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Journal of

Applied
Crystallography
ISSN 1600-5767

Moroccan ornamental quasiperiodic patterns constructed by
the multigrid method
Youssef Aboufadil, Abdelmalek Thalal and My Ahmed El Idrissi Raghni

J. Appl. Cryst. (2014). 47, 630–641

c International Union of Crystallography
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Many research topics in condensed matter research, materials science and the life sciences make use of crystallographic methods to study crystalline and non-crystalline matter with neutrons, X-rays and electrons. Articles published in the Journal of Applied Crystallography focus on these methods and their use in identifying structural and diffusioncontrolled phase transformations, structure-property relationships, structural changes of
defects, interfaces and surfaces, etc. Developments of instrumentation and crystallographic apparatus, theory and interpretation, numerical analysis and other related subjects are also covered. The journal is the primary place where crystallographic computer
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J. Appl. Cryst. (2014). 47, 630–641

Youssef Aboufadil et al. · Moroccan ornamental quasiperiodic patterns

research papers
Journal of

Applied
Crystallography

Moroccan ornamental quasiperiodic patterns
constructed by the multigrid method

ISSN 1600-5767

Youssef Aboufadil,* Abdelmalek Thalal* and My Ahmed El Idrissi Raghni*
Received 8 November 2013
Accepted 23 January 2014

# 2014 International Union of Crystallography

Department of Physics, LSM, Faculty of Science, University of Marrakech-Semlalia, Boulevard
Prince My Abdellah, Marrakech 40000, Morocco. Correspondence e-mail:
youssefaboufadil@yahoo.fr, abdthalal@gmail.com, elidrissiraghni@ucam.ac.ma

The similarity between the structure of Islamic decorative patterns and
quasicrystals has aroused the interest of several crystallographers. Many of
these patterns have been analysed by different approaches, including various
kinds of ornamental quasiperiodic patterns encountered in Morocco and the
Alhambra (Andalusia), as well as those in the eastern Islamic world. In the
present work, the interest is in the quasiperiodic patterns found in several
Moroccan historical buildings constructed in the 14th century. First, the zellige
panels (ﬁne mosaics) decorating the Madrasas (schools) Attarine and Bou
Inania in Fez are described in terms of Penrose tiling, to conﬁrm that both panels
have a quasiperiodic structure. The multigrid method developed by De Bruijn
[Proc. K. Ned. Akad. Wet. Ser. A Math. Sci. (1981), 43, 39–66] and reformulated
by Gratias [Tangente (2002), 85, 34–36] to obtain a quasiperiodic paving is then
used to construct known quasiperiodic patterns from periodic patterns extracted
from the Madrasas Bou Inania and Ben Youssef (Marrakech). Finally, a method
of construction of heptagonal, enneagonal, tetradecagonal and octadecagonal
quasiperiodic patterns, not encountered in Moroccan ornamental art, is
proposed. They are built from tilings (skeletons) generated by the multigrid
method and decorated by motifs obtained by craftsmen.

1. Introduction
‘In quasicrystals, we ﬁnd the fascinating mosaics of the Arabic
world reproduced at the level of atoms: regular patterns that
never repeat themselves.’ This is what the Nobel Committee
wrote (Royal Swedish Academy of Sciences, 2011) regarding
the Nobel prize in Chemistry attributed to Shechtman in 2011
for his discovery, in an aluminium manganese alloy, of quasicrystals with an icosahedral ordered phase, the diffraction
pattern of which exhibits ﬁne well resolved diffraction spots
like in crystals but distributed according to the icosahedral
symmetry that is prohibited by both the two- and the threedimensional periodic lattice.
The simplest way of visualizing a two-dimensional quasicrystal is that followed in another context by Penrose (1974),
who invented a nonperiodic tiling of a plane by two types of
tiles, generated by a deterministic construction using inﬂation,
the diffraction pattern of which consists of peaks that are
distributed on ﬁgures with pentagonal symmetry. This tiling,
which quickly became the archetype of quasicrystals, has been
explained in the multigrid method developed by De Bruijn
(1981) in the plane and generalized by Duneau and Katz
(Duneau & Katz, 1985; Katz & Duneau, 1986), who showed
that a quasiperiodic tiling could be obtained by the cut-andprojection method from a higher-dimension space.
The similarity between the zellige panels (ﬁne mosaics)
found in historical monuments and mausoleums and quasi-

630

crystal diffraction patterns (Fig. 1) shows that Moroccan
master craftsmen have also built, since the 14th century,
decorative motifs that are encountered in quasiperiodic
patterns. Several authors have analysed, using different
approaches, various kinds of quasiperiodic patterns encountered in Moroccan–Andalusian decorative art and in the
eastern Islamic world. Makovicky (1992, 2004, 2007, 2008,

