otherwise choose either a rectangle or a rhombus by plotting the lattice of
points and drawing the primitive cell as described above. If a rhombus is chosen,
it is extended to a rectangle. The extended rectangle is called a centered cell.
Figure 3.8 gives an example of a tiling for each type of lattice. The points denoted by red
circles are a lattice of points of the tiling. The green figure is the primitive cell of the
tiling. The type of primitive cell of each tiling is written below the tiling. The rhombus in
the periodic tiling in Figure 3.8 is obtained as a result of Method 3(b) mentioned above.
The dotted rectangle in the periodic tiling is the extended rectangle. Note that the tilings
consist of one or two type of tiles. Generally, the notations for the symmetry groups of
periodic tilings applies to tilings with one or more types of tiles.
Parallelogram
Rectangle
Rhombus
Square
Hexagonal cell
Figure 3.8. An example of a tiling for each type of lattice.
We will now study the notations of the symmetry groups for periodic tilings. Each
symmetry group of a periodic tiling is named in the form of a four-symbol notation qrst.
The notation was derived by crystallographers who use it to classify crystals. The four symbol notation of a periodic tiling is denoted in the following way:
1. Plot the lattice of points for the periodic tiling and determine the primitive cell.
2. q =
p if the periodic tiling has primitive cell
c if the periodic tiling has centred cell
3. r = the highest order of rotation of the tiling
4. s =
m if there is reflection perpendicular to an edge of the cell
g if there is glide reflection perpendicular to an edge of the cell
1 otherwise
Symbol s denotes a symmetry perpendicular to an edge of the cell and we call the edge of
the cell x-axis. Note that for symbol s, we may have a case where we have a choice
between m, g and 1 depending on the edge of the cell chosen as the x-axis. Then, we
make a choice of the x-axis that gives us m, g or 1 in that order. For example, if a
periodic tiling has both reflection and glide reflection perpendicular to different edges of
the cell then we denote s by m. The edge perpendicular to the axis of reflection
corresponding to m is then the x-axis. In symmetry group p4gm there is no reflection
perpendicular to the edges of the primitive cell but there is glide reflection perpendicular
to the edges of the primitive cell. Thus, symbol s is denoted g in p4gm. In p31m, there is
no reflection and no glide reflection that are perpendicular to the edges of the primitive
cell. Thus, symbol s is denoted 1 in p31m.
5. t =
m if there is reflection at an angle θ (≠ 90°) to the x-axis
g if there is glide reflection at an angle θ (≠ 90°) to the x-axis
1 otherwise
Symbol t denotes a symmetry at an angle θ (≤ 180) to the x-axis. Let n be the highest
order of rotation of the periodic tiling. In particular θ must take the following values:
θ = 180° if n = 2
θ = 45° if n = 4 and
θ = 60° if n = 3 or 6
Figure 3.9 shows the lattice unit for each of the 17 periodic groups and the symmetries
present in each periodic group. In Figure 3.9 under each lattice unit the four-symbol
notation of the periodic group is given in parentheses. The short form of the periodic
group is given above the four-symbol notation. Note that no symbol in the third and the
fourth positions of the four-symbol notation indicate that the periodic group has no
reflection and no glide reflection. Symmetry groups cm and cmm are the only periodic
groups that have centered cells.
Figure 3.9. Lattice units of each periodic group and the symmetries in each
periodic group.
Let us consider the positions of the axes of reflection and glide reflection at the center of
rotation of the highest order in the lattice unit of each symmetry group
Note that for each order of rotation in the table above there is a lattice unit that has no
reflection and no glide reflection. Moreover, for each order of rotation there is a lattice
unit where the axes of reflection and glide reflection are in the positions of the dotted
Possible positions of axes
Positions of axes
Highest
reflection and glide
Symmetry
reflection and glide
order of
reflection at each center
group
reflection at each center
rotation
of rotation of the highest
of rotation of the highest
order
order
2
p211
no reflection and
no glide reflection
p2mm
p2mg
p2gg
no axis of glide
reflection passing
through centers of
rotation of order 2
c2mm
3
p311
no reflection and
no glide reflection
p3m1
p31m
4
p411
no reflection and
no glide reflection
p4mm
p4mg
6
p611
p6mm
no reflection and
no glide reflection
lines. And another lattice unit with axes of reflection and glide reflection in the positions
of the undotted lines or both the dotted and undotted lines. Only in the highest order of
rotation 2 can we see two symmetry groups, pmm2 and cmm2, that have the axes of
reflection passing through the center of rotation in the same way. However, as seen in
Figure 3.9, in pmm2 the axes of reflection are perpendicular to the edges of the lattice
unit. But in cmm2 both the axes of reflection and glide reflection are not perpendicular to
the edges of the lattice unit. Thus, in cmm2 we extend the lattice unit to the dotted
rectangle in order to make the axes of reflection and glide reflection perpendicular to the
edges of the extended cell. The dotted rectangle cell is called the centered cell.
Similarly, let us consider symmetry groups with no rotation (p111, p1m1, p1g1, c1m1).
Symmetry group p111 has only translation. Symmetry group p1m1 and p1g1 have axes
of reflection and glide reflection, respectively, that are perpendicular to the edges of the
lattice unit. Symmetry group c1m1 has both axes of reflection and glide reflection that
are not perpendicular to the lattice unit. Thus we extend the lattice unit to a rectangle in
order to make the axes of reflection and glide reflection perpendicular to the edges of the
lattice unit. The dotted rectangle is called the centered cell.
Earlier in Method 3(b) in determining the type of primitive cell, we will obtain a rhombus
if there is no rectangle cell. The rhombus obtained is then extended to a rectangle and the
extended rectangle is called the centered cell. The reason for extending the rhombus
obtained from Method 3(b) to a centered cell is as follows. As seen in Figure 3.9 only
periodic tilings with either symmetry group c1m1 or c2mm will obtain a rhombus due to
Method 3(b). In these two symmetry groups as we mentioned earlier, the axes of
reflection and glide reflection are not perpendicular to the edges of the rhombus (lattice
unit). Thus, we form the centered cell to make the axes of reflection and glide reflection
perpendicular to the edges of the centered cell. Thus, the symmetry group of periodic
tilings with a centered cell is c1m1 or c2mm.
