Mathematical Tools for Computer

```Mathematical Tools for Computer-Generated
Ornamental Patterns
Victor Ostromoukhov
Ecole Polytechnique F&eacute;d&eacute;rale, Lausanne, Switzerland
ornamental patterns, with a particular attention payed to Islamic patterns. The
article shows how, starting from a photo or a sketch of an ornamental figure, the
designer analyzes its structure and produces the analytical representation of the
pattern. This analytical representation in turn is used to produce a drawing which
is integrated into a computer-generated virtual scene. The mathematical tools for
analysis of ornamental patterns consist of a subset of tools usually used in the
mathematical theory of tilings such as planar symmetry groups and Cayley
diagrams. A simple and intuitive step-by-step guide is provided.
1 Introduction
It’s very common to see around us all kinds of ornamentations in form of repetitive
patterns: floor tilings, wallpaper designs, ornamental brickwork or yet more patterns
on our clothing. It was probably the Arabs, and more particularly the Moors, who
developed the most acute sensitivity towards ornamental designs. The historical monuments left by the Moors are covered with intricate arabesques which are very often
composed of geometrical patterns, floral motifs and stylish scripts [Abas&amp;Salman
1995], [Hargittai 1986], [Hargittai 1989].
Translational
Unit
Computer-Generated Drawing
Further
Transformations
Analysis
➤
Cayley
Diagram
Generation
Analytical Represenation
Photo or Sketch
➤
➤
Fundamental
Region
With the advent of sophisticated computer graphics tools able to generate very complex virtual scenes, the need for computer-aided ornamental design becomes greater
and greater. An appropriate integration of ornamental patterns into synthetic images
means that these patterns are adequately represented and may be freely generated by
means of a set of simple primitives. Figure 1 presents a typical workflow for producing
computer-generated Islamic patterns. The first stage of the workflow enables the informal graphical material to be analyzed (hand-maid drawings, photos, sketches etc.) and
an appropriate analytical representation to be built. Based on this analytical representation, a set of drawing procedures (primitives) is used to produce the infinite analytical
drawing, which may go through further transformations before being integrated into a
computer-generated virtual scene.
A large number of articles and books has been devoted to the analysis of ornamental
patterns and tilings [Grunbaum&amp;Shephard 1987], [Washburn&amp;Crowe 1988]. In a
recent book [Abas&amp;Salman 1995], a relatively simple and clear method for the analysis of Islamic geometrical patterns was presented. This method is based on the classification according to planar symmetry groups, very popular in the western scientific
milieu. Another recent article [Grunbaum&amp;Shephard 1993] explores the applicability
of Cayley diagrams for analysis of the interlaced patterns in Islamic and Moorish art.
This article limits the analysis to two particular symmetry groups: p4mm and p6mm,
by far the most common in Moorish design.
Although the group-theoretical approach is not unique, and has been criticized several
times [Grunbaum 1986], it still represents a very useful analysis tool. In addition, such
an instrument provides a good basis for computer-generated imagery: it defines the set
of objects and the set of actions on these objects needed for manipulating arbitrary planar ornamental patterns.
This article presents mathematical tools needed for the analysis as well as for the generation of two-dimensional ornamental patterns. In section 2 the basis of classification
according to symmetry groups is presented. In section 3, we extent the analysis to Cayley diagrams. In section 4 we present the strand analysis applied to all 17 planar symmetry groups. Finally, section 5 contains a simple and intuitive step-by-step guide
intended for persons who would like to incorporate an ornamental pattern into a computer-generated image.
Although the explanation is provided for Islamic designs from well-known sources
[Bourgoin 1879], it can easily be extended to other fields of application, such as floral
ornamentation, abstract geometrical patterns and many others.
2 Symmetry Groups of two-dimensional patterns
In Electronic Publishing, Artistic Imaging and Digital Typography , Lecture Notes in Computer Science 1375, Springer Verlag, pp. 193-223, 1998.
Numbers measure size, groups measure symmetry...
This enigmatic sentence opens the treatise on the subject [Armstrong 1988]. What is
symmetry? Hundreds and thousands of articles and books are devoted to this subject
with somehow fuzzy boundaries, trying to delimit the border between mathematics and
art, between the exact scientific quantification and intuitive qualification, between the
notions of measure and order.
Webster gives us the following definition for Symmetry:
193
194
Fig. 1. A typical workflow for producing computer-generated Islamic patterns. Further
transformations may include: interlacing, coloration, integration in a virtual scene,
illumination, texture mapping etc.
.
1: balanced proportions; also: beauty of form arising from balanced proportions
2: the property of being symmetrical; especially: correspondence in size, shape, and relative
position of parts on opposite sides of a dividing line or median plane or about a center or
axis -- compare BILATERAL SYMMETRY, RADIAL SYMMETRY
3: a rigid motion of a geometric figure that determines a one-to-one mapping onto itself
4: the property of remaining invariant under certain changes (as of orientation in space, of
the sign of the electric charge, of parity, or of the direction of time flow) -- used of physical phenomena and of equations describing them
All this is too vague to be used in algorithms. Instead, we shall use the mathematical
notions, which are much more precise.
The symmetry operations for the object, also referred to as isometries, are the rigid
motions which leave the distances between different parts of the object unchanged. It
can be shown that different symmetries of a figure or a pattern nicely match the mathematical notion of group [Armstrong 1988]. A group is a set G together with a multiplication on G which satisfies three axioms:
notation in the last edition of the International Tables for Crystallography [Hahn 1995]
(which is not identical to the notation of another widely-cited source: International
Tables for X-Ray Crystallography [Henry&amp;Lonsdale 1952]). An interesting historical
review of different systems of notation can be found in [Shattschneider 1978].
Every periodic pattern may be associated to a lattice of points. The points of this lattice
are inter-related by two translational vectors, V 1 and V 2, which leave the whole lattice
unchanged under the operation of translation by a linear combination nV 1+mV 2,
where n and m are integers. Consequently, any point in the repetitive pattern will
remain unchanged under the same translation. This translational lattice is one of the
basic characteristics of every symmetry group, and the smallest area of the repetitive
pattern which remains unchanged under the translation by two translational vectors, V 1
and V 2 is called lattice unit or translational unit. Fig. 2 shows five principal types of
lattice units.
