you mention Muslims. like everything else, there is a complex history there that goes beyond a pretend religious insularity. Islamic extension (al-mad al-islami) first reached the Syrian lands where overlapped Byzantine Christianity with a hoard of other eastern churches and religious traditions reflecting the myriad regional and historical interfaces with the region (Sassanid, Egyptian, Yemenite, greco-roman, Byzantine...etc). The establishment of the first Islamic empire, the Ummayad's, incorporated these elements including those architectural. it would thus be incorrect to try and posit a solely western or eastern, Christian or Islamic history either in the context of architecture or indeed in the larger history of people. Let us not forget that Byzantine history itself included a tradition of iconoclasm and the empire was ruled on two occasions by iconodules. Thus a geometric cross fertilisation is very feasible in place of a mainstream figural one although if I recall correctly Byzantine iconoclasm might have been instated in Byzantium after Islamic iconoclasm was instated in syria-palestina. Anyway, the ummayads then spread outwards from damascus to north africa (with its own rich admixture of mediterranean, berber and african cultures) and thereafter reaching Iberian lands. This is the origin of the Moorish Arabs. 

So, if you were to take seriously this interest you have, the best route is to trace trajectories of travel and influence across cultures, lands and religions as opposed to limiting yourself to fake dichotomies (west and east, Christianity and Islam, Europeans and Arabs). In fact, did you know that a culturally rich period of the ancient Roman history was presided over by Syrians (septimus Severus and family) and that the very first Roman emperor to be sympathetic to the spread of Christianity was Phillip the Arab? And the history of Islamhas its fair share of blonde northeners and Europeans. 

To reiterate some of what tammuz said:  It is a challenging, but fun quest and as your (hopefully unbiased) curiosity digs deeper into the archeology of this architectural typology you will find it is due to a lot of cultural cross-breeding spanning Northern Africa, the Iberian peninsula, Mediterranean, Near East, et al. (which by the way are Western classifications).

Might be a spoiler alert, but in my past searching I have not come to the conclusion that the intricate designs, whether they be structural or not, are not from pure idea. Quite the opposite, actually.

Also, another interesting thing about domes is how they resolved the cube (square) to dome (circle) transition.

Good luck.

Thank You for your comments, suggestions and wishes.

As I am informed, groin vault was first used by Romans and became widespread in Romanesque architecture, earlier churches being mostly vaulted (if they were vaulted at all) with simple barrel vaults. It was also used in Byzantine architecture, mostly for galleries, and sometimes it involved a more complicated outline (for instance, when octagonal galleries surounding the dome space were layed on them, it was necessary to cover a non-squared space by adding the third vault-element on the wider side, like in San Vitale in Ravenna. In Charlemagne's Palatine Chapel, simplified solution was used: between squared spaces there were triangle-shaped spaces with kind of a groin vault matching its shape).

I didn't actually mean that they invented groin or ribbed vault, but I am questioning idea that somehow they influenced the concept of complex vaults, ie. "creativity" in vault design that appears later in Gothic architecture. What I cannot find is predecessor nor succesor of this form, because I don't know for any method of "rounding" the square shape of the domed space except pendentives and squinches (with their variants) used before them. I wouldn't say at all that the origin od these is Middle-Eastern, because Islamic domes of those areas resemble earlier Byzantine, Roman, Syriac and Persian domes (simple-structured oval domes laying on pendentives or squinches), and it remained like that long after these Moorish domes appear (they were actually never introduced to Middle East).

Also, they don't seem to have any direct succesor, because no one inherited the form, it just "died" with them. I only found an information on the Internet (though I'm suspicious about that) that they were studied by Brunelleschi in 1402.

http://www.abc.es/cordoba/20130109/sevp-maqsura-duda-expresion-bella-20130109.html

(it's in Spanish)

What, let me say so, "drives me crazy" is how they came to combine vaults, ribs (sc. arches) and domes in same form? I will surely continue in research.

P. S. I'm not biased, nor unaware of the fact of cultural exchange. I realise that similar architectural forms and elements were used by different countries, religions and civilisations. I call those domes "Moorish" because they seem to appear only in their buildings, and I speculated "Islamic" influence because geometrical design was highly appreciated in Islamic Art, at least in decoration (actually, square-to-circle patterns are often imployed as decorative elements in mosques).