Apply to Escher’s periodic tilings
We will use Escher’s periodic tilings to show how to obtain some of the symmetry
groups given a tiling (see Figure 3.10). Colours of the tilings are ignored when finding
the symmetry group. Note that all the examples given of Escher’s periodic tilings are
marked tilings. This simply means that there is a marking on each tile. Symmetry of a
marked tiling is an isometry which not only maps the tiles of the tiling onto tiles of the
tiling. It also maps each marking on a tile of the tiling onto a marking on the image tile.
Therefore, when finding the symmetry group of an Escher’s tiling we have to make sure
that the symmetries map the tiling onto itself and also map the markings onto themselves.
In Figure 3.10 remarks on deriving the correct symmetry group are given below each
drawing. Symbols used are as follows:
•
•
•
•
•
•
red dots – lattice of points
black parallelogram – primitive cell
dotted parallelogram – centred cell
green lines – axis of reflection
purple dotted lines – axis of glide reflection
number in bracket beside symmetry group – Escher number of drawing in
Table 1 in the appendix
p1 (18)
•
•
Parallelogram cell
Only translation
p3m1 (69)
• Hexagonal cell
• Highest order of rotation is three. Red dots are
lattice points and centers of rotation of order
three.
• There is reflection axis perpendicular to the
left and right edges of the cell. Any one of them
is the x-axis. So third symbol is m.
• There is no reflection or glide reflection at an
angle 60° to the x-axis. So fourth symbol is 1.
p2mg (89)
We will look at this tiling as two separate
tilings (top and bottom). This is because the
fins of the fishes are shaped differently for
the top tiling and the bottom tiling. But both
tilings have the same symmetry group p2mg.
• Rectangular cell
• Highest order of rotation is two. Red
dots are lattice points and centers of
rotation of order two.
• There is reflection axis perpendicular
to the left and right edges of the cell. Any
one of them is the x-axis. So third symbol
is m.
c1m1 (91)
• Centered cell
• Highest order of rotation is one.
• There is reflection axis perpendicular
to the top and bottom edges of the cell.
Any one of them is the x-axis. So third
symbol is m.
• There is no reflection or glide
reflection at an angle 180° to the x-axis.
So fourth symbol is 1.
Figure 3.10. Escher’s periodic tilings and their symmetry groups.
4. Isohedral types
4.1 What is an isohedral tiling?
Two tiles T1, T2 of a tiling T are said to be equivalent if the symme try group S(T)
contains a transformation that maps T1 onto T2 . The collection of all tiles of T that are
equivalent to T1 is called the transitivity class of T1 . If all tiles of T form one transitivity
class we say that T is an isohedral tiling. If T is a tiling with exactly k transitivity classes
then T is called a k- isohedral tiling. If a tiling T admits only the identity symmetry then
every tile in T is a transitivity class on its own. Figure 4.1 gives examples of an isohedral
tiling and a 2- isohedral tiling. In the 2- isohedal tiling the tiles congruent to tile one form a
transitivity class. The tiles congruent to tile two form another transitivity class.
1
2
isohedral tiling
2-isohedral tiling
Figure 4.1. Examples of an isohedral tiling and a 2-isohedral tiling.
What is a monohedral tiling?
In a monohedral tiling all the tiles of the tiling are congruent (directly or reflectively) to
one fixed set. This means that all the tiles in the monohedral tiling have the same size and
the same shape. Figure 4.2 gives examples of monohedral tilings.
Figure 4.2. Two examples of monohedral tilings.
In both monohedral and isohedral tilings the tiles have the same shape and size. However,
in isohedral tiling there is a symmetry that makes all the tiles equivalent. In monohedral
tilings there may not exist a symmetry that maps one tile to another. The distinction
between monohedral and isohedral tilings may seem slight but it is very significant. The
monohedral tilings in Figure 4.2 are isohedral. However, the monohedral tilings (a) and
(b) in Figure 4.3 are not isohedral. This is because there is no symmetry of the tiling that
maps tile 1 onto tile 2 in each tiling. On the other hand, monohedral tiling (c) in Figure
4.3 is 2-isohedral. This is because (c) consists of congruent tiles. Thus, it is monohedral.
Moreover, in (c) the tiles of type 1 belong to one transitivity class and the tiles of type 2
belong to another transitivity class. Therefore, (c) has two transitivity classes and thus it
is 2-isohedral.
1
1
2
2
(a)
(b)
1
2
(c)
Figure 4.3. (a) and (b) are monohedral but not isohedral tilings. (c) is both monohedral
and 2- isohedral.
Apply to Escher’s drawings
In Table 1 in the appendix, the column for the number of motifs is equal to the number of
different kinds of marked tiles in a marked tiling or the number of different kinds of
unmarked tiles in an unmarked tiling. All of Escher’s periodic tilings with one motif
(from Escher number 1 to137) are monohedral tilings. We say that every monohedral
tiling in Escher number 1 to 137 are isohedral except for 137 (ignoring 131 and ignoring
the colours of the tilings). Therefore, all of Escher’s periodic tilings from 1 to 137 have
isohedral types (IH) except for 137. Figure 4.4 shows Escher’s tiling 137. Tiling 137 is
monohedral because all the tiles (ghost figures) are congruent. However, 137 is not
isohedral because there is no symmetry of the tiling that maps tile 1 onto tile 2. Thus,
tiles in 137 do not belong to one transitivity class.
2
1
Figure 4.4. Escher’s tiling with Escher number 137.
Gru/nbaum and G.C.Shephard [2] classified isohedral tilings into isohedral types. The
notation used is IH followed by a number as seen in Table 1 in the appendix under
isohedral types. Table 4.1 gives all the classifications of the isohedral tilings into their
isohedral types depending on the following: topological type, incidence symbol,
symmetry group and induced tile group. In Section 3.2.3 we learned how to find the
symmetry group of a periodic tiling. In the following Section 4.2 we will learn how to
find the topological type, induced tile group and the incidence symbol of an isohedral
tiling. Then by matching them with Table 4.1 we will obtain the isohedral type of the
tiling. Colours of isohedral tilings are ignored in the classification. Therefore, when
looking at Escher’s tilings in this chapter treat them as being uncoloured. The aim of this
chapter is to find the isohedral type when given an isohedral tiling.
