17 have to define and select individual elements from the complex, often continuous pattern. Patterns can form series in which one pattern evolves into another, via a series of transitional patterns. Character of these changes determines the type of series (Makovicky 1989):
expansion-reduction series in which certain elements expand in area/importance at the expense of others which are being reduced;
accretional series in which a certain type of elements is multiplied without changing relationship to the rest of the pattern;
intercalation series in which additional elements are being introduced – intercalated – between the existing elements. They differ from the original ones. Omission series is the opposite sequence of changes.
complication/simplification series in which the existing elements receive or loose ornamentation appendages, change their shapes in a quantitatively expressible way –can be called (de)baroquization of the pattern;
element-substitution series in which a certain, increasing percentage of original, identical elements are being replaced by a new (e.g. differently coloured) element; pattern-reduction or crystallographic shear series in which pattern is being reduced on selected crystallographically defined lines by a desired interval, connected with an exactly defined mutual shift of the pattern portions adjacent to such a line of reduction.
MATERIAL AND STYLE
Material used has profound influence upon the patterns created and their symmetry. On the one end of spectrum are brickwork patterns in which one or several basic brick shapes and a limited number of sizes were used, eventually bevelled on edges when necessary. This characterizes the old, wonderful Seljuq brickwork of Central Asia, Iran and Turkey, with a plentiful use of brick ribs and recessed fields in order to achieve a play of sunlight on these uncoloured patterns (e.g., Makovicky 1989)(Figs. 43-47). Discovery of turquoise glaze led often to two-coloured, two-tier, interlacing patterns. At the other end of Islamic world, the marble-and-brick ornaments of Cordoba (about 936-1008 C.E.) represent another flowering of this style (Makovicky and Fenoll 1997)
18 (Fig. 48).
Sadly, the technological advances in the field of glaze, e.g., as the haft rengi tiles of Safavid Iran, led to a substantial reduction in the level of two-dimensional ornamental crystallography, replaced by opulent floral designs.
The other end of the spectrum is represented by hand-fashioned colour-glazed ceramic tiles (zalij of Morocco), which allow virtually any shape and any symmetry – mostly ‘uncoloured’ or only dichroic, in spite of their often rich gamut of colours. Furthermore it is plaster (Fig. 19), less frequently stone, which also allow variations not normally observed in bricks. Here often limitation is dictated by the exposure of these elements to the weather (or, also, play of sunlight), a factor foreign to the smooth surface of ceramic tiles.
Different techniques are reflected especially in different statistics of plane group of symmetry used. Here we present diagrams for Cordoba (Makovicky and Fenoll 1997), a Seljuq Kharraqan Tower in Iran (Makovicky, in preparation) and the general statistics from Bourgoin (1973) who presents primarily ceramic tile patterns and some stucco patterns.
KASBAH DE TELOUET
In the mid-nineteenth century the family of El Glaoui’s were clan leaders controlling the age-old route from Marrakech to the Draa and Dades valleys; they were limited to their domain. However, in the hard winter of 1893, Sultan Moulay Hassan was passing on the way back from a disastrous military campaign and found himself at the mercy of Glaoui’s brothers for food and shelter for himself and his army. Their reception and care were handsomely rewarded: they became caids for the whole region and received lots of arms the sultan was forced to leave behind. By 1901 they were sole rulers of the area and when French occupied Morocco in 1912 they pledged their loyalty to the new rulers. They became pashas of Marrakech and caids for the region of Atlas and desert cities. As could be expected, they turned their income and power into building a residence appropriate to their new standing.
19 The kasbah of Telouet, built by them was abandoned only in 1956 but many of its dark-red earth bulidings crumble fast as such constructions do without constant care. It is a labyrinth of rooms, doors and connecting passages. It is the reception rooms we come to visit, with iron-grilled windows, fine carved ceilings and zelij and plaster ornaments. Although they are 19th and early 20th century creations, they follow the traditional style and they are richer in character than the bulk of Marrakech ornaments of any age. They also give insight into lives of local warlords who, in spite of their new-found influence remained faithful to their safe, clan core area.
Pay attention to the following (and everything else you discover): Any zero-dimensional ornaments and their symmetry?
Frieze ornaments and their true 1-D or truncated 2-D character..
Are the plaster ornaments one- or two-sided (plane- or layer groups of symmetry)? Can you reconstruct the 2-D ornaments on the doors and similar objects?
Plane groups of zellij ornaments..
Relationship of site symmetry in plane- or frieze patterns and symmetry of the motif in this site..
Dichroic and polychromatic groups...?
Anatomy of octagonal ornaments: correlation/counterplay between a quasilattice and polygonal representation?
Anything interesting among the hangings?
Were they contrasting various symmetries for better effect?
Get ‘supervisors’ to take a look at your conclusions...
.
C
APTIONST
OT
HEF
IGURESFig. 1 Plane group of symmetry cmm with general positions and various types of special positions (on m, 2mm and 2).
Fig. 2 Pattern p6 of raised bricks with general positions (d) and diverse special positions (a-c). Central Asia.