Parallelogramm
(a) the multiplication is associative, that is (xy)z = x(yz) for any three elements of G,
Rectangular
Rhombic
Square
Hexagonal
(b) there is an identity element e, such that xe = x = ex for every x in G,
(c) each element x of G has an inverse x-1 which belongs to the set G and satisfies x-1x = e = xx-1.
Many useful properties derived from the abstract group theory can be applied to artistic
creations from different centuries. This analysis tool is very useful for understanding
the relationship between different parts of repetitive patterns. As we shall see later, it is
especially beneficial when manipulating the drawing by computers.
The idea of the possibility of classification or ornamental patterns by means of symmetry groups was first suggested by Polya and further promoted by Speiser and Weyl
[Grunbaum 1984], [Weyl 1952]. Satisfactory for many tasks of analysis, the group-theoretical approach shows its limitations when the authors tries to apply this analysis to
all artistic production (see the discussion in [Grunbaum 1984]). In fact, the ancient artists and craftsmen did not know any group theory, and this “lack” did not diminish
their creativity.
For the purpose of computer-generated ornaments, the symmetry group analysis is a
precious tool. As we shall see later, each symmetry group has its proper elementary
object - fundamental region - and a set of elementary actions of this object which are
needed in order to produce the whole pattern. It is precisely what is generally required
for an algorithmic representation of visual objects: the set of elementary objects and
the set of actions on them.
2.1 Classification of two-dimensional patterns
There are five basic transformations which form the basis of any symmetry group
[Armstrong 1988]: identity transformation (or do-nothing), translation, mirror-reflection, glide-reflection and rotation.
There are seventeen planar symmetry groups [Shubnikov&amp;Koptsik 1974], [Grunbaum&amp;Shephard 1987], often referred to as two-dimensional crystallographic groups.
Unfortunately, there is no unanimity with respect to their notation. We adopt here the
195
V2
V2
V2
V1
V1
V2
V1
V2
V1
V1
Fig. 2. Five types of lattices used for classification of two-dimensional symmetry groups.
Notice the alternative representation by the “centered cell” containing two translational
units, in the case of the rhombic lattice.
Intuitively, one can find the lattice unit by looking for the “center” of any arbitrary
“reference point” of each of the figures which form the repetitive pattern.
Different sources propose different more-or-less complex methods for identifying the
symmetry group to which a particular pattern belongs. One of the most compact and
comprehensive ones has been proposed in [Abas&amp;Salman 1995, p. 108]. This method
can be resumed as a set of questions about the pattern (as graphically shown in Fig. 3):
1 - Is there rotational symmetry about some point and, if so, what is the smallest angle
through which the pattern coincides with itself?
2 - Are there any mirror reflection lines?
3 - Are there mirror reflection lines in more than one direction?
4 - Are there any glide reflection lines? Do the glide reflection lines coincide with mirror reflection lines? Do centers of rotation lie on mirror reflection lines?
Although it looks relatively easy and straightforward, this mode of identification
requires acute observation and some skills, as recognized by its author.
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6-fold symmetry?
NO
YES
Mirror reflection?
YES
p6mm
4-fold symmetry?
NO
NO
YES
p6
Mirror reflection?
YES
NO
Mirror lines in 4 directions? p4
YES
p4mm
NO
3-fold symmetry?
NO
YES
2-fold symmetry?
Mirror reflection?
YES
NO
YES
p4gm
NO
p3
Centers of 3-fold
Mirror reflection?
rotation only on
NO
YES
mirror lines?
A second mirror? Glide reflection?
NO
YES
p2mg
Rhombic lattice?
YES
NO
c2mm
p2mm
cm
p2
c2gg
NO
YES
Rhombic lattice?
NO YES
YES
NO
p31m YES
p3m1
Mirror reflection?
NO
pm
Glide reflection?
YES
pg
NO
p1
Fig. 3. Planar symmetry group selector.
3 Cayley diagrams and their practical usage
Cayley diagrams are one of the most important graphical representations of groups. A
Cayley diagram is a graph which shows the consecutive states of the identity element
(mandatory element of the group) under a sequence of transformations (multiplications, in group-theoretical jargon). The vertices of the diagram are the consecutive
states, and the edges of the diagram are the transformations. Cayley diagrams of the
finite groups are finite, and those of infinite groups are infinite. For example, the Cayley diagram of the dihedral symmetry group D3 which describes the symmetry of a
finite three-fold point-symmetrical figure with three mirror-reflection axes (symmetry
group of the isosceles triangle) is presented in Fig. 4.
Fig. 4. The Cayley diagram of the
symmetry group of
the isosceles triangle.
Dihedral group D3
{e, r, r2, m, mr, mr2}
m
e
r: clockwise rotation
m: mirror-reflection
r2
mr
r
mr
2
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3.1 Cayley diagrams of symmetry groups of two-dimensional wallpaper patterns
Cayley diagrams of symmetry groups of two-dimensional wallpaper patterns are more
complicated. They represent an infinite graph which follows the structure of the pattern. The unit element e in these graphs is always the fundamental region of the symmetry group. Fig. A1(b)-A17(b) show the Cayley diagrams of each of seventeen
symmetry groups of two-dimensional wallpaper patterns. In these Cayley diagrams,
only the transformations which relate the adjacent fundamental regions are presented.
Please note that in most cases, this representation is not the only one possible. Nevertheless, the diagrams shown in Fig. A1(b)-A17(b) show the minimal set of transformations needed to cover the whole plane.
As illustration of the usage of Fig. A1(b)-A17(b), let us consider the pattern of the
symmetry group p2mg as shown in Fig. A7. The fundamental region of this group is a
rectangular region shaded in dark gray in Fig. A7(a) and A7(c). The fundamental
region with an asymmetrical figure inside and the sides marked by lowercase letters
a,b,c and d is shown in Fig. A7(d). Fig. A7(e) shows all possible relations between
adjacent fundamental regions. The interrelations through the sides b and d of the fundamental region are translations; the interrelations through the sides a and a are twofold rotations; the interrelations through the sides c and c are mirror-reflection. Shown
in Fig. A7(b) is the resulting Cayley diagram. Each fundamental region is associated
with the node of the diagram, and all three types of relations between adjacent fundamental regions are shown using three different types of edges. This figure shows the
relationship between the geometric structure of the pattern, namely its subdivision into
fundamental regions, and the corresponding Cayley diagram.
