Imaginary Cubes and Their Puzzles

Imaginary cubes are three dimensional objects which have square silhouette projections in three orthogonal ways just as a cube has. In this paper, we study imaginary cubes and present assembly puzzles based on them. We show that there are 16 equivalence classes of minimal convex imaginary cubes, among whose representatives are a hexagonal bipyramid imaginary cube and a triangular antiprism imaginary cube. Our main puzzle is to put three of the former and six of the latter pieces into a cube-box with an edge length of twice the size of the original cube. Solutions of this puzzle are based on remarkable properties of these two imaginary cubes, in particular, the possibility of tiling 3D Euclidean space.