20 on the motif of the ‘maple leaf’ p4gm (fig. a). Figs. a and b: Sala de la Barca, The Alhambra.
Fig. 4 Uncoloured and dichroic operations of symmetry (two-fold axes 2, mirror planes m and glide planes g.
Fig. 5 Selected dichroic plane groups derived from the plane group pmm. Fig. 6 Selected dichroic plane groups derived from the plane group cmm.
(c and d): general and special positions, respectively, of the group cm’m’. Fig. 7 Dichroic pattern in p4’g’m. Grand Mosque of Cordoba.
Fig. 8 Dichroic group p4’g’m. Special position on a neutral background. Shahrisabs, Uzbekistan.
Fig. 9 Dichroic group pC’ 4 derived from a p4 mosaic. Special positions on 4-fold
axes and general positions.
Fig. 10 Dichroic group pC’m. Baños de Alhambra.
Fig. 11 Four-coloured wave superposed on a dichroic pattern p4’g’m, which in turn is based on the p4gm tiling. Patio de Mexuar, The Alhambra.
Fig. 12 Dichroic pattern p4’g’m coloured by a three-coloured wave. Black tiles are nodes of the wave.
Fig. 13 Large-scale dichroic pattern c’mm based on a p4gm tiling.
Fig. 14 Pattern with a sequentially dichroic coloration . Stage 1: white/coloured; stage 2: coloured tiles → grey and black. Salon de Comares, The Alhambra. Fig. 15 Interlaced version of the ‘maple leaf’ pattern. Layer group p-4b2. Fig. 16 Pattern with a layer group p422.
Fig. 17 Interlaced Islamic pattern with a layer group p622.
Fig. 18 Pattern with 10-fold rosettes, layer group c222. A very common Islamic pattern.
Fig. 19 Square pattern p4, traced out in plaster. The Alhambra, Puerta de Vino (before 1300). Two types of stripes with a frieze group pmm2 situated between broader stripes p1m1 in two orientations.
Fig. 20 Another OD variant of the Puerta de Vino pattern, constructed by the action of two-fold axes situated in the pmm2 stripes.
Fig. 21 Islamic pattern based on the 14-fold rosettes. Stripes have frieze symmetry p2, with inter-stripe operations being either m (resulting in a pgm pattern) or 2 (resulting in a cmm pattern). Alternatively, one can choose stripes with
21 symmetry close to pmm2 with two different inter-stripe operations (these being two different sets of two-fold axes).
Fig. 22 Intergrowth of two patterns based on 12-fold rosettes. Adopted from Bourgoin (1973).
Fig. 23 Statistics of plane groups of symmetry in Islamic patterns in the pattern collection of Bourgoin (1973).
Fig. 24 Statistics of plane groups of symmetry for the brick-and-marble patterns from the Great Mosque of Cordoba (Makovicky and Fenoll 1997).
Fig. 25 Statistics of one-dimensional groups of symmetry in the same mosque (Makovicky and Fenoll 1997).
Fig. 26 ‘Kite and dart’ version of a decagonal cartwheel quasicrystalline pattern. Five- and ten-fold discs and Conway worms are accentuated. Reinterpreted from Grünbaum and Shepherd (1987).
Fig. 27 A quasiperiodic tiling from the tomb tower of Maragha, NW Iran, with elements corresponding to those in fig. 26 accentuated by shading. Slightly simplified. Fig. 28 A decagonal quasiperiodic cartwheel pattern. Note Ammann bars with widths
equal to 1 and τ, respectively, in a quasiperiodic succession. Museum of the Alhambra.
Fig. 29 Structure of the quasicrystalline pattern of the panel in the Museum of The Alhambra expressed as an assembly of PM1 decagons of Makovicky (1992), see Makovicky et al. (1998). Based on a free rotation of decagonal elements resulting in ‘averaged’ rosettes.
Fig. 30 Quasicrystalline decagonal cartwheel pattern from Palais Royal, Fez, Morocco.
Fig. 31 A cmm pattern of decagonal rosettes. The Alhambra and elsewhere. Fig. 32 Quasilattice of the octagonal quasiperiodic pattern of the Patio de las
Doncellas , Los Reales Alcazares, Sevilla. Width of intervals is unity and √2, respectively.
Fig. 33 An octagonal pattern from the Salon de Comares, The Alhambra, with a separate central disc and frequent phason shifts (frequently interchanged unit and √2 stripes).
Fig. 34 Primary Ammann bars superimposed upon an aperiodic octagonal tiling. Modified from Socolar (1989).
22 Fig. 35 Secondary Ammann bars superimposed on the same tiling as in Fig. 34. Fig. 36 Quasicrystalline octagonal tiling with Ammann’s vertex and edge decorations
and with a system marking lines derived from them by Makovicky and Fenoll (1996).
Fig. 37 A quasiperiodic pattern with a system of marking lines superimposed; its configuration indicates quasiperiodicity.