Type Topological
type
Incidence
symbol
Symmetry
group
Induced tile
group
Type
Topological
type
Incidence
symbol
Symmetry
group
Induced tile
group
Table 4.1. Classification of isohedral types of isohedral tilings.
4.2 Identifying isohedral type
To be able to identify the isohedral type of an isohedral tiling we need to first learn the
meanings of the following terms:
•
•
•
Topological types
Incidence symbols
Induced tile group
4.2.1 Topological types
The valence of each vertex of a tile is the number of edges that meet at the vertex. Let T
be a tile of an isohedral tiling T with k vertices and with valencies j1 , j2,…jk. In an
isohedral tiling all the tiles will have the same valencies. The topological type of
the isohedral tiling is [j1 … jk]. To standardize the notation for the topological type [j1 …jk ]
we do the following:
1. If two or more consecutive numbers ji take the same value, use superscripts to
abbreviate the symbols. For example, let the number of vertices of a tile in an
isohedral tiling be five. If j1 to j3 have value 3 and j4 to j5 have value 4 then the
topological type = [3³.4²].
2. If the smallest valence of the tile has a superscript then write the smallest valence
with the superscript first. Otherwise, write the smallest valence of the tile first.
We will now illustrate how to find the topological type of isohedral tilings. Figure 4.5
shows Escher’s marked isohedral tilings 16 and 117 as indicated below the tilings. In
tiling 16 the tiles are dogs. In tiling 117 the tiles are crabs. The red dots denote the
vertices of a tile in each tiling. The valence of each vertex in a tile is denoted in the tiling.
In tiling 16 the valencies of a tile are 3,3,4,3,4. Thus, the topological type of 16 is
[33434]. After standardizing the notation we have [32 .4.3.4] as the topological type of 16.
Similarly, in tiling 117 the valencies of a tile are 4,4,4,4. Thus, the topological type of
117 after standardizing is [44 ].
4
3
3
3
4
16
4
4
4
4
117
Figure 4.5. Escher’s isohedral tilings number 16 and 117.
Isohedral tilings will only have one of the following 11 topological types: [36 ], [34 .6],
[3³.4²], [3².4.3.4], [3.4.6.4], [3.6.3.6], [3.12²], [44 ], [4.6.12], [4.8²], [6³]. A detailed proof
of this is given by Grunbaum and G.C.Shephard [2]. Escher’s isohedral tilings admit all
the topological types except the topological types [3.12²] and [4.6.12]. This is because an
isohedral tiling of topological type [3.12²] or [4.6.12] consists of 12 tiles meeting at some
of the vertices of each tile. Escher restricted his isohedral tilings to lesser than 12 tiles
meeting at a vertex of each tile.
4.2.2 Incidence symbols
The incidence symbol [L;A] consists of the tile symbol L and the adjacency symbol A.
Given an isohedral tiling T the following method tells us how to obtain the tile symbol of
T:
Tile symbol
Step 1: Place a small arrow labeled with the letter a near to and parallel to an edge of a
tile. The tile and the edge are chosen arbitrarily.
Step 2: Apply all the symmetries in the symmetry group S(T) on the arrow labeled a.
This will carry the arrow labeled a onto the other tiles. Since the tiling is isohedral, the
arrow a will be associated with at least one edge of every tile in T. Following cases may
arise depending on the symmetry group of T:
(i)
(ii)
arrow a is assigned to two or more edges of the same tile.
arrow a is assigned to the same edge of the same tile in the reversed
direction. Then we replace the small arrow by a double- headed arrow.
And we say that a is assigned to an unoriented edge.
Step 3: If all the edges of all the tiles are labeled a then our labeling is finished.
Otherwise, choose an edge of a tile that is not labeled and place a small arrow labeled
with another letter b. Then repeat step two to the arrow b. When all the edges of all the
tiles are labeled proceed to step four.
Step 4: Write the sequence of letters on the edges as we go around a tile. Then
(i)
add a superscript + to a letter when the arrow associated with the letter points
in the direction in which we are going around the tile.
(ii)
add a superscript – to a letter when the arrow associated with the letter points
in the direction opposite to that in which we are going around the tile.
(iii)
if the edge associated with a letter is unoriented then no superscript is added to
the letter.
The sequence of letters with the superscripts is the tile symbol of T.
Figure 4.6 illustrates the method used to find the tile symbol of a marked isohedral tiling
(a). In (b) we carry out step one and place an arrow labeled a on one of the edges of a tile.
Translation and diagonal reflection are the only symmetries of the tiling. In (c) we carry
out step two and apply translation and diagonal reflection on the arrow a. In (d) we repeat
step one and two using arrow b. Thus, in (d) we stop since all the edges of all the tiles are
labeled. Finally we carry out step four. We start with a in a tile and move anticlockwise
around the tile. Add the superscripts to the letters in a tile accordingly. The tile symbol is
a+b+b-a-.
Figure 4.6. Illustration of finding the tile symbol of a marked
isohedral tiling.
Figure 4.7 also illustrates the method used to find the tile symbol of a marked isohedral
tiling (a). We give this example to show the case where unoriented edges arise. In (b) the
arrow a is a double-headed arrow. This is because vertical reflection is a symmetry of the
tiling. When vertical reflection is applied to arrow a it is assigned to the same edge of the
same tile in the reversed direction. Thus, we replace the small arrow by a double-headed
arrow. And we say that a is assigned to an unoriented edge. This refers to step 2(ii). In (c)
we carry out step two and apply rotations of order two and four to a. In (d) we repeat step
one and two using arrow b. b is assigned to an unoriented edge because horizontal
reflection assigns b to the same edge of the same tile. Since all the edges are unoriented
the tile symbol has no superscripts. The tile symbol is abab.
