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An argument for this; is that they cannot rely on representational art, as is done in Hindu & Christian tradition. It would be disobeying the Quranic command against idols.

In, a positive sense, was it done after the assimilation of the Greek philosophical tradition once the Islamic civilization began to form and move out of Arabian peninsula, and began assimilating the achievements of other civilizations bordering it? After all, the Plato, the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here". Falsafa was introduced "from the ninth century onward, owing to Caliph al-Ma'mun and his successor, Greek philosophy was introduced among the Arabs, and the Peripatetic school began to find able representatives among them; such were Al-Kindi, Al-Farabi, Ibn Sina (Avicenna), and Ibn Rushd (Averroës). all of whose fundamental principles were considered as criticized by the Mutakallamin. Another trend, represented by the Brethren of Purity, used Aristotelian language to expound a fundamentally Neoplatonic and Neopythagorean world view".

Pythagoras, in his mystic vision, taught that "God geometrises continually". One can speculate, whether the early Islamic philosophers/architects, inheriting this idea, used it to represent the non-representational divine realm. Early Islamic architecture between 621-661 AD, "was characterized by humbleness & simplicity"; presumably reflecting the submissiveness of Islam, and not necessarily the poverty of means.

Edit

A tesselation, geometrically speaking, is a decoration "of a two-dimensional plane using the repetition of a [single] geometric shape with no overlaps and no gaps"; generally it can be extended infinitely in all directions.

An Arabesque consists of "surface decorations based on rhythmic linear patterns of scrolling and interlacing foliage, tendrils or plain lines...[it has] formalistic continuity and development in decorative plant forms from Ancient Egyptian art and other ancient Near Eastern civilizations through the classical World...[They are often combined with] geometric patterns...[Some use] patterns that are made up [purely] of straight lines and regular angles but are clearly derived as a whole from curvilinear arabesque patterns; the extent to which these too are described as arabesque varies between different writers...Many arabesque patterns disappear at (or "under" as it often appears to a viewer) a framing edge without ending,... and thus can be regarded as infinitely extendable outside the space they actually occupy; this was certainly a distinctive feature of the Islamic form".

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According to the Wikipedia entry on Arabesque (what the art form is called): this style of decoration was an Islamic development of Byzantine architecture of the time. Through Byzantium's Leventine outposts the early Muslims would have had ample contact with Byzantine art. Coupled with the fascination Muslims scholars had with math and geometry, the result was an art form that was uniquely Islamic.

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I'm sorry to have to disagree, arabesques are not geometric tesselations, but of course they're a part of Islamic decoration; they can be combined though. –  Mozibur Ullah Dec 10 '12 at 21:04
    
As far as I can tell, Arabesque does include geometric tessellation. Unless we both define tessellation differently? –  System Down Dec 10 '12 at 21:11
    
I think different writers are using tesselation - particularly those in mathematics and art history -in different ways. I'll edit my question to illustrate what I mean; but your link has led me to the right answer anyway. Thanks. –  Mozibur Ullah Dec 10 '12 at 21:22
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