Can uncorrelated surrounding sound sources be used to generate extended diffuse sound fields? By definition, targets are a constant sound pressure level, a vanishing average sound intensity, uncorrelated sound waves arriving isotropically from all directions. Does this require specific sources and geometries for surrounding 2D and 3D source layouts? As methods, we employ numeric simulations and undertake a series of calculations with uncorrelated circular/spherical source layouts, or such with infinite excess dimensions, and we point out relations to potential theory. Using a radial decay 1/r^b modified by the exponent b, the representation of the resulting fields with hypergeometric functions, Gegenbauer polynomials, and circular as well as spherical harmonics yields fruitful insights. In circular layouts, waves decaying by the exponent b=1/2 synthesize ideally extended, diffuse sound fields; spherical layouts do so with b=1. None of the layouts synthesizes a perfectly constant expected sound pressure level but its flatness is acceptable. Spherical t-designs describe optimal source layouts with well-described area of high diffuseness, and non-spherical, convex layouts can be improved by restoring isotropy or by mode matching for a maximally diffuse synthesis. Theory and simulation offer a basis for loudspeaker-based synthesis of diffuse sound fields and contribute physical reasons to recent psychoacoustic findings in spatial audio.