Construction and fabrication of reversible shape transforms

We study a new and elegant instance of geometric dissection of 2D shapes: reversible hinged dissection, which corresponds to a dual transform between two shapes where one of them can be dissected in its interior and then inverted inside-out, with hinges on the shape boundary, to reproduce the other shape, and vice versa. We call such a transform reversible inside-out transform or RIOT. Since it is rare for two shapes to possess even a rough RIOT, let alone an exact one, we develop both a RIOT construction algorithm and a quick filtering mechanism to pick, from a shape collection, potential shape pairs that are likely to possess the transform. Our construction algorithm is fully automatic. It computes an approximate RIOT between two given input 2D shapes, whose boundaries can undergo slight deformations, while the filtering scheme picks good inputs for the construction. Furthermore, we add properly designed hinges and connectors to the shape pieces and fabricate them using a 3D printer so that they can be played as an assembly puzzle. With many interesting and fun RIOT pairs constructed from shapes found online, we demonstrate that our method significantly expands the range of shapes to be considered for RIOT, a seemingly impossible shape transform, and offers a practical way to construct and physically realize these transforms.