The KA +/- lA +/-double dagger Ali PaAYa Mosque in Istanbul was completed in the late sixteenth century and it is today regarded as one of architect Sinan's most important works. Like many of Sinan's designs it has been the subject of detailed analysis in the past, with historians offering theories to explain its visual properties and how they differ from those of his previous works. However, such theories have remained largely untested because of the difficulty of quantitatively analysing the building's intricate formal, ornamental and material properties. The present paper uses computational fractal analysis to overcome this problem. This method, as it is applied in the paper, derives a non-integer measure of the characteristic complexity of three hierarchical layers in visual information in the facades of the KA +/- lA +/-double dagger Ali PaAYa Mosque. Through this process the paper not only provides important quantitative results which can be used to test past theories about the building, but these new measures are then compared with those that have previously been developed, using the same method, for Sinan's Suleymaniye Mosque.