Tessellations: Islamic Tile Patterns and M. C. Escher(1)
“Number is the tune to which all things move,
and as it were make music; it is in the pulses
of the blood no less than in the starred curtain
of the sky. It is a necessary concomitant alike
of the sharp bargain, the chemical experiment,
and the fine frenzy of the poet. Music is number
made audible; architecture is number made
visible; nature geometrizes not alone in her
crystals, but in her most intricate arabesques.”(2)
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| An early 8th century, repeating tile tessellation from the Umayyad Palace of Khirbat al-Mafjar near Jericho, Palestine. (AramcoWorld, Volume 66, Number 6, November/December 2015, Special Insert) |
There is a similarity of patterned design across many Islamic arts and crafts and throughout all Islamic historic periods and across geographic regions. These patterns are found in ceramic tile and pottery designs,
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| A tiled Mihrab from Karamanoglu, c. 1432. (From the Tiled Kiosk--Çinili Köşk--Topkapi Palace, Istanbul; Photo credit: Michael Padwee, 2011) |
wood and stone carvings,
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| Wood carving with an 8-pointed star rosette pattern, door of a Moroccan Mosque in Fez. (Photo credit: Chris Boundy; https://www.travelblog.org/Photos/3581213) |
architecture and painting,
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| Mahmut Paşa Mausoleum, Istanbul, Turkey, built by Atik Sinan in 1474 using First Period Iznik tiles. (Photo credit: Michael Padwee, 2011) |
and textiles, among others.
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| "Star Ushak" Carpet, late 15th century. Woven in the Ushak region of western Turkey, "Star Ushak" carpets were made for regional consumption and for export throughout Europe. (Metropolitan Museum of Art, New York, New York, Gift of Joseph V. McMullan, 1958, Accession number 58.63; http://www.metmuseum.org/toah/works-of-art/58.63/) |
“Islamic art comprises geometric prints and tessellations which connect to form infinitely repetitive designs. They are beautiful in their simplicity, creating shapes and patterns that simultaneously satisfy and intrigue the eye of the observer. They invite the observer to contemplate the nature of the relationship between beauty and geometry[... .]”(3)
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| Detail of a tile panel with a repeating vegetal pattern in the Rüstem Paşa Camii (Rustem Pasha Mosque) in Istanbul built in 1560 by the architect Mimar Sinan. (Photo credit: Michael Padwee, 2011) |
Muslim intellectuals believe that "this relationship between beauty and geometry is an expression of the fundamental truths of Tawhid and Mizan, without committing Shirk (idolatry – the unforgivable sin). Tawhid, the ultimate unity of all things, emanating from the Oneness of God, is depicted in the patterns which tessellate into infinity. The fact that these tessellating designs are found to be so pleasing to the eye is evocative of the order and balance of the universe, expressed through the laws of geometry. For the Islamic artists, beauty and truth come together in geometric designs to express the perfection of God.”(4) Thus, the preference for geometry is directly related to the religious essence of Islamic culture. Such forms convey an aura of spirituality, but with no symbolic significance that attacks the beliefs of Islam. They also believe that symmetry (5) evokes a transcendent beauty by being able to stimulate and free the intellect.(6)
“There is a certain disregard for scale in Islamic art that derives from this perception. Similar kinds of patterning, for instance, might be found on a huge tile panel or on a bijou ornament. This is because decorative effects, in an Islamic context, are never mere embellishments, but always refer to other, idealised states of being. In this view, scale is almost irrelevant. For similar reasons Islamic ornemanistes (7) usually opted for a-centric arrangements in patterning, avoiding obvious focal points – a preference that resonates with the Islamic perception of the Absolute as an influence that is not ‘centred’ in a divine manifestation (as in Christianity), but whose presence is an even and pervasive force throughout the Creation.”(8)
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| A simple periodic tile panel made from hexagonal Canakkale tiles surrounded by hexagonal interlocking rosettes and rectangular border tiles. (From the Tiled Kiosk--Turkish: Çinili Köşk--is a pavilion set within the outer walls of Topkapı Palace, Istanbul; Photo credit: Michael Padwee, 2011) |
