Of the mosques and other structures created by the Medieval Spanish Muslims, none quite surpassed the majesty
of The Alhambra -- the ``red one'' in Arabic, honored as the ``Jewel of Moorish Spain'' -- originally built by the Nasarides in Granada, beginning in
about the 13th century.  It was both a palace and citadel set upon a hill, protected by a forest, and housed gardens, fountains and a mosque.   
Though under Muslim control for over two centuries, in 1492 it was conquered by the armies of King Ferdinand II and Queen Isabella I,
then soon remade into a Christian court, complete with a Franciscan monastery.  Despite the refashioning, 
much of the original art of the Nasarides survived the ages.

Among such arts, revealing great sophistication and craftsmanship, were the arabesques -- 
decorative tiles -- painted with dazzling geometric shapes, and fitted together 
to form endlessly repeating patterns.  Perhaps such forms took inspiration from a Muslim-influenced belief in the 
ascendence of cosmic laws, uniform and perfect as the motion of the planets and stars, extending out to the very boundaries
of the world.

Understanding why Medieval European Muslims chose to focus on such abstract forms, rather than the
human body (as in other European cultures), has been a question many have
asked.  Perhaps, as with their beliefs about physical law, the answer also lies in religion:  ``Angels do not enter a house in which 
there are pictures,'' says one hadith, a kind of oral tradition in Islam; while another says that on Judgement Day all artists 
will be called before god to breathe life into their works, and failing that, as they are not divine, they will be 
condemned to Jahannam -- hell.  Though these hadiths were intended for idolatry artists, perhaps they worried the artisans of the
Alhambra enough to avoid portraits.

Whatever the reasons for their subjects, history has proved their choices good ones, for not only 
do we even now recognize their timeless beauty, but they have influenced and enriched many generations of artists and mathematicians
down through the centuries.  Perhaps most notable among these was M. C. Escher, the Dutch sketch artist whose tessellation drawings arose 
from a transformative vacation he took to Alhambra in 1922.  

And for mathematicians, with the development of ``higher algebra'', tilings such as those in Alhambra became an interest.
With time, they were analyzed and classified according to the different types of symmetries they enjoyed:  Any regular tiling has associated to it 
certain symmetries, such as translation, rotation, reflection, and glide-reflection; for example, ``translation symmetry'' refers to the fact that if 
all your tiles are squares of width 1 foot on a side, then if you took an infinite wall (infinite in all directions) covered with these tiles, and 
moved it to the left by 1 foot, it would appear to be the exact same wall as before.  In fact, there are an infinite number of symmetries corresponding 
to each tiling, as every tiling has an infinite number of translational symmetries alone.  Yet, using the mathematics of ``group theory'', this 
infinite list can be compactly represented by a short, finite set of ``generators'', which one can think of as a kind of ``genetic code'' for the tiling.

By the late 19th and early 20th century a stunning discovery was made regarding this ``genetic code'' by Evgraf Fedorov in 1891, and 
independently by George Polya in 1924:  It turns out that there are exactly 17 different types; that is, there are only 17 different
``species'' of tilings!   There are three dimensions of space, five Platonic solids, ... and exactly 17 kinds of tilings.  There 
is something transcendent in the definiteness of such things.

Among Alhambra's secrets that came to light not so long ago was that fact that it contains all the 17 species along its ornamented walls.  
Perhaps no other structure in the world from the medieval era or earlier can make such a boast -- if others exist they are very few in number.  
It of course makes one wonder whether Alhambra's tilers, and perhaps also Moorish artists from even earlier ages, had such knowledge.  
Certainly, they were not aware of ``group theory'', but perhaps they had an intuition about the limitations of their craft.   If so, then perchance 
could they have left other insights for us to re-discover in their works?  Only time will tell...

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UPDATE:  According to John Baez, in an article on his website (http://math.ucr.edu/home/baez/week267.html), Alhambra actually only 
contains 13 of the 17 symmetries.  If so, then that is still very impressive, and still makes Alhambra unique among structures (as far as
is known) from the medieval period or earlier.