We introduce a design methodology for nonimaging, single-reflection mirrors with polygonal inlet apertures that generate a uniform irradiance distribution on a polygonal outlet aperture, enabling a multitude of applications within the domain of concentrated photovoltaics. Notably, we present single-mirror concentrators of square and hexagonal perimeter that achieve very high irradiance uniformity on a square receiver at concentrations ranging from 100 to 1000 suns. These optical designs can be assembled in compound concentrators with maximized active area fraction by leveraging tessellation. More advanced multi-mirror concentrators, where each mirror individually illuminates the whole area of the receiver, allow for improved performance while permitting greater flexibility for the concentrator shape and robustness against partial shading of the inlet aperture.