Imagine that we have a procedure for drawing the content of the fundamental region all graphical objects inside it. Now, in order to fill the whole plane with this fundamental region, using the selected symmetry group, one has to walk through the Cayley diagram following the edges and, each time the vertex is encountered, put another copy of
the fundamental region, in appropriate form (translated, mirror-reflected, glidereflected or rotated).
For further reading about Cayley diagrams you may refer to the books related to
groups and symmetry [Armstrong 1988], [Budden 1972], [Grossman&amp;Magnus 1964],
[Farmer 1992].
4 Strand analysis
An interesting problem of strand analysis may occur when manipulating the strandbased patterns, very frequent in Muslim art. The revealing article by Grunbaum and
Shephard [Grunbaum&amp;Shephard1993] describes an original method of analysis using
Cayley diagrams. Many important properties of the pattern can be derived from a simple analysis of its fundamental region. The article analyses only two symmetry groups:
p4mm and p6mm. Let us recall some of the propositions of [Grunbaum&amp;Shephard
1993]:
• the pattern has n different strands, where n is the number of different tracks in the
fundamental region (see the case study below)
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• if the group element path in the Cayley diagram of one particular track in the fundamental region is finite, the corresponding strand in the pattern is finite (a
loop). Inversely, infinite sequences correspond to infinite strands in the pattern
(see the case study below)
• the strands in the pattern have the same induced symmetry group as the corresponding group element paths in the Cayley diagram
We shall show here how a very similar analysis can be applied to all seventeen symmetry groups of the wallpaper patterns.
4.1 Case study: the symmetry group p6mm
We shall first consider a relatively simple case: a pattern which belongs to the symmetry group p6mm. For this case study, we shall consider the mosaic tilework from the
courtyard of the Attarine Medeza, Fez. The outline representation of this pattern is
shown in Fig. 5(a). Applying the classification presented in section 2.1, we can reliably
conclude that this pattern belongs to the symmetry group p6mm. Its fundamental
region is shown in Fig. 5(c). The Cayley diagram and interrelations between fundamental regions are presented in Fig. 5(b) (copied from Fig. A17) and Fig. 5(d), respectively. We can state that the fundamental region is related to its neighbors only by
mirror-reflections. For this particular case, a simple strand analysis rule may be
applied. The key idea of this method is that we first analyze the behavior of the strands
in the fundamental region only, by applying simple fundamental region boundary traversal rules. Then, we transpose the sequence of fundamental region boundary traversals observed in the fundamental region to the Cayley diagram, where we draw the track
of the specific strand, Finally, we apply the rules of correspondence between the Cayley diagram and the pattern itself which were mentioned in the previous section
(cycling, number of strands etc.). All phases of this analysis may be meant to be taking
place simultaneously: when the strand walks inside the fundamental region, we stay on
the vertex of the Cayley diagram; at the same time we walk inside a small region – replica of the fundamental region – in the pattern itself. When we traverse the boundary of
the fundamental region, we walk along an edge of the Cayley diagram, and at the same
time we go to another small region in the pattern, adjacent to the previous one.
Let us apply these principles:
• we follow one particular strand in the fundamental region, e.g. the strand
ABCDEF, in Fig. 5(a);
• when the strand encounters the side of the fundamental region, it bounces, like at
points B and D, Bouncing is one particular case of the fundamental region
boundary traversal rules, applicable in the case of mirror-reflection from the
boundary;
• when we arrive at the end of the strand, we turn back and continue the track in the
opposite direction: ABCDEFFEDCBABCDEFFEDCBA...
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(a)
b
b
c
c
a
a
c
c
b
b
aa
aa
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b
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(b)
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(d)
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B
a
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c
(c) Fundamental J
Region
c
a
M
a
L
b
KI
N
A
H
O
E
G
Q
P
C
D
Fig. 5. (a) Schematic representation of the mosaic tilework from the courtyard of the
Attarine Medeza, Fez, (b) its Cayley diagram, (c) its fundamental region and (d) three
possible interrelations between fundamental regions.
200
R
• we mark the track on the Cayley diagram: when the strand walks inside the fundamental region, we stay at the vertex of the Cayley diagram; when the strand
touches the side of the fundamental region, we move through the corresponding
edge of the Cayley diagram. For example, the strand ABCDEFFEDCBA... will
generate the following sequence on the Cayley diagram:
bb,V,cc,V,aa,V,cc,V,aa,V,cc,V,bb...
where V stands for vertex, aa - walking along the edge marked as aa etc.
This particular strand in the fundamental region and the corresponding track in the
Cayley diagram are shown as a heavy dashed line in Fig. 5(c) and in Fig. 5(b). The corresponding strand in the pattern itself has been equally shown as a bold dashed line in
Fig. 5(a).
The track in the Cayley diagram is an infinite periodic figure, consequently the corresponding strand in the pattern is infinite and periodic. One may draw 12 distinct tracks
in the Cayley diagram which all correspond to the same walk in the fundamental
region, therefore there are 12 distinct strands in the pattern, all related by the rules of
the induced symmetry group. Refer to Fig. A17(a) to see all mirror reflection axes and
all centers of rotations.
Finally, a very similar analysis may be applied to the strand GHIJKLMNOPQRG...
This strand is a cycle inside the fundamental region; nevertheless, it is an open infinite
periodic track in the Cayley diagram and, consequently, an infinite periodic strand in
the pattern. This second strand, as well as the corresponding track in the Cayley diagram, is shown as a heavy solid line in Fig. 5(a), (b) and (c). Similarly, there are 12 distinct strands of this type, all related by the rules of the induced symmetry group.
According to the proposition in [Grunbaum&amp;Shephard 1993], the pattern in Fig. 5(a)
is composed uniquely of these two families of 12 strands each.
In can easily be noticed that the same method works in case of symmetry group p4mm
(as explained in [Grunbaum&amp;Shephard 1993]), as well as for symmetry groups p2mm
and p3m1. In all four cases the fundamental region is related to the adjacent fundamental regions uniquely by mirror-reflections (we can refer to these four cases as mirrorreflections-only symmetries).
