We present explicit equations for three different mappings between the disk and the square. We then use these smooth and invertible mappings to convert the Poincaré disk into a square. In doing so, we come up with three square models of the hyperbolic plane. Although these hyperbolic square models probably have limited use in mathematics, we argue that they have artistic merit. In particular, we discuss their use for aesthetic visualization of infinite patterns within the confines of a square region.