Figure 1
(a) Zellige panel of the Madrasa Attarine (Fez). (b) The distances
between two symmetrical elements relative to the centre of the pattern
are in the ratio of the golden mean.

doi:10.1107/S1600576714001691

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2011), Makovicky & Fenoll Hach-Alı´ (1997), Makovicky et al.
(1998), Castera & Jolis (1991) and Castera (1996, 2003)
studied the octagonal and decagonal patterns, Rigby (2005),
Lu & Steinhardt (2007), Saltzman (2008) and Al Ajlouni
(2012) were interested in the decagonal patterns, Makovicky
& Makovicky (2011) studied the dodecagonal structure, and
Bonner & Pelletier (2012) constructed a sevenfold pattern. In
another context, Whittaker & Whittaker (1988) proposed
heptagonal and enneagonal tiling, and Franco et al. (1996) also
proposed enneagonal quasiperiodic tiling.
In this work we shall ﬁrst show, using the multigrid method,
that some patterns encountered in Moroccan monuments built
by the craftsmen’s method called Hasba (Thalal et al., 2011;
Aboufadil et al., 2013) are underlined by a quasiperiodic tiling.
We then give examples of the well known decagonal patterns
that adorn the Madrasas (schools) Attarine and Bou Inania
(Fez), and a variant of the dodecagonal pattern of the
tympanum at the entrance of the mausoleum of Moulay Idriss
(Fez). These patterns have been widely analysed by Makovicky et al. (1998), Makovicky (2004) and Makovicky &
Makovicky (2011). We shall then build new octagonal quasiperiodic patterns from periodic patterns that adorn the
Madrasas Ben Youssef (Marrakech) and Bou Inania (Fez).
Finally, we shall use the multigrid method to construct
heptagonal, enneagonal, tetradecagonal and octadecagonal
quasiperiodic patterns not encountered in Moroccan
geometric art.

2. The multigrid method
Several methods are used to construct quasiperiodic tilings.
One of the easiest, called the multigrid method and formulated by Gratias (2002), is discussed in this section.
We construct a set of equidistant parallel lines, characterized by a vector V orthogonal to the lines and of length equal
to the period. We copy this set N times by performing each

time a n rotation in the plane; N is a generic value greater
than 2, and it corresponds to the rotation order of the grid in
relation to the others. We then obtain a grid, called a multigrid,
constituted by N families of lines; each family is characterized
by an orthogonal vector Vn (Fig. 2a).
From a central point, we place the vector Vn and its opposite
and we trace orthogonal lines passing by their extremities. We
obtain an irregular convex polygon and we trace the segments
joining the centre point to the vertices of the polygon. By
drawing a small circle around the central point, we obtain N
sectors, each based on the polygon edge (Fig. 2b). In each
sector, we choose a vector Wn of arbitrary length and direction, and we associate it with the corresponding line family.
We then construct parallelograms Pi, j formed by the vectors
Wi and Wj taken in pairs. These parallelograms will constitute
the paving tiles.
We assume that the families of lines are distributed in a
generic way, so that the lines intersect in pairs and there are no
triple or multiple intersections. With each intersection Ii, j
between lines Li and Lj, we associate the parallelogram Pi, j
engendered by the vectors Wi and Wj (Figs. 3a and 3b). The
tiles are arranged according to the sequence of intersection of
the lines (Fig. 3c) and the whole tiling is obtained by this
process (Fig. 3d). The perfectly ordered tiling obtained is a
quasiperiodic lattice (except for the case N = 2, which corresponds to a periodic lattice).
It is shown (Jaric, 1989) that these tilings are the result of
cutting a periodic N-dimensional object in an irrational
direction. This construction leads to remarkable results when
choosing a symmetric multigrid constructed from a single
family of lines copied N times by the same angle of rotation
n = 2/N and taking Wn equal to the vector Vn.
The multigrid method is suitable for many variations in the
choice of grid spacing sequences, their relative positions and

Figure 2
(a) A multigrid is the superposition of N > 2 sets of equidistant parallel
lines in N distinct directions of the plane. In this case, N = 4. Periods in
each direction are not necessarily equal. (b) Vectors Vn and their
opposites are traced from the same centre to obtain a convex polygon
generated by lines perpendicular to the vectors and passing by their
extremities. We arbitrarily choose in each sector a vector Wn associated
with the same family of lines.
J. Appl. Cryst. (2014). 47, 630–641

Figure 3
(a) Intersection of lines. (b) Each intersection, Ii, j, of the lines Li and Lj of
the multigrid is associated with the parallelogram Pi, j. (c) The paving is
obtained by placing adjacent parallelograms in the exact order of
intersection of the lines in the multigrid. In the case of multiple
intersections, we move a family of lines by an inﬁnitesimal amount to lift
the degeneracy. (d) The corresponding tiling.

