# Polycube with distinct rectangular faces

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.

• Isn't your first picture a polycube where all faces are distinct rectangles? – Joel Reyes Noche Sep 23 '16 at 0:43
• Do you perhaps mean that all face dimensions are distinct? That is, for example, only one rectangle side is of length 1? – Joel Reyes Noche Sep 23 '16 at 0:45
• The first picture has two 1x2 rectangles. – Ed Pegg Sep 23 '16 at 1:30
• Oh, so you're looking for a polycube where the "end faces" are different. (I'm sorry, but I was confused by what you meant by "face.") – Joel Reyes Noche Sep 23 '16 at 3:58