Quasiperiodic ornamental patterns represent only a small percentage of patterns when compared to the entire body of periodic patterns. Decagonal pattern is known since twelfth century Iran and fourteenth century western Islam (Andalusia and Morocco). A rich spectrum of octagonal patterns exists at the latter localities (fourteenth century and later), whereas a sole example of a dodecagonal pattern comes from Morocco. Later copies exist in all these regions. My most recent studies were concentrated upon the Andalusian and Moroccan regions, in which the fourteenth century (and later) wall mosaics occur as uninterrupted coatings of entire walls so that the motif of individual panels had to be adjusted to secure continuity of their underlying bar-and-band structure. In Andalusia, the tetragonal structure of the panels and their complexes were locally adjusted to become octagonal quasiperiodic. Only two geometric types of such octagrids were derived in Andalusia, in agreement with the rarity of quasiperiodic ornaments in general. In Morocco, before the panel substructure became heavily masked by an overflow of rosettes of several sizes, the mosaic panel was based on an octagonal quasiperiodic grid and ornamental rosettes were placed in it, disposed in the form of concentric octagons. As a prominent example, the octagonal motif of the Nejjarine Fountain and its plaster encasement will be discussed.