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the vectors Wn that deﬁne the shape of the tiles. In all cases, we
obtain uniform tilings with beautiful decorative elements. In

the following, we shall use a symmetric multigrid, and the
spacing sequences will be speciﬁed in each case.

Figure 4
(a) Fibonacci sequence. (b) A 5-grid (Ammann grid). (c) Penrose tiling.

Figure 5
(a) A panel of the Madrasa Attarine (Sijelmassi, 1991). (b) The inﬁnite quasiperiodic pattern. (c) A ﬁnite part of the quasiperiodic pattern.

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3. Analysis of quasiperiodic patterns of some historical
monuments by the multigrid method
3.1. Decagonal panels of the Madrasas Attarine and Bou
Inania

Makovicky et al. (1998) were the ﬁrst to study and explain
the decagonal pattern from Madrasas Attarine and Bou
Inania. The purpose of this section is to show, through these
two well known examples, that the multigrid method can be
adapted to study quasiperiodic patterns. We use the Penrose
tiling (Fig. 4c) obtained, by the same process as explained in
the previous section, from a 5-grid, which is a superposition of
ﬁve identical grids rotated by an angle = 2/5 to one another
around the ﬁvefold axis passing through the centre (black
point) (Fig. 4b). The parallel lines of the grids are disposed in a
Fibonacci sequence (Fig. 4a).

3.1.1. Madrasa Attarine. The zellige panel (ﬁne mosaic) of
the Madrasa Attarine (MA) (Fig. 5c) has the symmetries 5 and
10. We deﬁne in it two rhombic tiles with vertex angles 36 and
72 , as shown in Fig. 5(a). If we arrange the tiles according to a
Penrose tiling, we obtain an inﬁnite quasiperiodic pattern
(Fig. 5b). The MA panel appears as a ﬁnite part of this
quasiperiodic pattern (Fig. 5c).
As already mentioned by several authors, we can notice the
similarity of the decagonal pattern to the diffraction pattern of
the quasicrystal AlMn (Shechtman et al., 1984). The ratio of
the distances between two symmetrical elements in the case of
the panel (Fig. 1) is equal to the golden number, = (1 + 51/2)/
2 = 1.61803398875 . . . .
3.1.2. Madrasa Bou Inania. The zellige panel of the
Madrasa Bou Inania (MBI), which is a variant of the MA
panel, is also extracted from an inﬁnite decagonal quasi-

Figure 6
(a) A panel of the Madrasa Bou Inania. (b) The inﬁnite quasiperiodic pattern. (c) A ﬁnite part of the quasiperiodic pattern.
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periodic pattern obtained from a Penrose tiling. The two
rhombus tiles have the same dimensions as in the MA panel.
The difference between them is the tile decoration (Fig. 6).
Both MA and MBI panels are extracted from inﬁnite
quasiperiodic patterns. These panels, originally designed by
craftsmen, can be built from decorated tiles arranged in a
Penrose tiling. The difﬁcult part of such a construction is the
tile decoration. Indeed, the juxtaposition of the tiles should
lead to a harmonious model. The transition from one tile to

another must satisfy the artistic criteria required by the
craftsmen.
3.2. Construction of octagonal quasiperiodic patterns from
known periodic patterns

In this section, we present quasiperiodic patterns built from
known periodic patterns. For instance, we chose the patterns
drawn on the wooden gates of the Madrasas Bou Inania
(MBI) and Ben Youssef (MBY). They have the symmetry

Figure 7
Wooden gates at (a) Madrasa Bou Inania (Fez) and (b) Madrasa Ben Youssef (Marrakech).

Figure 8
(a) Octagonal tiling. (b) Square and rhombic decorated tiles. (c) An octagonal quasiperiodic pattern.

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groups p4mm and c2mm, respectively, and the repeat units are
eight- and 16-fold rosettes, respectively (Fig. 7).
To build an octagonal quasiperiodic pattern, we reproduce
the square and rhombic tiles traced in Fig. 8. The ornament of
the rhombic tiles consists of two eightfold rosettes and petals
of a 16-fold rosette distributed over the four corners of the
rhombus: two petals at small angles (45 ) and six petals at
large angles (135 ). The square tile has four eightfold rosettes
and four elements of a 16-fold rosette spread over the four
corners of the square. The two types of tile arranged in the
plane according to octagonal tiling give an octagonal quasiperiodic pattern composed of eight- and 16-fold rosettes
(Fig. 8).