Fig. 38 The dome of the Sala de dos Hermanas in the Alhambra. Bold lines indicate two systems of ornamental decorations incorporated in the assembly of adarajas (wooden tiles of the stalactite vault). Makovicky and Fenoll (1996). Fig. 39 Transcription of tiles of the dome of La Sala de Dos Hermanas into the
square-and-lozenge octagonal tiling. Grey elements: ‘rotating octagons’ which allow a multiplicity of orientational interpretations.
Fig. 40 Marking lines applied to the (transcribed) dome of La Sala de Dos Hermanas. Their configuration indicates frequent departures from quasiperiodicity. Fig. 41 A system of Ammann bars (unity [white] and √2 [grey]) applied to the tiling
of the dome of La Sala de Dos Hermanas. Note occurrences of √2-√2 and multiple 1-1-1 contacts not allowed in a quasiperiodic octagonal tiling. Fig. 42 Three inflation stages of an octagonal quasiperiodic tiling of
squares-and-lozenges.
Fig. 43 Pattern of raised bricks, Djami Mosque in Gurgan, N Iran (12th century). Plane group p2 resulting from affine transformation of a p4 pattern.
Fig. 44 A large-scale p4mm pattern of raised bricks from the caravanserail Ribat-e-Sharaf, Iran.
Fig. 45 A p4gm pattern of raised bricks. Minaret of the Qalan Mosque, Bukhara, Uzbekistan (Seldjuk, 1127).
Fig. 46 A complex swastika pattern of raised, stacked bricks, plane group p4. Gunbad i’Alavyian, Hamadan, 13th century.
Fig. 47 A p6mm pattern of raised, partly fashioned bricks. Tympanum of a tomb-tower at Kharraqan, Iran, 1067-68.
Fig. 48 (a through c) A homologous series of patterns based on swastikas and squares with single, double and triple separation lines, respectively. The Grand Mosque of Cordoba.
23 procedure, the resulting configuration is shown in the lower portions of the figure. Grand Mosque of Cordoba.
S
ELECTED REFERENCESABAS, S. J. & SALMAN, A. S. (1995): Symmetries of Islamic Geometrical Patterns. Singapore: World Scientific.
BELOV, N.V. & TARKHOVA, T.M. (1956): Groups of colour symmetry. Crystallography 1, 4-13.
BOURGOIN, J. (1973): Arabic Geometrical Pattern and Design. Dover, New York. CASTÉRA, J.-M. (1996): Arabesques. Art Décoratif au Maroc. ACR Edition, Paris. COXETER, H.S.M. (1987): A simple introduction to coloured symmetry. Int. J.
Quantum Chemistry 31, 455-461.
GRÜNBAUM, B. & SHEPHARD, G. C. (1987): Tilings and Patterns. Freeman & Co., New York.
KLEIN, C. (2002): Mineral Science, 22nd
edition. Wiley & Sons, New York LU, P.J. & STEINHARDT, P.J. (2007): Decagonal and quasicrystalline tilings in
medieval Islamic architecture. Science 315, 1106-1110.
MAKOVICKY, E. (1986): Symmetrology of art. Coloured and generalized symmetries. Computers & Mathematics with Applications 12B: 949-980.
MAKOVICKY, E. (1989): Ornamental brickwork. Theoretical and applied
symmetrology and classification of patterns. Computers & Mathematics with Applications 17: 955-999.
MAKOVICKY, E. (1992): 800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of aperiodic tiling it inspired. In: Fivefold Symmetry, ed. I Hargittai, 67-86. World Scientific, Singapore.
MAKOVICKY, E. & FENOLL HACH-ALÍ (1996): Mirador de Lindaraja: Islamic ornamental patterns based on quasi-periodic octagonal lattices in Alhambra, Granada, and Alcazar, Sevilla, Spain. Bol. Soc. Española de Mineralogía 19: 1-26.
24 MAKOVICKY, E. & FENOLL HACH-ALÍ, P. (1997): Brick-and-marble ornamental
patterns from the Great Mosque, and the Madinat-al-Zahra palace in Cordoba, Spain. I. Symmetry, structural analysis and classification of two-dimensional ornaments. II. Symmetry and classification of one-dimensional patterns. Boletín Soc. Española de Mineralogía 20: 1-40 and 20: 115-134.
MAKOVICKY, E. & FENOLL HACH-ALÍ, P. (1999): Coloured symmetry in the mosaics of Alhambra, Granada, Spain. Boletín Soc. Española de Mineralogía 22: 143-183. MAKOVICKY, E. & MAKOVICKY, M. (1977): Arabic geometrical patterns - treasury for crystallographic teaching. Neues Jahrbuch für Mineralogie, Monatshefte 2: 58-68. SCHNEIDER, G. (1980): Geometrische Bauornamente der Seldschuken in Kleinasien.
Reichert Verlag, Wiesbaden.
SOCOLAR, J.E.S. (1989): Simple octagonal and dodecagonal quasi-crystals. Phys.Rev. B 39, 10519-10551.
WASHBURN, D. & CROWE, D. (1998): Symmetries of Culture. (3rd Ed.) Seattle: University of Washington Press.
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