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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.

rectangular polycube

What polycubes exist where all faces are distinct rectangles? Here are the six orthographic views of one solution:

polycube with distinct rectangles

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.

thirty rectangles

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  • $\begingroup$ Isn't your first picture a polycube where all faces are distinct rectangles? $\endgroup$ – Joel Reyes Noche Sep 23 '16 at 0:43
  • $\begingroup$ Do you perhaps mean that all face dimensions are distinct? That is, for example, only one rectangle side is of length 1? $\endgroup$ – Joel Reyes Noche Sep 23 '16 at 0:45
  • $\begingroup$ The first picture has two 1x2 rectangles. $\endgroup$ – Ed Pegg Sep 23 '16 at 1:30
  • $\begingroup$ 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.") $\endgroup$ – Joel Reyes Noche Sep 23 '16 at 3:58

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