We present a novel mapping for converting the Poincaré disk to a square. Although this mapping does not produce results as aesthetically appealing as the conformal mapping, it can be used as a computationally inexpensive substitute to produce satisfactory hyperbolic art. Our mapping is orders of magnitude faster to calculate than the conformal mapping, thereby making our mapping suitable for interactive hyperbolic visualization. We also present some other artistic uses such as the conversion of rectangular paintings into oval regions.