The main goal of this article is to present some geometric constructions that have been performed on the sphere by a medieval Persian mathematician, Abul Wafa al-Buzjani, which is documented in his treatise On Those Parts of Geometry Needed by Craftsmen. These constructions, which have been illustrated as flat images, could be considered the bases of the arts and designs that artists and artisans have created on both the exterior and interior surfaces of a dome. Therefore, such a dome art design is a result of cooperation between mathematicians and artists. This article also shows that the construction of the icosahedron on a sphere presented in that treatise is not mathematically correct. However, the construction of the spherical dodecahedron is exact. The article also presents flat images of constructions of some Archimedean solids according to the treatise.