The Myth:
It is frequently contended that the Alhambra palace in Granada, Spain contains, amongst its various ornamental patterns examples with all 17 types of periodic plane symmetry groups (i.e. those containing two directions of translational symmetry). A related myth is that the 17 symmetry groups were known, at least in an empirical way, to Islamic artisans.
The Reality:
This is actually a topic of much debate, and not having been there myself, I have to rely on the relative merits of the arguments put forth by others. The most compelling argument I'm aware of is due to Branko Grünbaum1 in an article [Gr06] written in response to Montesinos' [Mo87]. The crux of Grünbaum's argument is two-fold: (1) He went there, spent several days looking, and only found 13 symmetry groups represented and (2) attempts to find all 17 are marred by inconsistencies about the role of color in the patterns (ignoring them or using them as convenient, but without consistency in methodology). While the first point shouldn't be particularly convincing to anyone, after all, there are a lot of ornaments in the Alhambra, perhaps he just didn't notice some, the second is critical as it speaks to the role of evidence. Evidence of absence is usually impossible to obtain, whereas fantastic claims (and wouldn't it be fantastic if all 17 were in evidence?) require at the very least good evidence. Given the internal inconsistencies that appear in the published accounts by those supporting the view that the Alhambra bears exemplars of all 17 symmetry types in its ornamentation, we reject the hypothesis.
Grünbaum also raises another very interesting point, whether or not it is possible to find examples of all 17 types somewhere amongst the ornaments used in classical Islamic art using a consistent methodology, this doesn't mean that Islamic artisans were in fact aware of these symmetry groups in even a rudimentary form. Not only is group theory a much more recent invention, and so the language of groups was unavailable to these artisans, the conceptual standpoint of group theory ignores the usual practices and oral traditions surrounding how patterns are created in most cultures, namely via a system of local rules about how elements fit together as opposed to being conceived of as an instantiation of a particular kind of symmetry.
References:
[Gr06] B. Grünaum. What symmetry groups are present in the Alhambra? Notices of the AMS, 53(6):670–673, 2006.
[Mo87] J. Montesinos. Classical tesselations and three-manifolds. Springer, New York, 1987.
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(1) In the interests of full disclosure, Branko Grünbaum was my thesis advisor, so perhaps I'm biased. I would like to think however that while this may have some influence on my opinions in this matter, the influence hasn't clouded my thinking too much.