There are many types of ornamentation that we generally recognize as “Islamic” patterns, yet “the whole range of Islamic patterns represents an amalgam of many different styles, some simply adapted and absorbed from classical sources and from various cultures with which Islam came into contact during its early expansion.”(9) In his article, “Islamic Star Patterns”, A. J. Lee states that “in their simplest form all Islamic geometrical patterns are examples of periodic tilings (or tessellations) of the two dimensional plane, consisting of polygonal areas (10) or cells of various shapes abutting on neighboring cells at lines termed the edges of the tiling, and with three or more cells meeting at points termed the verticies, or nodes, of the tilings.”(11)
In 1905 Islamic scholar E. H. Hankin wrote of his research into the creation of Islamic tile patterns, and of the serendipitous discovery of a four-century-old working drawing of an arabesque pattern in a palace of the Mogul Emperor Akbar that led him to his conclusions. Hankin describes four classes of elementary patterns used in Islamic tilings:
a) the space to be decorated is divided into squares where the dividing lines are at right angles, such as a chess board, and parts of these squares help form the pattern;
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| ”An elaborate variation of the ‘swastika’ motif derived from the elementary square grid.” (David Wade, Pattern in Islamic Art, 1976; # PIA 015 accessed at http://patterninislamicart.com/drawings-diagrams-analyses/6/pattern-islamic-art/pia015) |
b) the lines that divide the space to be decorated are drawn at 60 degree angles forming equilateral triangles. The overall patterns are hexagonal in shape and can form 6-sided stars;
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| Some of the many patterns using a basic hexagonal motif. (David Wade, Pattern in Islamic Art, 1976; # PIA 023 accessed at http://patterninislamicart.com/drawings-diagrams-analyses/6/pattern-islamic-art/pia023) |
c) the third type of patterns is derived from octagons; the complicated octagonal patterns usually have octagons of two sizes and can also form 8-pointed stars;
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| The octagon and some of its variations. (David Wade, Pattern in Islamic Art, 1976; # PIA 034 accessed at http://patterninislamicart.com/drawings-diagrams-analyses/6/pattern-islamic-art/pia034) |
and d) arabesques are the fourth class of patterns and occur when the dividing pattern lines run in more than four directions.(12)
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| An interlacing arabesque pattern with its construction diagram. (David Wade, Pattern in Islamic Art, 1976; # PIA 0100 and PIA 079 accessed at http://patterninislamicart.com/drawings-diagrams-analyses/6/pattern-islamic-art/pia100 and http://patterninislamicart.com/drawings-diagrams-analyses/6/pattern-islamic-art/pia079) |
In all Islamic tile patterns, “[the] artist simply uses the full but confined surface of the material in the chosen technique in a particular mathematical structure, thus creating a balanced and often rhythmic image consisting of human, floral or abstract figures in a seamless, pattern-like design. The repetition of this pattern offers a sense of tranquillity, movement or even infinity... .”(13)
A.J. Lee uses star patterns to illustrate geometric patterns in the formation of Islamic tessellations.
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| Drawings of star patterns (clockwise from UL): two early rectilinear star patterns; a colinear link between two 9-pointed stars; a parallel link between two 10-pointed stars; an eighth century pattern with curvilinear 8-pointed stars; 12-pointed stars on a grid of squares; and 12-pointed stars on a triangular grid. (A.J. Lee, “Islamic Star Patterns”, Muqarnas, Vol. 4, 1987, pp. 182-197; http://www.jstor.org/stable/1523103) |
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| Detail of a tessellated 8-pointed star tile pattern with interlacing band decorations. (Garden path in the Alhambra, Granada, Spain; Photo credit: Michael Padwee, 2006) |
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| Section of a 15th century tiled pavement from Seville. (Museu de Cerámica i de les Arts Decoratives, Barcelona, Spain; Photo credit: Michael Padwee, 2006) |
“[...The] most typically ‘Islamic’ of all star motifs...is the geometrical rosette... .