In the fourteen symmetry groups other than the mentioned p6mm, p4mm, p2mm and
p3m1, the edges of Cayley diagrams may be labeled by the combination of two different letters such as ac or bd. Accordingly, the bouncing operation from the sides of the
fundamental region in the case of mirror-reflections-only symmetries can be replaced
by the operation of “fundamental region sides traversal”. The correspondence between
the operation of “fundamental region sides traversal” operated on an isolated fundamental region can be related to the corresponding operation of walking along the corresponding edge in the Cayley diagram. It is important to underline that every side of
every fundamental region meets the other sides of the adjacent fundamental regions in
a unique combination. Therefore, when side x of a fundamental region is traversed and
we reach the adjacent fundamental region through its side y, this operation corresponds
to walking along the edge xy in the Cayley diagram, which can clearly be identified.
The only complication with respect to the case of mirror-reflections-only symmetries
resides in the less-intuitive continuity condition of the strands with respect to the operation of “fundamental region sides traversal” (bouncing in the case of mirror-reflections-only symmetries). Fig. 6 summarizes the behavior of different strands during the
fundamental region sides traversal for six different types of interrelations between fundamental units which may occur in the planar symmetry groups.
c
201
b
❶❸❷
❶ ❸❷
d
b
c
c
a
c
bd
❶ ❷❸
❶ ❷❸
d
b
❷
❸
❶
b
d
❸
❷
❶
a
c
a
a
c
b
a
d ❶❸ ❷
c
❶
dd
4.2 Extension of the strand analysis to all 17 planar symmetry
groups
Before starting to extend the method explained in the previous section to the rest of
symmetry groups of the wallpaper patterns, let us once more examine Fig. 5(b) and (c).
It can be observed that the bouncing of the strand from the side a of the fundamental
region in Fig. 5(c), as explained previously, corresponds to the walk in the Cayley diagram along the edge marked as aa. The same is valid for bouncing from the sides b and
c (correspondingly walks along the edges bb and cc). Edges of Cayley diagrams are
marked by double-letters since they cross two boundaries: we join the “centers” of
each fundamental region to all adjacent fundamental regions, as it can be clearly visible in Figs. A1(b) to A17(b).
d
b
d
a
c
a
b
b
❷
❶
c
❷
❸
❶
a
d
❶
❷
❶ ❷❸
❸
❷
❶
d
a b
a
d
c
❶a
a
b
c
b
c
Translation
between
fundamental
regions
Glide-reflection
between
fundamental
regions
Mirror-reflection
between
fundamental
regions
2-fold rotation
between
fundamental
regions
3-fold rotation
between
fundamental
regions
4-fold rotation
between
fundamental
regions
Fig. 6. The strand continuity condition during the fundamental region sides traversal for
six different types of interrelations between fundamental units which may occur in the
planar symmetry groups.
4.3 Case study: the symmetry group c2mm
Let us consider another case study: the Islamic pattern taken from [Bourgoin 1879,
plate 97]. Its outline representation is shown in Fig. 7(a). This belongs to the symmetry
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group c2mm, according to the classification presented in section 2.1 Its fundamental
region is shown in Fig. 7(c), and its Cayley diagram – in Fig. 7(b). Fig. 7(d) illustrates
interrelations between fundamental regions. The fundamental region is related to its
neighbors by two types of relations: by mirror-reflections about the sides a, b and d,
and by two-fold rotation about the center of the side c. Consequently, the fundamental
region boundary traversal rule by bouncing presented in section 4.1 should be replaced
by a more elaborate rule chosen from the six rules presented in the previous section.
(a)
Fundamental
Region
O c A E P
N B
(c)
M
R
Q F
H
L
S
X
d
W
G
b
C
Y
U
T
K
I
a
D
V
J
a
center of two-fold
rotation between two
b fundamental regions
❷
d
❶ ❶
❸
(b)
c
c
c
c
c
c
bb
c
c
bb
a
a
a
a
dd
c
c
bb
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dd
a
a
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c
(d)
a
c
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d
b
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a
a
dd
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a
❷
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d
a
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dd
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c
bb
a
a
a
a
❸
❶ ❶
bb
dd
bb
c
c
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c
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dd
a
a
dd
c
c
c
c
dd
a
a
dd
bb
c
c
bb
dd
a
a
c
c
c
c
bb
dd
a
a
c
c
b
d
2-fold
rotation
between
fundamental
regions
a
Fig. 7. (a) Islamic pattern from [Bourgoin 1879, plate 97] (b) its Cayley diagram, (c) its
fundamental region and (d) interrelations between two fundamental regions through the
side c (two-fold rotation about the center of the side); the above figure shows the strand
labeling on this boundary.
203
5 Step-by-step Guide
Let us summarize the material presented in the previous sections in the form of a stepby-step guide, intended for a person who would like to incorporate an ornamental pattern into a computer-generated image.
c
c
c
Let us develop the strand analysis for this particular case:
• Let us follow the strand ABCD in the fundamental region. When the strand
ABCD encounters the side a of the fundamental region, it bounces and turns
back, like at point D.
• When the strand ABCD encounters the side c of the fundamental region, it simply
turns back. It is due to the fact that the center of the two-fold rotation is precisely at point A, the center of the side c, as shown in Fig. 5(d).
• The strand AEFGHIJKLMNO in Fig. 5(c) bounces from the sides b (point G), a
(point J) and d (point M), before arriving at point O on the side c, where the
fundamental region boundary traversal rule of type “two-fold rotation about the
center of the side c” should be applied. At point O, the strand traverses the
boundary between fundamental regions and continues at point P; the strand
marked as ❶ in Fig. 5(d). Then, the strand PQRSTUV bounces from the sides d
before arriving at point V (corner), where it bounces from both sides b and a.
The strand returns back and continues in the opposite direction: VUTSRQPONMLKJIHGFEA...
• Finally, the strand WXY bounces from sides b and d.
• These three different strands ABCD, AEFGHIJKLMNOPQRSTUV and WXY in
the fundamental region are shown as a dashed bold line, a bold line and a dotted
bold line. Their tracks in the Cayley diagram are marked by the same styles, as
well as the corresponding strands in the pattern itself.
• From our analysis, we can conclude that our pattern contains three different types
of strands shown in Fig. 5(a).
Given: a sketch of a photo of a planar ornamental pattern.
Find: all information needed to integrate this pattern into a computer system.
