We have defined the infinity boxes in general dimensions, which are mathematical objects that allow rays of light to continuously reflect. When the collision points of these rays are are positioned where they would appear to the viewer (along the inititial ray), then the mirror pattern will be revealed. These mirror patterns can be used to create mirror-tilings of two- and three-dimensional Euclidian space. We have answered the question whether a curved infinity box can create patterns that are nearly identical to exact tilings of Euclidian space. To obtain the answer, we set up the fundamental theory of rays and infinity boxes, and computed the radii of the curved mirror faces that exhibit certain properties. These properties are the alignmnent of internal angles, and the alignment of image points in an infinity box. We have simulated infinity boxes in two- and three-dimensions to visualise the patterns from these infinity boxes curved with these radii. Furthermore, we connected the theory of infinity boxes the theory of dynamical billiards by stating that infinity boxes visualise solutions of dynamical billiards. Lastly, we analysed the connection between the dodecahedron infinity box, and concluded that curving the mirrors of the dodecahedron infinity box results in a slight approximation of the interior of a stereographically projected 120-Cell. All the numerical methods used to visualise infinity boxes have been thoroughly explained, and the computer program that we created called MiRai, is available over the internet.