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| 12- and 6-rayed geometrical rosettes and a 6-rayed rosette seen as a discrete motif. “The prototype for the general n-rayed rosette almost certainly consisted of a 6-pointed star surrounded by six regular hexagons... . [...The] prototypical 6-rayed rosette seems to have been used for the first time as a distinct motif on the Arab-Ata mausoleum (978) at Tim, in Uzbekistan... .” (A.J. Lee, “Islamic Star Patterns”, Muqarnas, Vol. 4, 1987, pp. 188, 183; http://www.jstor.org/stable/1523103) |
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| Мавзолей Араб-Ата в селении Тим (the Arab-Ata Mausoleum), Tim, Uzbekistan. (Photo credit: TripAdvisor) |
Dutch artist Maurits Cornelis Escher (1898-1972) and his wife, Jetta, visited the Alhambra in Granada, Spain in 1922, and again in 1936. On these trips Escher saw Islamic-patterned tile installations first hand, and he was inspired by the patterns he saw. “[Escher] was fascinated by the effect of colour on the visual perspective, causing some motifs to seem infinite–an effect partly caused by symmetry [see endnote 5]. Despite his efforts in the...years [following 1922], Escher failed to understand the principles behind tessellation. Only between 1937 and 1942 [did] he succeed..in doing so, after he had visited the Alhambra for [the] second time... .”(14)
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| Tessellations, arabesques and calligraphy on a wall of the Myrtle Court in the Alhambra. (Photo credit: By Jebulon - Own work, Public Domain, https://commons.wikimedia.org/w/index.php?curid=11234546) “In the Alhambra (14th C), Spain..., geometric pattern is perfectly integrated with biomorphic design (arabesque) and calligraphy. These are the three distinct, but complementary, disciplines that comprise Islamic art. They form a three-fold hierarchy in which geometry is seen as foundational. This is often signified by its use on the floors or lower parts of walls, as shown in the image above.” (Richard Henry, “Geometry – The Language of Symmetry in Islamic Art”, Art of Islamic Pattern, p. 3; http://artofislamicpattern.com/resources/educational-posters/) |
“Escher...was fascinated by every kind of tessellation—regular and irregular—and took special delight in what he called ‘metamorphoses,’ in which the shapes changed and interacted with each other, and sometimes even broke free of the plane itself."(15) Escher was very successful at depicting the real world in a 2-dimensional plane as well as at translating the principles of regular division onto a number of 3-dimensional objects such as spheres, columns, and cubes. Also, some of his prints combine both 2 and 3-dimensional images.(16)
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| Tiled panel in the Queen’s Chamber in the Alhambra. (Photo credit: Michael Padwee, 2006) |
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| Tile installations, the King’s Chamber in the Alhambra. (Photo credit: Michael Padwee, 2006) |
Escher experimented with simple repetition of a plane and mirror images to create his own tessellations. "In many of Escher’s tessellations, there is a clear reference to the geometric shapes and principles used in the Islamic art that he saw during his visit [to the Alhambra]. In particular, Escher’s circular tessellations reflect the Islamic principle of eternity, as..., in theory the pattern can [be] repeated infinitely as the circle expands.”(17)
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| Detail of tessellated, glazed ceramic tiles forming colorful geometric patterns in the Alhambra. (Photo credit: By Dmharvey - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=1980257) |
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| A starfish tessellation by M.C. Escher.(17) (Photo from Pinterest via http://edtech2.boisestate.edu/meganhoopesmyers/502/virtualtour/akvtour.html) |
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| For Escher, “the infinite regular repetition of the tiles in the hyperbolic plane, growing rapidly smaller towards the edge of the circle, [...] allow[ed] him to represent infinity on a two-dimensional plane.” (Hyperbolic tessellation (18): Circle Limit III, 1959; Photo Credit: By Fair Use, https://en.wikipedia.org/w/index.php?curid=7516812) The process of creating hyperbolic drawings by hand was extremely difficult and tiresome for Escher (19) much more so than creating Euclidian tilings. Escher did not have the help of computers, which would have greatly aided him at the time.(20) |
Islamic tile patterns were created using geometric grids, and "Escher mastered his skills on geometric grids and used them as the basis for his sketches, later improving them with additional designs, mainly animals such as birds, lions, and reptiles."(21) The following set of drawings illustrates how Escher could modify the geometric grids found in Islamic art to form one type of tessellation:
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| The tessellation pattern is created by cutting portions of the pattern as in (b) and (d), and mounting them to the correct locations of the pattern considering the rotations and reflections as in (c) and (e). Finally, the pattern is rendered in tile as illustrated in (f). (21) |
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| "Metamorphosis II", Woodcut, 1940, 7.6 in × 153.3 in. (By Official M.C. Escher website., Fair use; http://www.mcescher.com/gallery/switzerland-belgium/metamorphosis-ii/) "[...The] concept of this piece is to morph one image into a tessellated pattern and then slowly alter that pattern eventually to become a new image. The process begins left to right with the word metamorphose (the Dutch form of the word metamorphosis) in a black rectangle, followed by several smaller metamorphose rectangles forming a grid pattern. This grid then becomes a black and white checkered pattern, which then becomes tessellations of reptiles, a honeycomb, insects, fish, birds and a pattern of three-dimensional blocks with red tops. These blocks then become the architecture of the Italian coastal town of Atrani (see Atrani, Coast of Amalfi). In this image Atrani is linked by a bridge to a tower in the water, which is actually a rook piece from a chess set. There are other chess pieces in the water and the water becomes a chess board. The chess board leads to a checkered wall, which then returns to the word metamorphose."(https://en.wikipedia.org/wiki/Metamorphosis_II) |
"There is a beauty in discovery. There is mathematics
in music, a kinship of science and poetry in the descrip-
tion of nature, and exquisite form in a molecule.