Figure 4.7. Illustration of finding the tile symbol of a marked isohedral tiling.
We will now do the labeling for Escher’s tiling 117 (see Figure 4.8). In Escher’s tiling it
is difficult to look at edges since they are very much contoured. We will use the vertices
(red dots) to guide us since between two red dots denoted as vertices there has to be an
edge. Moreover, in Escher’s tiling we will only do the labelling for one tile since the
labeling for all the tiles is the same. Also we will do the labeling for the opposite side of
each edge of the tile. Then we find that the tile symbol of tiling 117 is ab+cb-.
c
a
b
b
b
b
c
a
117: Tile symbol: ab+cbFigure 4.8. Escher’s isohedral tiling 117 and its tile symbol.
Adjacency symbol
1. Let w (where w represents a,b,c,d,e,or f) be the first letter in the tile symbol. Let x
(where x represents a,b,c,d,e,or f) be the letter on an adjacent tile corresponding to
the same edge where w is labelleled. This means that the edge is labelled w on
one side and x on the other side. Thus, the first component of the adjacency
symbol is x.
(a) Add the superscript + to x if the arrows corresponding to x and w are in
the opposite direction.
(b) Add the superscript – to x if the arrows corresponding to x and w are in
the same direction.
(c) No superscript is added if the edge w is unoriented. Since edge w is also
the edge x, if edge w is unoriented then edge x is also unoriented. This
means that w and x each correspond to double-headed arrows.
2. Take the second letter in the tile symbol that is distinct from w. Let this letter be
y. Let z be the letter on an adjacent tile corresponding to the same edge where y is
labelleled. This means that the edge is labelled y on one side and z on the other
side. Thus, the second component of the adjacency symbol is z. Then add or do
not add the superscript + and – in the same manner that was mentioned in 1(a) to
1(c).
3. The third component of the adjacency symbol is added in the same way using the
third distinct letter from w and y (if any) in the tile symbol. We stop adding letters
to the adjacency symbol when all the distinct letters in the tile symbol have been
considered. Thus, the adjacency symbol is a sequence of letters with or without
superscripts. The number of letters in the adjacency symbol is equal to the number
of distinct letters in the tile symbol.
We will illustrate how to find the adjacency symbol using the isohedral tilings we saw
earlier in Figure 4.6 and 4.7. In Figure 4.6 a is the first letter in the tile symbol. The letter
opposite a is b which are both in the same direction. Therefore, the first letter in the
adjacency symbol is b-. The second distinct letter in the tile symbol is b. The letter
opposite b is a which are also in the same direction. Therefore, the second letter in the
adjacency symbol is a-. Only letters a and b are in the tile symbol. Thus, all the distinct
letters in the tile symbol are used. Therefore, the adjacency symbol is b- a-. Similarly, we
follow the method for Figure 4.7 and find that its adjacency symbol is ba. Furthermore,
when we do the method for the Escher’s tiling 117 in Figure 4.8 we will find that its
adjacency symbol is cb+a.
Now we can combine the tile symbol L and the adjacency symbol A. We will then get the
incidence symbol [L;A]. For example, as we saw earlier, the tile symbol for Escher’s
tiling 117 is ab +cb- and its adjacency symbol is cb+a. Therefore, the incidence symbol for
Escher’s tiling 117 is [ab +cb-; cb+a].
4.2.3 Induced tile group
Let T be a tile of a tiling T. The induced tile group of T in T is the group of symmetries
of a tile T which are also symmetries of the tiling T. The induced tile group is denoted
S(TT). Note the following:
•
•
All the symmetries of T that map T onto itself are clearly symmetries of T.
Symmetries of T are not necessarily symmetries of T that map T onto itself.
In Section 3.1 we saw that symmetry groups of a tile S(T) are either cyclic groups or
dihedral groups. The induced tile group of a tile S(T T) also consist of symmetries of a
tile. But the additional criterion is that in the induced tile group the symmetries of the tile
must also occur in the symmetry group of the tiling that contains the tile. Therefore, the
induced tile group of a tile in a tiling is either a cyclic group or a dihedral group (S(TT)
= cn or dn, n ≥1). For induced tile groups d1 and d2 of a tile we have to distinguish two
cases.
For S(TT) = d1:
If the axis of reflection is vertical or horizontal with respect to the tile
then the induced tile group is denoted d1(s).
If the axis of reflection is parallel to the diagonal of the tile then the
induced tile group is denoted d1(l).
For S(TT) = d2 : If both axis of reflection are vertical and horizontal with respect to
the tile then the induced tile group is denoted d2(s).
If both the axis of reflection are parallel to the diagonals of the tile
then the induced tile group is denoted d2(l).
The symbol s stands for “short’ and the symbol l stands for “long”. This is because
vertical and horizontal lines of a figure (s) are shorter than the diagonal lines of the figure
(l). The induced tile group for the crab tile in Escher’s isohedral tiling 117 in Figure 4.5 is
d1(s). This is because the crab tile admits vertical reflection only and vertical reflection is
also a symmetry of the tiling 117.
For an isohedral tiling with asymmetric tiles the induced tile group of the asymmetric tile
is c1 (S(TT) = c1). This is because the asymmetric tile admits only the identity
symmetry (S(T) = c1). Therefore, the only symmetry of the asymmetric tile that is also a
symmetry of the isohedral tiling is the identity symmetry. In Figure 4.5 the induced tile
group for the dog tile in Escher’s isohedral tiling 16 is c1. This is because the dog tile is
asymmetric.
We have found the following information on the Escher’s tiling 117:
Topological type = [44 ]
Incidence symbol = [ab+cb-; cb+a]
Induced tile group = d1(s)
Using what we have learned in Chapter 3, we can find out that its symmetry group is
pmg. Therefore, matching all this information with Table 4.1 (shows classification of
isohedral types) we will find that the isohedral type of Escher’s tiling 117 is IH66. In this
way, we can find the isohedral types of an isohedral tiling.