Phase I: Analysis
Step 1: Find the translational unit. Look for the “center” of any arbitrary “reference point” of each figure which forms the repetitive pattern. Check that
this unit is minimal. It’s common to select the double of the translational
unit, which may introduce an erroneous analysis.
Step 2: Define to which symmetry group among the seventeen available twodimensional crystallographic groups your pattern belongs. Use the questionnaire presented below.
Step 3: Identify the fundamental region and the Cayley diagram associated with
this symmetry group, using Figures A1- A17. It may be useful to redraw
separately the fundamental region of your particular pattern, marking all
details, as shown in Fig. 5.
204
Step 4: In certain cases, patterns may contain significant continuity between the
fundamental regions, e.g. strand-based Islamic patterns. In these cases, the
strand analysis may be applied. Refer to Figs. A1(e) - A17(e) for the
schemes of interrelations between fundamental regions and to Fig. 6 for
the strand continuity condition. Follow the analysis presented in section 4
Strand analysis.
Generation
Step 5: Implement the primitive DrawFR() which puts inside the fundamental
region all graphical information in a conventional format that may subsequently be transformed by the primitives in step 6.
scenes. For reasons of space, the subject of frieze symmetries has been deliberately left
We used Islamic patterns to illustrate the presented analysis concepts and techniques.
Nevertheless, the same analysis, or a very similar one, may be used in other fields of
application, such as floral ornamentation, abstract geometrical patterns and many others.
Phase II:
Step 6: According to the symmetry group of your pattern, implement the needed
primitives among the six possible:
References
1. [Abas&amp;Salman 1995] S.J. Abas &amp; A.S. Salman, Symmetries of Islamic Geometrical Patterns,
World Scientific, 1995
2. [Armstrong 1988] M.A. Armstrong, Groups and Symmetry, Springer Verlag, 1988.
3. [Bourgoin 1879] J. Bourgoin, Elements de l'art arabe, Fermin-Didot, Paris, 1879. Reprint
available: Arabic Geometrical Pattern and Design, Dover, 1974.
TranslateFR()
MirrorReflectFR()
4. [Budden 1972] F.J. Budden, The Fascination of Groups, Cambridge University Press, 1972.
GlideReflectFR()
5. [Emmer 1993] M. Emmer (ed.), The Visual mind: art and mathematics, MIT Press, 1993.
Rotate2FoldFR()
6. [Farmer 1996] D.W. Farmer, Groups and symmetry: a guide to discovering mathematics,
Providence, AMS, 1996.
Rotate3FoldFR()
Rotate4FoldFR()
Step 7: Following the Cayley diagram of the symmetry group of your pattern (refer
to Figs. A1(b) - A17(b)), implement the cycle which passes by all the
nodes of the diagram and which applies the needed primitives of the step 6
on the edges of the diagram. This operation fills the whole plane with your
pattern (the parameters of the cycle delimit the spread).
Phase III: Further transformation
Steps 8++: If needed, further transformations may be applied: interlacing, coloration, illumination, texture mapping etc.
Attention: this analysis does not include seven one-dimensional symmetry groups
(also known as frieze groups) or 230 three-dimensional symmetry groups (usually
referred to as three-dimensional crystallographic groups).
7. [Grossman&amp;Magnus 1964] I. Grossman &amp; W. Magnus, Groups and their Graphs, The Mathematical Association of America, 1964.
8. [Grunbaum 1984] B. Gr&uuml;nbaum, The Emperor's New Clothes: Full Regalia, G string, or
Nothing, The Mathematical Intelligencer, Vol. 6, No. 4, pp. 47-53, 1984.
9. [Grunbaum&amp;Shephard 1987] B. Gr&uuml;nbaum, G. C. Shephard, Tilings and Patterns, W. H.
Freeman and company, New York, 1987.
10. [Grunbaum&amp;Shephard 1993] B. Gr&uuml;nbaum, G. C. Shephard, Interlaced Patterns in Islamic
and Moorish Art, in [Emmer 1993], pp. 147-155.
11. [Hahn 1995] T. Hahn (ed.), International Tables for Crystallography, Fourth Edition, Vol. A,
Reidl Publishing Co., 1995.
12. [Hargittai 1986] I. Hargittai (ed.), Symmetry, Pergamon Press, 1986.
13. [Hargittai 1989] I. Hargittai (ed.), Symmetry 2, Pergamon Press, 1989.
6 Conclusions
This article summarizes different techniques for analyzing the symmetry of ornamental patterns dispersed through the vast literature in the fields of crystallography, chemistry, mathematics and history of art. Certain aspects of this analysis, such as the
classification according to planar symmetry groups, are relatively well-known and
largely used. Other analysis tools, such as Cayley diagrams and representation by fundamental regions, are less known, and deserve a broader diffusion. We re-explain certain basic techniques introduced in [Grunbaum&amp;Shephard 1993] and [Abas&amp;Salman
1995]. The original contribution of this article resides in the extension of the strand
analysis using Cayley diagrams to all 17 planar symmetry groups.
We provide a simple and intuitive step-by-step guide intended for computer graphics
persons who would like to incorporate ornamental patterns into artificial images and
205
14. [Henry&amp;Lonsdale 1952] N.F.M. Henry &amp; K. Lonsdale (eds.), International Tables for X-Ray
Crystallography, Vol. 1, Kynock Press, 1952.
15. [Shubnikov&amp;Koptsik 1974] A. V. Shubnikov, V. A Koptsik, Symmetry in science and art, Plenum Press, New York, 1974.
16. [Schattschneider 1978] D. Schattschneider, The Plane Symmetry Groups: Their Recognition
and Notation, Amer. Math. Monthly, Vol. 85, pp.439 - 450, 1978.
17. [Washburn &amp;Crowe 1988] D.K. Washburn &amp; D.W. Crowe, Symmetries of culture: theory and
practice of plane pattern, Donald W. Crow, Seattle: University of Washington Press, 1988.
18. [Weyl 1952] H. Weyl, Symmetry, Princeton University Press, 1952.