Attempts to place different disciplines in different camps
are revealed as artificial in the face of the unity of
knowledge. All literate men are sustained by the
philosopher, the historian, the political analyst, the
economist, the scientist, the poet, the artisan and the
musician." (Glenn T. Seaborg, scientist, Nobel Laureate)
NOTES:
1. I am not a mathematician, but my discussion today necessarily deals with mathematical concepts. I apologize in advance for any lack of explanatory ability on my part in the use of those concepts. For a database of Islamic tile patterns along with explanations, please see http://patterninislamicart.com/about. Also, for a brief explanations of the different types of tessellations, see "Tessellation: The Geometry of Tiles, Honeycombs and M.C. Escher" by Robert Coolman, http://www.livescience.com/50027-tessellation-tiling.html.
2. Claude Bragdon, “Ornament from Mathematics Part I: The World Order” in Architecture and Democracy, Alfred A Knopf, New York, 1918, p. 78.
3. “Escher meets Islamic Art”; https://alevelreblog.wordpress.com/ 2013/09/17/escher-meets-islamic-art/.
4. Ibid.
5. “One way to describe symmetry is to say that it is harmony or beauty of form that results from balanced proportions. More technically, symmetry is a correspondence between different parts of an object. For a geometric object, symmetry is a correspondence between pairs of points that are equally positioned about a point, line or plane.” (http://mathstat.slu.edu/escher/index.php/ Introduction_to_Symmetry)
6. A Google translation from the Portuguese of http://chocoladesign.com/padronagens-na-cultura-islamica.
7. An ornemanist ( sometimes called an "ornamental") is an artist or craftsman who conceives or performs ornamentation, ornaments . We meet ornamentalists in architecture , sculpture , cabinetmaking , typography ... (A Google translation of https://fr.wikipedia.org/ wiki/Ornemaniste)
8. David Wade, “The Evolution of Style”; http://patterninislamicart. com/background-notes/the-evolution-of-style.
9. A.J. Lee, “Islamic Star Patterns”, Muqarnas, Vol. 4, 1987, p. 182; http://www.jstor.org/stable/1523103.
10. From: http://mathstat.slu.edu/escher/index.php/ Fundamental_Concepts. “A polygon in the plane is a closed figure made by joining line segments. The segments may not cross, and each segment must connect to exactly one other segment at each endpoint.

The left figure is not closed, and the figures in the middle are not made of line segments. The figure on the right is not a polygon, since its sides intersect each other.
Vertex
A vertex of a polygon is a point where two sides come together.
A fundamental characteristic of any polygon is the number of sides it possesses, which is the same as its number of angles."
11. A. J. Lee, p. 183.
12. E. H. Hankin, M.A., “On Some Discoveries of the Methods of Design Employed in Mohammedan Art”, Journal of the Society of Arts, March 17, 1905, pp. 461-465; accessed at http://patterninislamicart.com/drawings-diagrams-analyses/3/methods-design.
13. Aya Johanna Dani Ëlle Durst Britt, “Optical Illusion as a Bridge to Infinity: Escher Meets Islamic Art”, Al.Arte, July 25, 2013, p. 4; http://www.alartemag.be/en/en-art/optical-illusion-as-a-bridge-to-infinity-escher-meets-islamic-art/.