5. Heesch types
5.1 Classification of isohedral tilings into Heesch types
Heinrich Heesch, a German mathematician, classified isohedral tilings with no reflection
according to the type of asymmetric tiles that each isohedral tiling consists of. He found
that only 28 different types of asymmetric tiles (called Heesch types) form isohedral
tilings with no reflection symmetry. The 28 Heesch types are given in Table 5.1.
Table 5.1 Heesch’s table classifying the 28 Heesch types.
The Heesch’s table was derived by first considering the seven symmetry groups of
periodic tilings that do not contain reflection. The seven symmetry groups are p1, p2, p3,
p6, p4, pg and pgg. Secondly, he found the kind of asymmetric tile that an isohedral tiling
with one of the seven symmetry groups can have for each topological type. The single
numbers in the top row in Table 5.1 denote the number of vertices of each tile in the
columns corresponding to the number. The sequences of numbers in the top row in Table
5.1 denote the topological type without using superscripts. The following show the
relation between the sequence of numbers in the top row and the topological types with
notations as given in Section 4.2.1:
333333 = [36 ]
6434 = [3.4.6.4]
63333 = [34 .6]
4444 = [44 ]
43433 = [3².4.3.4]
666
= [6³]
44333 = [3³.4²]
884
= [4.8²]
6363 = [3.6.3.6]
12,12,3 = [3.12²]
Note that only ten topological types are given in Table 5.1. The topological type [4.6.12]
is missing. This is because referring to Table 4.1, we see that isohedral tiling with
topological type [4.6.12] will admit only one symmetry group that is p6m. Isohedral
tiling with symmetry group p6m admits reflection. Since Heesch only classified isohedral
tilings with no reflection symmetry topological type [4.6.12] was not considered.
Each of the 28 asymmetric tiles is denoted by a sequence of letters as seen in Table 5.1.
The letters of a tile denote the isometries that relate the edges of the tile. Following show
the isometry that each letter denotes:
T = translation
C = half-turn
C3 = 120º rotation
C 4 = 90º rotation
C 6 = 60º rotation
Each letter in the sequence corresponds to an edge of the tile. The letter denotes the
isometry that maps that edge to an adjacent edge or to an opposite edge. Edges with a
centre of half- turn are related to themselves. The sequence of letters characterizing a tile
is obtained by traveling clockwise around its boundary and associating to each edge the
appropriate letter. For example, Figure 5.1 shows an asymmetric tile from Table 5.1
corresponding to symmetry group pgg and topological type [3³.4²]. Let us denote the
edges of the tile one to five. Edges one and three are mapped to each other by translation.
Therefore edge one and edge three are denoted by T. Edge 2 has a centre of half- turn
symmetry and thus is denoted by C. Edges four and five are mapped to each other by
glide reflection. Therefore, edge four and edge five are denoted by G. Thus, we get the
sequence of letters TCTGG.
3
4
2
5
1
Figure 5.1. An asymmetric tile from Table 5.1 with Heesch type TCTGG.
In some cases, a pair of edges of a tile is related by glide reflection. And another pair of
edges in the same tile is also related by glide reflection. In such cases, we denote the
edges in one of the pairs G1 and the edges in the other pair are denoted G2 . For example,
we look at the tile G1G2 G1G2 in Figure 5.2. Edges one and three are related by glide
reflection and are denoted G1 . The axis of glide reflection for G1 is the vert ical dotted
line. Edges two and four are also related by glide reflection and are denoted G2 . The axis
of glide reflection for G2 is the horizontal dotted line. Thus, we get the sequence of letters
G1G2 G1G2 .
3
G1
G2
G2
2
4
1
G1
Figure 5.2. An asymmetric tile from Table 5.1 with Heesch type G1G2G1G2.
Apply to Escher’s periodic tilings
Let us illustrate how to find the Heesch type of an Escher’s tiling. Figure 5.3 shows
Escher’s drawing corresponding to Escher number one in Table 1 in the appendix. This is
a marked isohedral tiling with symmetry group p2 containing no reflection. All the
marked tiles are congruent shapes of an animal. The colours of the tiles are to be ignored.
Focusing on one of the tiles and noting the vertices of the tile by black dots, we see that
this is a tiling of topological type [36]. To obtain the Heesch type, we repeat the process
mentioned above. The edges of a tile are denoted one to six. Edges one, two, four and
five have a center of half-turn each and thus are denoted C. Edges three and six are
related by translation and are denoted T. Thus, we get the sequence of letters TCCTCC.
C
1
C2
3
T
4
C
T
6
5C
Figure 5.3. Escher’s tiling corresponding to Escher number1 and its Heesch type.
5.2 Relation between Heesch types and isohedral types
The isohedral type of each of the 28 Heesch types is given in Table 5.2.
