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Quasicrystalline patterns in western Islamic art: problems and solutions

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Abstract

The quasiperiodic ornamental patterns of western Islam in Spanish Andalusia and in Morocco were produced by a modification of the multigrid method instead of tile composition, which was used for central discs of some Moroccan panels. The infrequent decagonal patterns are based on quasiperiodic sequences of unit and tau bars in the Alhambra and Fez. The much larger body of octagonal quasiperiodic patterns has quasiperiodic sequences of unit and √2 bars, so-called S and L bars, with frequently present, and ornamentally used, bar fragmentation and interchange. The octagonal patterns from the Alcázar in Sevilla, the Alhambra in Granada, and from Cordoba, are based on conscious modifications of overall tetragonal tiling schemes composed of elongate hexagonal and several types of isometric Alicatado tiles in bar-like arrangements. Bands, based on simple and on complexly fragmented bar sequences, alternate in differently organized schemes. Two different modifications of ‘octagonalization’ were used, producing several octagonal tiling varieties each. The octagonal wall mosaics in Morocco are panels largely based on octagonal bar sequences with different degrees of bar fragmentation and interchange, and with progressively loser composition rules with time. These are masked by large ornamental rosettes, especially by an oversize central rosette, which may entirely obscure the bar background of the panel. In both areas, the bar-and-band sequences point to a conscious mathematical design and organization of tiling and ornamental patterns used by their creators.

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Acknowledgements

Friendly comments and suggestions of two anonymous referees and of the editor were appreciated. Dedicated to Milota Makovicky and Puri Fenoll for their support.

Funding

No funding was received for the current study.

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Authors and Affiliations

  1. Institute for Geoscience and Natural Resource Management, University of Copenhagen, Østervoldgade 10, 1350, Copenhagen, Denmark

    Emil Makovicky

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  1. Emil Makovicky
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All photographs, descriptions, text and ideas are author’s own. Quoted material is properly and fully acknowledged. Drawings are from author’s publications, graphic interpretations by author’s own hand.

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Correspondence to Emil Makovicky.

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Although many scientists are studying Islamic art, the approach used here appears to be unique, without competition. Therefore, it is not to be made public before publication in press.

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12210_2020_969_MOESM1_ESM.tif

Fig. S1 Octagrid of an undisturbed sequence of bars with unit and √2 bar thicknesses. Intersections of unit bars define decoration sites given in color. Bar sequence is copied from Fig. 1; bar switches and fragmentation are required for the outer portions of the pattern (TIFF 1964 kb)

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Fig. S2 Pentagrid of an undisturbed sequence of bars with unit and tau thicknesses copied from the pattern in the Alhambra Museum. Intersections of unit bars define decoration sites given in deeper color; incomplete sets of this kind in lighter color. The former are surrounded by full star-like configurations which are left uncolored. Bar switches and fragmentation are required to complete the sets in the outer portions of the pattern (TIFF 28929 kb)

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Fig. S3 The S-bar scheme of the Al-Attarin panel (Fig. 4). Arbitrary coloring of two selected S-bar orientations (TIFF 36496 kb)

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Fig.S4 Two selected S-bar systems of the decagonal-approximant panel from Fig. 5 accentuated by different colorings. The S-bar scheme of the panel suggests stacking of three semi-independent horizontal tiers (TIFF 75072 kb)

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Fig. S5 A system of horizontal bars and bands in the C-Type panel from the Alcázar, Sevilla. For explanations see Fig. 7 (TIFF 77437 kb)

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Fig. S6 Vertical/horizontal bar and band scheme for the Type E panel. The central SLSLS band is followed by (4S+3L) bands with oscillating character in the center and further modifications at the ends, encroaching upon the marginal unmodified S-L sequences. Critical points of the analogous D Type panel are superimposed here by 3 × 3 and 5 × 5 squares (in orange) (TIFF 27845 kb)

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Fig. S7 Diagonal bar and band scheme for the Type E panel (Fig. 16). More extensive space allowed to develop a band sequence as SLSLS – M band composed of (4S+3L) –unmodfied SLS-(3S+2L). See its effect on placement of ornamental stars by comparing to Fig. 13 (TIFF 27198 kb)

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Fig. S8 The underlying M-SLSLS-M- (horizontal & vertical) and MSM-SLSLS- (diagonal) bar sequences of the pattern from Fig. 19. Although both bar/band systems are present on all wall segments, different selected orientations were colored on them (TIFF 40695 kb)

