An often overlooked characteristic of Sinan's architectural production of the 16^th century is the use of low screening walls and windows carved of marble with pierced openwork or tracery to let in light and allow for air circulation. Typically these screens display polygonal networks often with intersecting polygons. This paper explores aspects of these carved patterns, which have antecedents in earlier Islamic architecture, attributing particular significance to this architectural feature in the works of Sinan and three successors whom he had trained. These slabs (and their counterparts in forged iron) may be considered études, serving as opportunities for the training of the royal corps of architects in what was then called the “science of geometry,” documenting a sustained concern with polygonal networks half a century before publication of Kepler's Harmonices Mundi in 1619.