Isohedral
type
IH1
IH2
IH3
IH4
IH5
IH6
IH7
IH21
IH23
IH25
IH27
IH28
IH31
IH33
IH39
IH41
IH43
IH44
IH46
IH47
IH51
IH52
IH53
IH55
IH79
IH84
IH86
IH88
Topological
type
[36 ]
[34 .6]
[33 .42 ]
[3².4.3.4]
[3.4.6.4]
[3.6.3.6]
[3.12²]
[44 ]
[4.8²]
[6³]
Incidence symbol
S(T) S(T | T)
Heesch type
[a+b+c+d+e+ f+ ;d +e+ f+a+b+c+ ]
[a+b+c+d+e+ f+ ;b -a-f+e-d-c+]
[a+b+c+d+e+ f+ ;c-e+a-f-b+d-]
[a+b+c+d+e+ f+ ;a+e+c+d+b+ f+ ]
[a+b+c+d+e+ f+ ;a+e+d-c-b-f+]
[a+b+c+d+e+ f+ ;a+e-c+ f-b-d-]
[a+b+c+d+e+ f+ ;b +a+d+c+f+e+ ]
[a+b+c+d+e+;e+c+b+d+ a+]
[a+b+c+d+e+;a+e+ c+d+b+]
[a+b+c+d+e+;a+e+d-c-b+]
[a+b+c+d+e+;a+d-e-b-c-]
[a+b+c+d+e+;a+e+d-c-b+]
[a+b+c+d+;b +a+d+c+ ]
[a+b+c+d+;d +c+b+a+ ]
[a+b+c+;a+c+b+ ]
[a+b+c+d+;c+d+a+b+ ]
[a+b+c+d+;c-d+a-b+]
[a+b+c+d+;b -a-d-c-]
[a+b+c+d+;a+b+c+d+ ]
[a+b+c+d+;c+b+a+d+ ]
[a+b+c+d+;c-b+a-d+]
[a+b+c+d+;c-d-a-b-]
[a+b+c+d+;b -a-c+d+]
[a+b+c+d+;b +a+d+c+ ]
[a+b+c+;a+c+b+ ]
[a+b+c+;a+b+c+ ]
[a+b+c+;b -a-c+]
[a+b+c+;b +a+c+ ]
p1
pg
pg
p2
pgg
pgg
p3
p6
p2
pgg
pgg
p4
p6
p3
p6
p1
pg
pg
p2
p2
pgg
pgg
pgg
p4
p4
p2
pgg
p6
TTTTTT
TG1 G1 TG2 G2
TG1 G2 TG2 G1
TCCTCC
TCCTGG
CG1 CG2 G1 G2
C 3 C3C3 C3C3 C3
CC 3C3C 6C6
TCTCC
TCTGG
CG1 G2 G1G2
CC 4C4C 4C4
C 3 C3C6 C6
C 3 C3C3 C3
CC 3C3
TTTT
TGTG
G1 G1G2 G2
CCCC
TCTC
CGCG
G1 G2G1 G2
CCGG
C 4 C4C4 C4
CC 4C4
CCC
CGG
CC 6C6
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
c1
Table 5.2. Heesch types and its corresponding isohedral types.
The matching of a Heesch type with the corresponding isohedral type was obtained by
using Table 5.1 and Table 4.1. In Table 4.1 we note that there are exactly 28 isohedral
types corresponding to the seven symmetry groups with no reflection (are p1, p2, p3, p6,
p4, pg and pgg). These 28 isohedral types also correspond to induced tile group c1. This
means that the 28 isohedral types correspond to isohedral tilings with asymmetric tiles
and no reflection. Similarly, in Table 5.1 we have exactly 28 Heeesch types. The 28
Heesch types also correspond to isohedral tilings with asymmetric tiles and no reflection.
Therefore, we can match each Heesch type with one of the 28 is ohedral types.
For the rows in black in Table 5.2 we can easily match the Heesch type with the isohedral
type. This is because there is only one Heesch and one isohedral type corresponding to
the same symmetry group and the same topological type. However, for the rows in red in
Table 5.2 it is not clear on which Heesch type matches with which isohedral type. For
example, there are two Heesch types TG1 G1 TG2 G2 and TG1 G2 TG2 G1 corresponding to
symmetry group pg and topological type [36 ]. Similarly, there are two isohedral types
IH2 and IH3 also corresponding to symmetry group pg and topological type [36 ]. Then
how do we know which isohedral type belongs to which Heesch type? To answer this we
look separately at isohedral tilings of IH2 and IH3 (see Figure 5.4). Note that the tilings
in Figure 5.4 consist of asymmetric tiles. This is because the induced tile group
corresponding to IH2 and IH3 is c1. Then we find the Heesch types of IH2 and IH3
directly and match them (see Figure 5.4). In the same way, we match the rest of the
Heesch types in the red rows with the isohedral types.
T
G
G
TG
G
TG1 G1TG2 G2
G G T
T
G G
TG1 G2 TG2 G1
Figure 5.4. Matching isohedral tilings of IH2 and IH3 with their Heesch types.
6. Escher’s system
Mathematicians’ aim is to logically analyze and classify periodic tilings according to
their symmetry. However, Escher’s aim was to discover various ways to create
interesting periodic tilings. The following describe the way in which Escher drew his
tilings:
1. He started by drawing a plane of congruent polygons. For example, in Figure 6.1
we see Escher’s periodic tiling corresponding to Escher number 9 in Table 1 (in
appendix). For this tiling he started by drawing a plane of congruent
parallelograms. The plane of congruent polygons for a tiling is called the
underlying plane of congruent polygons.
2. Then Escher drew a tile using an underlying polygon as a guide. The polygon will
contain most parts of the tile. The area of the tile will be equal to the area of the
polygon. For example in Figure 6.1 the birds are the tiles of Escher’s tiling 9.
Observe an underlying parallelogram and we will see that it will contain most
parts of either the red bird or the white bird. Thus, a bird tile was drawn using a
parallelogram as the guide. The area of a bird is equal to the area of a
parallelogram.
3. Finally, Escher drew the tiles for all the underlying polygons in a similar way.
Meaning the same parts of the tile are inside a polygon and the same parts of the
tile are outside the polygon for all the underlying polygons. But he drew the tiles
in a different position for particular underlying polygons. This created symmetries
for the tiling. For example in Figure 6.1, Escher filled the underlying
parallelograms with bird tiles. Notice that for each parallelogram the bird tiles are
drawn in the same way. But he drew the bird tiles on certain parallelograms with
their beaks pointing upwards. But in certain parallelograms he drew the birds in
the opposite direction with their beaks pointing downwards. In this way Escher
filled all the underlying parallelograms with bird tiles. He thus created a tiling of
bird with symmetries of translation and rotation of order two. Therefore, the
symmetry group of Escher’s tiling 9 is p2.
If Escher filled an underlying plane of congruent polygons with congruent tiles then
he creates an isohedral tiling. In Figure 6.1 Escher only used congruent bird tiles.
Thus, tiling 9 is an isohedral tiling.
Escher formed various ways to draw isohedral tilings with asymmetric tiles based on
Figure 6.1. Escher’s isohedral tiling 9.
different types of underlying congruent polygons. He drew a set of isohedral tilings with
asymmetric tiles using underlying planes of congruent quadrilaterals which come under
the systems called quadrilateral systems. Escher also drew another set of isohedral tilings
with asymmetric tiles using underlying plane of congruent “triangles” (meaning the
underlying polygons are made up of equilateral triangles). The second set of tilings come
under the systems called “triangle” systems. Note that we do not know whether Escher’s
classification is exhaustive.