206
Appendix A
FIG. A2(a) Symmetry group p2
FIG. A1(a) Symmetry group p1
Center of 2-fold
rotation
(b) Cayley diagram
(b) Cayley diagram
bd
bd
a
c
bb
bd
a
c
a
c
a
c
bd
bd
Translation
between
fundamental regions
a
c
c
a
dd
bd
a
c
bb
dd
a
c
a
c
a
c
c
a
Translation
between
fundamental regions
a
c
c
a
bb
dd
bb
dd
a
c
c
a
bb
c
a
a
c
bb
dd
2-fold rotation
between
fundamental regions
bb
dd
a
c
c
a
c
a
c
bd
bd
bd
a
c
bb
dd
bb
dd
b
d
b
d
a
c
bb
dd
c
a
c
a
c
a
c
c
a
a
c
c
a
c
a
a
c
a
c
bd
bd
bd
dd
b
d
bb
bb
dd
a
a
c
c
c
b
d
a
c
bd
d
b
a
(c) Translational unit
(d) Fundamental region
207
a
fundamental regions
c
a
c
b
d
dd
b
a
b
d
bb
dd
b
d
bb
d
a
(c) Translational unit
(d) Fundamental
region
c
a
a
c
(e) Relationship between adjacent fundamental regions
208
FIG. A4(a) Symmetry group pg
FIG. A3(a) Symmetry group pm
Mirror-reflection axis
Glide-reflection axis
(b) Cayley diagram
bd
(b) Cayley diagram
bd
bd
a
a
bd
a
a
c
c
c
c
a
a
bd
a
c
bd
Translation
between
fundamental regions
a
c
Glide-reflection
between
fundamental regions
bd
bd
b
bd
d
bd
bd
bd
bd
bd
b
d
bd
c
a
a
c
a
c
c
a
d
d
c
c
d
(d) Fundamental region
209
d
d
b
b
c
bd
a
a
c
a
c
b
a
a
b
b
c
c
b
d
a
c
c
a
a
bd
d
c
b
b
c
c
a
c
c
a
a
a
a
a
bd
a
c
c
a
c
c
c
a
a
(c) Translational unit
c
a
bd
a
c
c
a
bd
c
c
bd
c
a
bd
a
a
bd
a
c
bd
bd
bd
c
a
Translation
between
fundamental regions
bd
bd
bd
Mirror-reflection
between
fundamental regions
d
bd
b
b
a
fundamental regions
a
(c) Translational unit
a
(d) Fundamental
region
210
c
a
fundamental regions
FIG. A6(a) Symmetry group p2mm
FIG. A5(a) Symmetry group cm
Mirror-reflection axis
Center of 2-fold
rotation lying on a
mirror-reflection axis
Mirror-reflection axis
Glide-reflection axis
(b) Cayley diagram
bb
(b) Cayley diagram
bd
bd
bd
bd
a
a
bd
a
a
c
c
bd
bd
bd
bd
bd
a
a
bd
a
a
a
a
c
c
bd
bd
bd
a
a
c
c
bd
bd
b
a
a
c
c
a
a
a
a
dd
dd
b
bb
c
c
c
c
c
c
c
c
c
c
a
a
a
a
b
dd
a
c
c
c
(c) Translational unit
211
a
c
d
c
fundamental regions
a
c
b
b
a
b
d
d
a
(d) Fundamental
region
212
a
a
bb
d
a
(c) Translational unit
d
a
c
bd
d
a
c
b
dd
a
c
c
b
b
b
d
b
bb
d
(d) Fundamental
region
c
c
a
a
dd
bb
b
b
bd
d
b
dd
a
a
dd
d
c
d
b
dd
a
a
a
bb
dd
a
a
a
a
a
c
c
bd
bd
a
a
bb
c
b
dd
bb
d
bd
c
c
c
c
bb
dd
b
a
a
c
c
bd
a
a
c
c
b
c
c
dd
b
bd
dd
bb
a
a
a
a
bb
c
c
d
b
a
a
dd
dd
c
c
c
bd
bd
a
a
a
a
a
a
bb
c
c
Mirror-reflection
between
fundamental regions
dd
a
a
dd
a
a
b
c
c
bd
c
c
a
a
c
c
a
a
c
c
b
dd
a
a
Glide-reflection
between
fundamental regions
bd
a
a
c
c
c
c
c
c
b
bd
a
a
a
a
a
a
bd
bd
c
c
Mirror-reflection
between
fundamental regions
b
c
c
bd
bb
dd
b
a
fundamental regions
c
FIG. A7(a) Symmetry group p2mg
FIG. A8(a) Symmetry group p2gg
Mirror-reflection axis
Glide-reflection axis
Glide-reflection axis
Center of 2-fold
rotation
Center of 2-fold
rotation
(b) Cayley diagram
c
c
bd
bd
a
a
db
c
c
db
db
c
c
bd
db
bd
db
bd
db
c
c
a
c
bd
a
c
bd
a
c
c
a
db
d
b
a
c
c
a
c
bd
b
bd
a
c
d
a
c
c
a
db
a
c
db
db
db
d
b
d
a
a
b
bd
a
c
c
a
db
c
a
c
c
a
bd
c
d
db
a
c
c
a
db
a
c
c
a
db
bd
bd
b
bd
c
a
a
c
c
a
bd
c
a
d
a
c
c
a
c
c
a
a
db
db
bd
bd
a
c
c
a
db
db
db
c
c
d
a
c
db
Glide-reflection
between
fundamental regions
bd
c
a
db
c
c
a
a
a
c
c
a
c
a
b
bd
db
a
a
bd
c
c
bd
db
bd
bd
c
c
db
a
a
Translation
between
fundamental regions
bd
a
a
c
c
db
bd
c
c
a
a
bd
Mirror-reflection
between
fundamental regions
db
a
a
bd
c
c
bd
db
c
c
db
a
a
bd
2-fold rotation
between
fundamental regions
bd
a
a
c
c
db
(b) Cayley diagram
c
c
b
d
a
c
b
a
d
a
b
a
b
(d) Fundamental
region
d
c
d
bd
fundamental regions
(c) Translational unit
