Abstract
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.
Highlights
Imagine a three dimensional object which has square silhouette projections in three orthogonal directions. (a) (b)A cube has this property, but it is not the only object and there are plenty of examples like a cuboctahedron and a regular tetrahedron as Figure 1(a,b) shows
We have the observation that a minimal convex imaginary cube of C is a polyhedron such that all the vertices are on C-edges and each C-edge contains at least one vertex
(1) Two minimal convex imaginary cubes of C are equivalent if they have the same set of v-vertices
Summary
We have the observation that a minimal convex imaginary cube of C is a polyhedron such that all the vertices are on C-edges and each C-edge contains at least one vertex. (1) Two minimal convex imaginary cubes of C are equivalent if they have the same set of v-vertices. There are 256 subsets of the set of vertices of C and 23 subsets if we identify rotationally equivalent ones, and one can obtain a convex imaginary cube for each of them by selecting one e-vertex on each
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