6.1 Quadrilateral systems
Figure 6.1. Escher’s isohedral tiling corresponding to Escher number 9 in
Table 1 (in appendix).
In the quadrilateral systems Escher drew isohedral tilings with asymmetric tiles based on
congruent quadrilaterals as underlying polygons. The five types of quadrilaterals that
were considered and the letters (A, B, C, D, E) denoting each type of quadrilateral are
listed below. Note that arbitrary means the length of two adjacent sides and their enclosed
angle may be chosen in any way.
A
B
C
D
E
-
an arbitrary parallelogram
an arbitrary rhombus
a rectangle
a square
an isosceles right angled triangle
For example, for the Escher’s tiling 31 he used an underlying plane of congruent
rectangles (see Figure 6.2). However, as we saw earlier in Figure 6.1 he used an
underlying plane of congruent parallelograms for tiling 9.
Figure 6.2. Escher’s isohedral tiling 31.
6.1.1. Isohedral tilings with a minimum of two colours
Escher wanted his isohedral tilings in the quadrilateral system to have a minimum of two
colours. This means that he wanted to colour his isohedral tilings such that in each tiling
every tile shares a common boundary with surrounding tiles of contrasting colours and
every tile touches other tiles of the same colour at its corners. And only a minimum of
two contrasting colours are needed to satisfy this criteria. For example the Escher tiling
31 in Figure 6.2 is an isohedral tiling in the quadrilateral system with a minimum of two
colours, red and white.
For isohedral tilings in the quadrilateral system with a minimum of two colours Esche r
wanted to draw them with no three- fold rotation, no six-fold rotations and no reflection
symmetry. There are only five symmetry groups out of the 17 symmetry groups of
periodic tilings that have no three-fold rotation, no six-fold rotations and no reflection
symmetry. The five symmetry groups are p1, p2, pg, pgg and p4. For each symmetry
group, Escher drew the possible isohedral tilings with asymmetric tiles corresponding to
each one of the five types of quadrilaterals as the underlying plane of congruent
polygons.
Table 6.1 gives Escher’s overview diagram showing his classification of 24 types of
isohedral tilings in the quadrilateral system with a minimum of two colours. In the
overview diagram Escher took each individual underlying quadrilateral of a tiling to
represent a single tile. Thus, he satisfied the property that the area of each tile in a tiling is
equal to the area of each underlying polygon. To satisfy the other criteria that the tiles are
asymmetric, Escher used a hook symbol to mark all the tiles in each tiling. Therefore,
each tile is a marked underlying quadrilateral. The ten systems denoted by Roman
numerals I to X in the overview diagram represent the five types of symmetry groups in
the following way:
I
II, III
IV, V
VI, VII, VIII
IX, X
-
p1
p2
pg
pgg
p4
For each system, Escher notes the following two things (see Table 6.2):
1. The number and ol cation of two-fold and four-fold centers of rotation on the
boundary of every tile in a tiling.
2. The directions of translations and glide reflections in a tiling. There are two kinds
of directions- transversal direction and diagonal direction.
a) Transversal direction is parallel to an edge of the underlying quadrilateral
of a tiling. Translations and glide reflections in the transversal direction
move each tile to an adjacent tile of contrasting colour.
b) Diagonal direction is parallel to a diagonal of the underlying quadrilateral
of a tiling. Translations and glide reflections in the diagonal direction
move each tile to another tile of the same colour at its corner.
Table 6.1. Escher’s overview diagram of 24 types of isohedral tilings with asymmetric
tiles. The isohedral tilings are based on underlying quadrilaterals and they must have a
minimum of two colours.
Table 6.2. Direction of translation and glide reflection and location of rotation for each
system
Therefore, under tilings with one of the five types of symmetry groups there are two or
three systems. This is because each system in a symmetry group corresponds to the same
type of symmetry group but the positions and the directions of the symmetries are
different. For example, the isohedral tilings under systems I and II in Table 6.1
correspond to the same type of symmetry group, p2. But as seen from Table 6.2, the
isohedral tilings in system II admit translation in one transversal direction while the
isohedral tilings in system III admit translation in both diagonal directions. Similarly,
their differences in the number and location of two- fold centers is also seen in Table 6.2.
The symbol given to the 24 types of isohedral tilings in Table 6.1 with a minimum of two
contrasting colours is in the form IIIA. Escher’s tiling with a notation of this form has the
following properties:
1.
2.
3.
4.
The tiling is isohedral with asymmetric tiles.
The tiling can admit a minimum of two colours.
The tiling does not admit reflection.
The Roman numeral (I to X) refers to the type of symmetry group of the tiling
(see Table 6.2).
5. The superscript (A to E) refers to the type of underlying congruent quadrilaterals
of the tiling.
Escher’s tiling 9 given in Figure 6.1 corresponds to IIIA under the column of Escher
system in Table 1 (see appendix). Therefore, we can conclude that tiling 9 is an isohedral
tiling with asymmetric tiles (bird tiles). The tiling can admit a minimum of two colours
(red and white). The symmetry group of the tiling is p2. The tiling was drawn based on
an underlying plane of congruent parallelograms.
However, we come across Escher’s tiling with notation of the form IXD* (see Table 1 in
appendix under Escher’s tiling number 12). Meaning there is an additional superscript *
added to the notation. Escher’s tiling with notation of this form has properties one to five.
However, the tiling will have the following additional property: All the tiles in the tiling
admit reflection.
6.1.2 Relation between Escher’s systems, Heesch types and isohedral types
Escher’s 24 types of isohedral tilings (see Table 6.1) with notation of the form IIIA
(without *) have a relation with the Heesch types. This is because both the Escher’s
systems and the Heesch types classify isohedral tilings consisting of asymmetric tiles.
And both classify isohedral tilings with no reflection.
Note that the topological type of the underlying congruent quadrilaterals of an isohedral
tiling of one of the 24 types is equivalent to the topological type of that isohedral tiling.