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Fig. S9 The alicatado panel of the side altar in the Mesquita Aljama de Cordoba. The A-2 type pattern has been laid without narrow white spacers which dominate the wall panels at the Alcazares in Sevilla (TIFF 3553 kb)

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Fig. S10 A complex S + L bar scheme, with the difference between undisturbed and M-type bands (these are marked by yellow L tiles) blurred, and changing along band elongation. Note the stepped arrangement of S fragments for proper accommodation of star discs and bar sequences between them. The Fez pattern from Fig. 21 (TIFF 29527 kb)

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Fig. S11 A horizontal band scheme from the same Meknes dado as in Fig. 20. Alternation of simple M bands and SLS bands is prominent in the left-hand portion, tending to that of imperfect SLSLS and MSM bands in the right-hand part (TIFF 41942 kb)

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Fig. S12 A diagonal scheme of (predominantly) SLSLS and MSM/MMSM bands. Vertical and horizontal schemes are the same. Meknes panel # II (Fig. 24) (TIFF 41645 kb)

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Fig. S13 A detail of the bar sequence from Fig. 26 (Meknes panel # III). Unit bars are materialized as a sequence of parallel elongated hexagons or as sequences of small isometric tiles; √2-bars are formed by a stacking of inclined elongated hexagons. Try to find SLSL sequences and occasional SLSSLS sequences (TIFF 10290 kb)

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Fig. S14 Vertical bar and band sequence from the isolated panel No. III in the Meknes Mausoleum. Green boundary encircles a quasiperiodic area (TIFF 34449 kb)

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Fig. S15 Vertical bars and bands in the corner pattern Meknes IV. The scheme describes the panel in Fig. 28 (TIFF 70553 kb)

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Fig. S16 Semiregular sequence of horizontal SLSLS and SLS bands in alternation with MSM and M bands in the rims of the central ornamental disc from the Qasbah de Telouet, Morocco (Fig. 32) (TIFF 79490 kb)

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Fig. S17 A complicated bar and band scheme in the wall mosaic from the old city of Fez. Panel is reproduced in original colors in Fig. 2. Detailed discussion is in the text (TIFF 41666 kb)

Appendix

Appendix

1.1 Mathematics relevant for Islamic octagrids

Western Islamic octagonal quasiperiodic patterns described in this study represent two-dimensional octagrids with two quasiperiodically repeating well-defined values of line spacing. These intervals are conventionally denoted as S (short) and L (long) and for octagonal quasiperiodic tiling are in the ratio 1:√2. The Islamic artists always started the pattern by placing an S interval in the origin. Explanations concerning the decagonal patterns are in the introductory section.

S = 1 unit; L = (sqrt2) units; bar pairs: SS = 2 units; SL = 2.4142; LL = 2.8284; bar triples: LSL = 3.8284; SLS = 3.4142.

Higher bar combinations: SLSLS = 3 + 2sqrt2 = 5.8284 units; 3 M + S = 3S + 3L + S = 4S + 3L = 8.2426; MSM = 3S + 2L; 6S + 4L = 11.6568; 8S + 6L = 16.4852.

Band relations: sqrt2 × 3 = 4.2426 units; sqrt2 × 4 = 5.6568; sqrt2 × 5 = 7.0710; (3S + 2L) × sqrt2 = 3L + 4S; (4S + 3L) × sqrt2 = 6S + 4L; (8S + 6L) × sqrt2 = 12S + ; (6S + 4L) × sqrt2 = 8S + 6L (for diagonals of selected bar combinations).

Band ratios: (4S + 3L)/(3S + 2L) = 1.4142 = sqrt2; (6S + 4L)/(4S + 3L) = sqrt2; (8S + 6L)/(6S + 4L) = sqrt2; (6S + 4L)/(3S + 2L) = (sqrt2)2 = (8S + 6L)/(4S + 3L) = 2.

Bar composition of bands: 3S + 2L = SLSLS; 4S + 3L = SLSLSLS; 6S + 4L = SLSLSSLSLS; 8S + 6L = SLSLSLSSLSLSLS; 12S + 8L = SLSLSSLSLSSLSLSSLSLS; (SS pairs are in bold print).

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Makovicky, E. Quasicrystalline patterns in western Islamic art: problems and solutions. Rend. Fis. Acc. Lincei 32, 57–94 (2021). https://doi.org/10.1007/s12210-020-00969-9

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