213
d
d
a
bd
b
c
a
a
c
c
c
c
a
b
(c) Translational unit
(d) Fundamental region
214
b
a
c
fundamental regions
FIG. A10(a) Symmetry group p4
FIG. A9(a) Symmetry group c2mm
Mirror-reflection axis
Mirror-reflection axis
Glide-reflection axis
Glide-reflection axis
Center of 2-fold
rotation
Center of 2-fold
rotation lying on a
mirror-reflection axis
Center of 4-fold
rotation
Center of 2-fold
rotation
(b) Cayley diagram
(b) Cayley diagram
c
c
bb
a
a
c
c
dd
c
c
bb
c
c
a
a
dd
c
c
c
c
c
c
bb
a
a
bb
bb
c
c
c
c
bb
a
a
dd
c
c
a
a
dd
c
c
bb
c
c
b
b
b
b
a
d
c
a
d
d
a
da
cb
b
c
bc
a
d
d
a
c
b
b
c
da
a
a
cb
c
b
bc
c
b
bc
d
a
da
cb
b
c
a
d
d
a
da
c
b
b
c
a
d
d
d
c
c
c
c
da
c
b
a
d
d
a
cb
d
d
d
a
a
bb
c
c
bb
c
c
bb
a
a
dd
a
b
a
a
dd
c
c
da
a
d
d
a
b
c
bc
a
d
c
b
b
c
bc
d
c
c
da
cb
c
b
b
c
bc
d
dd
a
a
dd
a
a
bb
bb
da
4-fold rotation
between
fundamental regions
a
d
d
a
cb
c
b
b
c
a
d
d
a
da
c
b
d
a
a
a
bb
c
c
a
d
d
a
da
b
c
c
bb
a
a
dd
c
c
dd
a
a
dd
a
a
bb
bb
dd
a
a
bb
c
c
c
c
2-fold rotation
between
fundamental regions
b
a
a
dd
a
d
d
a
bc
cb
c
c
dd
a
a
bb
c
c
bb
a
a
dd
c
c
dd
a
a
c
c
bb
d
bc
Mirror-reflection
between
fundamental regions
d
a
a
bb
c
c
a
a
dd
dd
a
a
c
c
b
c
c
bb
a
a
dd
dd
a
a
bb
c
c
bb
a
a
bb
a
a
c
c
dd
a
a
bb
c
c
bb
c
c
bb
a
a
dd
dd
a
a
c
c
dd
a
a
bb
c
c
bb
c
c
bc
b
a
d
d
a
da
d
a
d
da
cb
cb
a
c
b
c
b
c
b
c
d
c
(d) Fundamental
region
c
b
c
a
215
b
a
b
d
(c) Translational unit
c
dd
a
c
bb
a
a
d
d
fundamental regions
a
d
a
(c) Translational unit
b
bc
a
(d) Fundamental region
216
d
fundamental regions
FIG. A12(a) Symmetry group p4gm
FIG. A11(a) Symmetry group p4mm
Mirror-reflection axis
Mirror-reflection axis
Glide-reflection axis
Glide-reflection axis
Center of 2-fold
rotation lying on a
mirror-reflection axis
Center of 2-fold
rotation lying on a
mirror-reflection axis
Center of 4-fold
rotation lying on a
mirror-reflection axis
Center of 4-fold
rotation
(b) Cayley diagram
(b) Cayley diagram
cc
cc
aa
cc
cc
aa
bb
a
a
a
a
cc
cc
aa
b
b
cc
a
a
cc
b
b
cc
aa
b
b
cc
aa
cc
a
a
cc
b
b
cc
cc
aa
cc
cc
aa
cc
aa
cc
a
a
aa
cc
bb
cc
aa
cc
aa
cc
ac
bb
ca
c
a
b
b
bb
cc
aa
bb
bb
cc
ac
b
b
cc
aa
cc
cc
aa
cc
cc
aa
a
a
a
bb
cc
a
a
c
c
b
c
bb
bb
ac
bb
ac
bb
bb
bb
ca
bb
ca
ac
ac
bb
bb
bb
ca
bb
ca
bb
bb
bb
ac
ca
bb
bb
ca
bb
ac
bb
ac
bb
ac
bb
bb
ac
ca
bb
bb
b
a
(c) Translational unit
(d) Fundamental region
217
c
a
bb
c
c
ac
bb
ca
bb
a
c
bb
bb
bb
4-fold rotation
between
fundamental regions
a
c
ac
bb
a
c
c
a
ca
Mirror-reflection
between
fundamental regions
bb
a
c
bb
ca
bb
c
b
a
a
c
ca
c
a
ac
bb
a
c
c
a
a
c
ca
ac
c
a
a
c
c
a
ca
ca
a
c
ca
bb
c
a
a
c
c
a
bb
ac
a
c
c
a
a
c
ca
bb
c
a
a
c
ac
ac
c
a
a
c
c
a
ca
bb
ac
a
c
c
a
bb
bb
a
c
c
a
a
c
ca
ac
c
a
a
c
c
a
bb
a
cc
aa
bb
b
bb
a
a
ca
c
a
b
b
ca
a
c
cc
aa
bb
bb
c
a
a
c
c
a
a
a
bb
ac
a
c
c
a
cc
aa
a
a
b
b
a
a
bb
cc
cc
a
a
cc
cc
a
a
Mirror-reflection
between
fundamental regions
a
a
b
b
bb
a
a
bb
cc
aa
b
b
cc
a
a
b
b
a
a
bb
bb
cc
aa
bb
b
b
bb
a
a
cc
aa
cc
a
a
b
b
cc
cc
bb
b
b
aa
a
a
b
b
cc
aa
a
a
bb
a
a
bb
cc
bb
b
b
bb
a
a
cc
aa
bb
a
a
bb
cc
c
b
c
a
fundamental regions
(c) Translational unit
c
b
a
(d) Fundamental region
218
b a
b
a
b
a
c
a b
fundamental regions
FIG. A14(a) Symmetry group p3m1
FIG. A13(a) Symmetry group p3
Mirror-reflection axis
Center of 3-fold
rotation
Glide-reflection axis
Center of 3-fold
rotation lying on a
mirror-reflection axis
b
(b) Cayley diagram
ab
ab
ab
ab
cd
c
d
cd
c
d
cd
c
d
cd
b
a
a
b
b
a
a
b
b
a
a
b
b
a
dc
dc
ab
dc
ab
c
d
a
b
c
d
c
d
a
b
b
a
dc
cc
b
b
ab
ab
cd
c
d
cd
c
d
cd
c
d
cd
b
a
a
b
b
a
a
b
b
a
a
