Consider a polycube where all faces are rectangles. In addition to a cuboid, there is the unfolding of a tesseract, the approximation of an octahedron by cubes, and the object below. The component cuboids each have two ends with the same rectangle, namely the repeated $1\times2$, $1\times3$, and $1\times5$ rectangles.
What polycubes exist where all faces are distinct rectangles? Here are the six orthographic views of one solution:
This solution has 30 rectangles, is bound by a size 16 cube, and has maximal rectangle edge 15. Can any of these numbers be reduced? Some raw data for the below solution is at Thirty Distinct Rectangle Polyhedron.