4. Construction of new quasiperiodic patterns

Figure 9
A tenfold pattern (Madrasa Ben Youssef, Marrakech).

The quasiperiodic patterns are obtained from a quasiperiodic
tiling, which is the skeleton of the pattern, decorated with
elements extracted from ornamental patterns constructed by a
method used by craftsmen, called Hasba.

4.1. Ornamental patterns constructed by the craftsmen’s
Hasba method

Figure 10
Continuity of ribbons in a tenfold pattern.

This method of construction of geometric patterns was
described exhaustively by Thalal et al. (2011). It leads to
patterns constituted of n-fold rosettes and n-star shapes, where
n is the number of rays, for instance n = 10 (Fig. 9).
Patterns obtained by the Hasba method have interlaced
ribbons of constant width, which represents the unit of
measure of the pattern. Even though they are hidden in the
above patterns, interlaced ribbons are naturally present. We
can reveal the ribbons by extending, according to certain rules,
the edges of the different shapes that constitute the patterns.
By marking out the hidden ribbons, a more artistic interlaced
pattern emerges from the initial one. The main condition
imposed by the Hasba method is that the ribbons should
always be continuous throughout the pattern (Fig. 10).
The n-fold rosettes are the elements used to decorate the
skeleton constructed by the multigrid method to obtain a
quasiperiodic pattern.

Figure 11

(a) Stacking of non-interpenetrating tenfold rosettes. (b) A tile with a vertex angle of 72 . (c) Stacking of interpenetrating tenfold rosettes. (d) A tile with
a vertex angle of 36 . (e) Ribbon continuity across adjacent tiles.
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Figure 12
(a) Dodecagonal tiling. (b) Square and rhombus tiles. (c) A dodecagonal quasiperiodic pattern.

4.2. Description of the method

The combination of the Hasba and multigrid methods leads
to quasiperiodic patterns that are not encountered in either
Moroccan decorative art or diffraction patterns.
A quasiperiodic pattern is a decorated quasiperiodic tiling
obtained from an N-grid with a rotation angle = 2/N. The
decorative elements of the tiles are extracted from a stacking

of n-fold rosettes, where the tile vertices coincide with the
centres of the rosettes (Figs. 11b and 11d).
The arrangement of the tiles obeys the criterion of continuity of the ribbons required by Hasba, as mentioned in x4.1.
Indeed, the ribbons must always be continuous across adjacent
tiles over the entire pattern (Fig. 11e).
Furthermore, the order p of symmetry of the rosettes used
to decorate the tiles must be compatible with its angles, which

Figure 13

Figure 14

(a) Decorated tiles. (b) A decagonal quasiperiodic pattern.

(a) Decorated tiles. (b) An octagonal quasiperiodic pattern.

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are multiples of the rotation angle of the N-grid. It follows
that the order p must satisfy the relation
p ¼ k2=;

ð1Þ

where k is an integer 1.
4.3. Application to known quasiperiodic patterns

The dodecagonal, decagonal and octogonal patterns have
already been described by several authors who used different
approaches (Makovicky & Makovicky, 2011; Makovicky, 1992,
2004, 2007, 2008, 2011; Makovicky & Fenoll Hach-Alı´, 1997;
Makovicky et al., 1998; Al Ajlouni, 2011; Castera & Jolis, 1991;
Castera, 1996, 2003). In this section, we present a variant of
these three patterns obtained by the combination of the Hasba
and multigrid methods.
4.3.1. Dodecagonal patterns. This quasiperiodic pattern is
built up by a square tile and two types of rhombic tiles having

respective vertex angles of 30 and 60 . The tiles are decorated,
and arranged according to a dodecagonal tiling obtained from
a 6-grid (Fig. 12).
4.3.2. Decagonal patterns. The decagonal tiling is obtained
from a 5-grid with a rotational angle = 2/5. It consists of two
rhombic tiles with vertex angles of 36 and 72 . The order of
symmetry of the rosette compatible with these angles is p = 10.
The tiles are then extracted from tenfold rosettes arranged in
two conﬁgurations. In the ﬁrst conﬁguration, the rosettes do
not overlap, and the tile with a vertex angle of 72 is obtained
by joining the centres of four rosettes (Fig. 11a). The second
conﬁguration, in which the rosettes are interpenetrating,
provides the tile with a vertex angle of 36 (Fig. 11c). The
ﬁlling of the central gap in the ﬁrst conﬁguration and the
overlapping arrangement of the second must satisfy the
continuity rules imposed by Hasba (Figs. 11b and 11d).
A quasiperiodic decagonal pattern is then constructed with
the two tiles according to Penrose tiling (Fig. 13). Other

