The conventional view holds that geometric star-and-polygon patterns adorning medieval Islamic architecture were designed using a straightedge and a compass, derived from ancient Greek mathematical techniques, which describes accurately a large number of examples. However, a wide variety of patterns throughout the medieval Islamic world with five- and ten-fold symmetric motifs, were conceived as tessellations of specific decorated polygons, called girih tiles, that appear in architectural scrolls from the 15th century, that documented practical architectural techniques and emerged in the context of dialogs between architects and mathematicians. These girih tiles were used like puzzle pieces to design a large number of geometric tiling patterns in architecture, beginning with 12th-century architectural monuments in the Anatolian Seljuk cultural context, in what is today western Iran and eastern Turkey. Interestingly, perhaps half of the extant Seljuk buildings with girih-tile tesselations had female sponsors, patrons or dedications—hinting at an important contribution from women that has not been previously explored adequately. Subsequently, these girih-tile tessellations spread throughout the medieval Islamic world, from North Africa to the Middle East and Central Asia, for half a millennium. Remarkably, in a few cases, these girih-tile tessellations exhibit mathematical principles of quasicrystallinity, that we in the West did not understand until the past few decades in the form of Penrose tilings.