The Mandelbox is a recently discovered class of escape-time fractals that in its standard form has the overall shape of a cube. In this paper we introduce an extension to the Mandelbox iterative algorithm called radial reflection that allows extensive control of the overall shape of the resulting fractal. We then explore how this technique can be used to generate new 3D and 4D fractals, and their use for a particular artistic video style we call hypermandalas.