Figure 15
(a) Heptagonal tiling. (b) Decorated tiles. (c) A heptagonal quasiperiodic pattern. (d) A ﬁnite part of the quasiperiodic pattern.
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conﬁgurations are possible, offering a wide variety of decagonal quasiperiodic patterns other than those described
above.
4.3.3. Octagonal patterns. The same process leads to
octagonal quasiperiodic patterns (Fig. 14). The decorated tiles
are obtained from eightfold rosettes and arranged in an
octagonal tiling.
4.4. New quasiperiodic patterns

In this section we present the heptagonal, enneagonal,
tetradecagonal and octadecagonal patterns, which are not
encountered in Moroccan geometric art.
Franco et al. (1996) proposed enneagonal tiling of the plane
using geometric shapes. Seven- and ninefold quasiperiodic
tilings very similar to ours were obtained by Whittaker &

Whittaker (1988), who used the method of projection of
N-dimensional space. In addition, another way of constructing
a sevenfold tiling was presented in another context by Bonner
& Pelletier (2012).
4.4.1. Heptagonal and tetradecagonal patterns. Both
patterns are underlain by a heptagonal tiling obtained from a
7-grid with a rotational angle = 2/7.
Heptagonal and tetradecagonal patterns are constructed by
three rhombic tiles with vertex angles of /7, 2/7 and 3/7,
and decorated by seven- and 14-fold rosettes, respectively
(Figs. 15 and 16).
4.4.2. Enneagonal and octadecagonal patterns. Enneagonal
and octadecagonal tilings are obtained from a 9-grid with a
rotational angle = 2/9. They are constituted of four rhombic
tiles with vertex angles of /9, 2/9, 3/9 and 4/9. The

Figure 16
(a) Tetradecagonal tiling. (b) Decorated tiles. (c) A tetradecagonal quasiperiodic pattern. (d) A ﬁnite part of the quasiperiodic pattern.

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decorated tiles extracted from the 18-fold rosette give an
enneagonal and an octadecagonal pattern, respectively, as
shown in Figs. 17 and 18.

5. Conclusion
The combination of the Hasba and multigrid methods
conﬁrms the previous conclusions of Makovicky et al. (1998)
that, since the 14th century, Moroccan and Andalusian
craftsmen have built quasiperiodic ornamental patterns. These
uncommon patterns in Moroccan geometric art, which are
different from the usual periodic patterns, were called by the
craftsmen mkhabal laaˆqol, which means ‘patterns that disturb
the mind’. Perhaps the nonperiodic distribution of rosettes
caused confusion in their minds, which were more used to
working with periodic arrangements.
The method presented in this paper allows reproduction of
all the traditional patterns found in historical monuments and

proposes their variants. It also offers the opportunity of
building new quasiperiodic patterns from periodic ones that
are much more common in Moroccan and Andalusian
geometric art. As the latest development in crystallography
after the discovery of quasicrystals, unconventional quasiperiodic patterns such as heptagonal, tetradecagonal, enneagonal, octadecagonal and other patterns will give new impetus
to Moroccan geometric art and open a new era of contemporary geometric art.
The multigrid method allows the construction of any type of
plane tiling. The method is very simple and lends itself very
well to computer calculation. This approach, which would be
in tune with modern science, will open new horizons for a
generation of artisans steeped in modern science and technology.
In addition, the relationship between mathematics and art is
also a powerful tool for teachers. These complex patterns offer

Figure 17
(a) Enneagonal tiling. (b) Decorated tiles. (c) An enneagonal quasiperiodic pattern. (d) A ﬁnite part of the quasiperiodic pattern.
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Figure 18
(a) Octadecagonal tiling. (b) Decorated tiles. (c) An octadecagonal quasiperiodic pattern. (d) A ﬁnite part of the quasiperiodic pattern.

students examples of the artistic application of symmetry and
geometry. The history of geometric design contains numerous
examples of collaborations between mathematicians and
artists (Tennant, 2003). The construction of Islamic tiling lies
on the interesting border between mathematics and art. These
constructions, which have a rich history involving both
mathematicians and artisans, could motivate students and
increase their interest in mathematics and crystallography.
The authors are grateful to Dr Denis Gratias (LEM–CNRS/
ONERA) for reading and checking the text, for the fruitful
discussions we had with him, and for his valuable contribution
regarding the multigrid method.

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