This is because Escher drew each tile using the underlying quadrilateral as the guide. A
plane of congruent quadrilaterals of types A to D have topological type [44 ]. Therefore,
the isohedral tilings with underlying quadrilaterals of types A to D have topological type
[44]. A plane of congruent isosceles right angled triangles (E) has topological type [4.82 ].
Therefore, the isohedral tilings with underlying isosceles right triangles (E) have
topological type [4.82 ].
In Escher’s Table 6.1, under symmetry group p1 all the isohedral tilings IA, IB, IC, ID
have topological type [44 ]. In Heesch’s Table 5.1, under symmetry group p1 and
topological type [4 4 ] the Heesch type of the asymmetric tiles in the isohedral tilings is
TTTT. Observe that in all the isohedral tilings IA, IB, IC and ID the Heesch type of the
asymmetric tiles which are the marked quadrilaterals (in Table 6.1) is TTTT. Therefore,
IA, IB, IC and ID correspond to TTTT. However, in Escher’s Table 6.1, under symmetry
group p2 there are two systems (II and III) of isohedral tilings with topological type [44 ].
They are IIA, II B, IIC, IID and IIIA, IIIB, IIIC, IIID. In Heesch’s Table 5.1, under symmetry
group p2 and topological type [44] there are two possible Heesch types for the
asymmetric tiles in the isohedral tilings: CCCC and TCTC. To match the Heesch type
with the corresponding system (I or II) of isohedral tilings in the Escher’s table with
symmetry group p2, we observe the asymmetric tiles of the isohedral tilings in system II
and III. Upon observation we find that system II corresponds to TCTC and system III
corresponds to CCCC. In the same way, we find the corresponding Heesch type for each
system I to X.
Table 6.3 gives all the 24 Escher’s isohedral tilings in Table 6.1 and their corresponding
Heesch types from Table 5.1. In Section 5.2 we discussed the relation between the
Heesch types and the isohedral types. And in Table 5.2 we see each Heesch type and its
corresponding isohedral type. Therefore, in Table 6.3 we insert the isohedral types for
each Heesch type and obtain the last column. Thus, we can also see from Table 6.3 the
corresponding isohedral type for each of the 24 of Escher’s tilings.
Escher’s system
IA, IB, IC, ID
IIA, IIB, IIC, IID
IIIA, IIIB, IIIC, IIID
IVB, IVD
VC, VD
VIB, VID
VIIC, VIID
VIIIC, VIIID
IXD
XE
Topological
type
[44 ]
Symmetry
group
p1
p2
pg
pgg
p4
2
[4.8 ]
System
(I to X)
I
II
III
IV
V
VI
VII
VIII
IX
X
Heesch
type
TTTT
TCTC
CCCC
G1G1 G2G2
TGTG
CCGG
CGCG
G1G2 G1G2
C 4C4 C4C4
CC4C 4
Isohedral
type
IH41
IH47
IH46
IH44
IH43
IH53
IH51
IH52
IH55
IH79
Table 6.3. Escher’s system and its corresponding Heesch type and isohedral type.
We will not study notations with hyphenation, such as
VC-IVB (see Table 1 in
appendix). And we also will not study notations of the form VC var (see Table 1 in
appendix). This is because these notations are derived from the 24 isohedral tilings in
Table 6.1 by tedious methods. Readers can refer to Schattschneider [4] for details on
these notations.
6.2 Triangle systems
Similar to the quadrilateral systems, the triangle systems also denote isohedral tilings
consisting of asymmetric tiles. And the isohedral tilings have no reflection. However, in
the “triangle” systems the tilings are drawn from underlying congruent equilateral
triangles. But in the quadrilateral systems the tilings are drawn from underlying
congruent quadrilaterals. Also, unlike the quadrilateral systems, in the triangle systems
isohedral tilings admit only rotations of order six, three and two. They do not admit
translation, reflection and glide reflection. The isohedral tilings in the triangle systems are
split into the following two systems:
1. A – Isohedral tilings that only admit rotations of order three.
2. B – Isohedral tilings that admit rotations of order two, three and six.
Each sequence of notation for the tilings in the triangle systems goes as follows: Number
of type of tiles (I or II), system (A or B), number of colours (2, 3, or 4), type (C,1, 2, 3,
4). Type C denotes that in the tiling the centers of rotations occur at the center of the tiles
of the tiling. The other types from 1 to 4 are not important. Also the tilings are coloured
in a way that no two adjacent tiles have the same colour. Similar to the quadrilateral
system, if the superscript * is added to the notation it means that the tiles in the tiling
admit reflection symmetry.
The Escher’s tiling corresponding to Escher number 55 has notation Tr I B3 type. Thus,
the tiling 55 has the following properties:
1. Tr means it belongs to the triangle system. Thus, it is an isohedral tiling consisting
of asymmetric tiles. The tiling only admits rotations.
2. Symbol I means the tiling consists of one type of tiles.
3. B3 means the tiling admits rotations or order two, three and six. The subscript
three denotes that the tiling consist of three colours.
Appendix:
Table 1. Symmetry classifications of Escher’s Periodic Drawings
References:
1. George E.Martin: Transformation geometry: an introduction to symmetry,
Springer-Verlag, New York, 1982.
2. B.Gru/nbaum and G.C.Shephard: Tilings and Patterns, W.H.Freeman and
Company, New York, 1987.
3. D.Schattschneider: The Plane Symmetry Groups: their Recognition and Notation,
American Mathematical Monthly, Volume 85, Issue 6 (Jun.-Jul., 1978), 439-450.
4. D.Schattschneider: Visions of symmetry: notebooks, periodic drawings, and
related work of M.C.Escher, W.H.Freeman and Company, New York, 1990.
5. C.H.MacGillvary: Symmetry aspects of M.C.Escher’s periodic drawings,
Oosthoek, Utrecht, 1965.
6. D.W.Crowe and D.K.Washburn: Symmetries of culture: theory and practice of
plane pattern analysis, University of Washington Press, Seattle, 1988.