14. Aya Johanna Dani Ëlle Durst Britt, p. 5.
15. E. H. Hankin, M.A., “On Some Discoveries of the Methods of Design Employed in Mohammedan Art”, Journal of the Society of Arts, March 17, 1905, pp. 461-465; accessed at http://patterninislamicart.com/drawings-diagrams-analyses/3/ methods-design.
16. Faith Gelgi, "The Influence of Islamic Art on M.C. Escher", Fountain Magazine, Issue 76 / July - August 2010; http://www.fountainmagazine.com/Issue/detail/The-Influence-of-Islamic-Art-on-MC-Escher.
17. Hankin, Op. Cit.
18. Part of an article defining and illustrating hyperbolic tessellations from David E. Joyce, “Hyperbolic Tessellations: Introduction”; http://aleph0.clarku.edu/~djoyce/poincare/ poincare.html:
“A regular tessellation, or tiling, is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex. No doubt, the tessellations of the Euclidean plane are well-known to you. They are: {3,6} in which equilateral triangles meet six at each vertex; {4,4} in which squares meet four at each vertex; and {6,3} in which hexagons meet three at each vertex. A notation like {3,6} is called a Schläfli symbol. There are infinitely many regular tessellations of the hyperbolic plane. You can determine whether {n,k} will be a tessellation of the Euclidean plane, the hyperbolic plane, or the elliptic plane by looking at the sum 1/n + 1/k. If the sum equals 1/2, as it does for the three tessellations mentioned above, then {n,k} is a Euclidean tessellation. If the sum is less than 1/2, then the tessellation is hyperbolic; but if greater than 1/2, then elliptic.
...The hyperbolic plane can not be metrically represented in the Euclidean plane, but Poincaré described ways that it can be conformally represented in the Euclidean plane. One of those is to represent the hyperbolic plane as the points inside a disk. For this representation, a straight line in the hyperbolic plane is represented as the part (in the disk) of a circle that meets the boundary of the disk at right angles.
...For instance, here is a representation of the tessellation of the hyperbolic plane by pentagons where four pentagons meet at each vertex, that is, the {5,4}-tessellation.
It may look like the sides of the pentagons are curved, but that's just because of the representation we're using. In the actual hyperbolic plane they would be straight. Also, the pentagon in the middle looks larger, but, again, that's due to the representation. You just can't put an infinite plane in a finite region without a lot of distortion.”
19. Escher’s hyperbolic tessellation, Circle Limit I, below, with two drawings: one showing the graphing of the triangular hyperbolic plane pattern, and the other showing the central “supermotif” for Circle Limit I. (Douglas Dunham, “Creating Repeating Hyperbolic Patterns—Old and New”, Notices of the AMS, Volume 50, Number 4, April 2003, pp. 453, 454; http://www.ams.org/notices/200304/ fea-escher.pdf--this article contains a complete mathematical explanation for the creation of this pattern.)
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| Circle Limit I, the Cayley graph of the group [6,4], and the central “supermotif ” for Circle Limit I. |
20. Jos Leys, “Mathematical Imagery, “Hyperbolic Escher”; http://www.josleys.com/show_gallery.php?galid=325.
21. Faith Gelgi, "The Influence of Islamic Art on M.C. Escher", Fountain Magazine, Issue 76 / July - August 2010; http://www.fountainmagazine.com/Issue/detail/The-Influence-of-Islamic-Art-on-MC-Escher.
22. Faith Gelgi, Ibid.
23. Faith Gelgi, Ibid.
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I'd like to thank my friend, Robert Ellis, a mathematics professor at New York University and a union representative for the Adjunct Professors in the UAW, for reading and commenting on this article, and photographer Chris Boundy for the use of his photo.
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The Historic Moyer House (1907) For Sale
In March 2013 I wrote about the history of Albert Moyer's concrete house and its Moravian tile decorations in South Orange, New Jersey. (https://tilesinnewyork.blogspot.com/2013/03/concrete-and-tiles-i-moyer-mercer-murosa.html) The current owners of the Moyer House are selling it, and the realtor has created a website with photos and a brief video of the house--including its tilework (http://324northridgewoodrd.com/), which is worth visiting.
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