b
b
a
dc
dc
ab
dc
c
aa
b
b
b
b
cc
c
c
b
b
a
a
c
c
c
c
b
b
b
a
a
a
c
c
b
aa
c
c
bb
a
a
cc
bb
a
a
cc
b
b
aa
c
c
bb
a
a
cc
a
a
cc
b
b
aa
c
c
bb
a
a
a
a
a
aa
b
b
b
b
b
b
aa
b
b
cc
b
b
aa
a
c
d
b
b
b
ab
d
219
c
c
b
b
aa
a
a
d
a
(d) Fundamental region
a
a
cc
a
a
c
c
bb
aa
b
b
Mirror-reflection
between
fundamental regions
a
c
c
d
b
b
b
b
bb
a
a
cc
aa
c
c
a
b
b
(c) Translational unit
a
b
ab
c
c
c
bb
c
c
b
dc
dc
ab
a
a
b
cc
a
a
c
c
b
aa
c
c
bb
b
b
a
a
b
a
a
cc
c
c
b
b
aa
c
c
b
b
aa
a
a
c
c
bb
a
ab
bb
a
a
a
a
c
c
b
aa
c
c
bb
b
b
(b) Cayley diagram
b
a
a
cc
c
c
dc
ab
c
c
b
b
aa
a
b
b
a
dc
ab
a
a
c
d
cd
c
c
b
ab
cd
a
b
b
a
dc
dc
ab
cd
3-fold rotation
between
fundamental regions
b
aa
a
c
b
b
aa
c
c
c
(e) Relationship between fundamental regions
(c) Translational unit
(d) Fundamental region
220
fundamental regions
FIG. A15(a) Symmetry group p31m
FIG. A16(a) Symmetry group p6
Mirror-reflection axis
Center of 2-fold
rotation
Glide-reflection axis
Center of 3-fold
rotation
Center of 3-fold
rotation
Center of 6-fold
rotation
Center of 3-fold
rotation lying on a
mirror-reflection axis
(b) Cayley diagram
cb
a
c
b
cb
a
a
c
b
cb
aa
c
b
cb
aa
c
b
cb
aa
cb
c
b
aa
cb
cb
c
b
aa
cb
b
c
c
b
c
b
aa
aa
cb
cb
aa
b a
cb
aa
cb
b
(d) Fundamental region
221
aa
aa
aa
c
a
a
bc
aa
c
b
aa
c
b
aa
bc
c
b
b
c
bc
bc
cb
aa
bc
aa
bc
aa
b
c
aa
c
b
bc
c
b
b
c
aa
bc
c
b
b
c
cb
aa
bc
aa
bc
3-fold rotation
between
fundamental regions
cb
a
a
aa
c
b
bc
c
b
b
c
cb
aa
bc
c
b
a
a
aa
cb
aa
c
b
bc
aa
b
cb
c
a
b
c
c
a
b
fundamental regions
aa
c
b
a
a
cb
bc
a
a
a
a
aa
aa
c
b
b
c
c
b
cb
cb
a
a
c
b
cb
bc
c
b
bc
c
b
bc
aa
a
a
c
b
cb
cb
2-fold rotation
between
fundamental regions
c
b
bc
a
a
c
b
cb
cb
a
a
b
c
c
(c) Translational unit
bc
cb
a
a
c
b
b
c
c
c
a
a
bc
c
b
bc
a
a
a
a
aa
a
a
bc
c
b
bc
bc
cb
c
b
bc
c
b
c
b
bc
b
c
aa
bc
a
a
a
a
cb
cb
bc
c
b
bc
b
c
aa
c
b
cb
aa
3-fold rotation
between
fundamental regions
cb
aa
a
a
a
a
c
b
bc
aa
cb
bc
cb
aa
c
b
a
a
aa
aa
b
c
cb
bc
b
c
c
b
cb
c
b
c
b
Mirror-reflection
between
fundamental regions
a
a
a
a
b
c
(b) Cayley diagram
cb
c
b
bc
bc
aa
cb
bc
a
a
c
b
a
a
a
bc
aa
cb
bc
cb
cb
c
b
bc
a
a
bc
cb
a
c
b
bc
(c) Translational unit
a
a
b
b
(d) Fundamental region
222
a
c
b
b
c
fundamental regions
FIG. A17(a) Symmetry group p6mm
Mirror-reflection axis
Glide-reflection axis
Center of 2-fold
rotation lying on a
mirror-reflection axis
Center of 3-fold
rotation lying on a
mirror-reflection axis
Center of 6-fold
rotation lying on a
mirror-reflection axis
(b) Cayley diagram
b
b
b
c
c
a
a
c
c
bb
bb
b
b
b
cc
cc
aa
aa
b
b
b
c
c
a
a
c
c
bb
bb
b
b
b
cc
aa
aa
aa
cc
cc
b
b
c
c
a
a
c
c
aa
aa
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
aa
b
b
b
b
cc
cc
aa
aa
b
b
b
b
c
c
a
a
c
c
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
aa
b
b
cc
cc
b
b
c
c
a
a
c
c
aa
aa
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
b
b
cc
aa
aa
aa
cc
cc
b
b
c
c
a
a
c
c
b
b
c
c
a
a
c
c
b
b
b
b
cc
cc
aa
aa
b
b
b
b
c
c
a
a
c
c
bb
bb
b
b
b
b
bb
aa
c
c
a
a
c
c
aa
bb
aa
b
b
aa
aa
aa
b
b
cc
cc
b
b
c
c
a
a
c
c
aa
aa
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
b
b
cc
aa
aa
cc
b
b
bb
aa
aa
b
b
cc
aa
aa
cc
b
b
c
a
(c) Translational unit
c
c
a
a
c
c
bb
b
b
aa
aa
cc
b
b
c
c
a
a
c
c
aa
b
b
b
(d) Fundamental region
223
b
b
c
c
a
a
c
c
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
aa
b
b
cc
cc
b
b
c
c
a
a
c
c
aa
aa
b
b
c
c
a
a
c
c
bb
bb
b
b
aa
b
b
cc
aa
cc
aa
aa
Mirror-reflection
between
fundamental regions
cc
b
b
c
c
a
a
c
c
b
bb
b
b
b
cc
aa
aa
cc
b
b
c
a
b
c
a
a
c
bb
a
b
b b
b
b
c
c
a
c
a