## 1 Introduction  
Imagine a three dimensional object which has square silhouette projections in three orthogonal directions. **Figure 1.** (**a**) A cuboctahedron; (**b**) a regular tetrahedron--the best-known imaginary cubes. 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 call an object an _imaginary cube_ if it has square projections in three orthogonal directions just as a cube has. Among imaginary cubes, there are two objects with some remarkable geometrical properties, which are a hexagonal bipyramid imaginary cube (Figure 2(a)) and a triangular antiprismoid imaginary cube (Figure 2(b)). A hexagonal bipyramid imaginary cube, which we call H in this paper, is composed of two copies of a regular hexagonal pyramid whose side faces are isosceles triangles with the height \(3/2\) of the base. A triangular antiprismoid imaginary cube, which we call T, is obtained by truncating the three vertices of one base of a regular triangular prism whose height is \(\sqrt{6}/4\) of an edge of a base, so that each truncated face passes through middle points of two edges of the base and a vertex of the opposite base. One can see that copies of these polyhedra scaled as in Figure 2 can be put into a cubic box with the side length 1 as Figure 3 shows, and they are imaginary cubes. As these examples show, it is sometimes difficult to realize at a glance that a given polyhedron is an imaginary cube, but once it is in a box, it becomes obvious by looking at it through the faces of the  
Figure 3: The objects H and T can be put into a cubic box. See also No. 5 and No. 8 of Table 1.  
Figure 2: Two imaginary cubes; (**a**) H (hexagonal bipyramid); (**b**) T (triangular antiprismoid). box. The author has had exhibitions at Kyoto University Museum and at other places and has displayed models of imaginary cubes. He asked guests to put them in a clear cubic box, and realized that it forms a good mathematical puzzle. In particular, it is the case when the object is a minimal convex imaginary cube, which is a convex imaginary cube such that if it is further truncated then it no longer has the same three square projections. All the four polyhedra in Figures 1 and 2 are minimal convex imaginary cubes. In the next section, we study minimal convex imaginary cubes and show that there are 16 (or 15 if we identify reflectively congruent ones) equivalence classes of minimal convex imaginary cubes with a natural representative in each class. Among imaginary cubes, H and T have remarkable geometrical properties as we show in Section 3. The author designed an assembly puzzle based on them, which is called the imaginary cube puzzle \(3\mathrm{H}=6\)T. The pieces of this puzzle are shown in Figure 4. We use three copies of H and six copies of T scaled as in Figure 2 and a cubic box with the side length 2. The objective of this puzzle is to put all the nine pieces into the box. Since H and T are imaginary cubes, it is immediate that one can put eight of them. Here, we have nine pieces and it forms a mathematical puzzle. In this paper, we add constraints to the ways the pieces are put into the box so that it only has solutions with good geometrical properties. One is that if two pieces meet, then their intersection is a face or an edge or a vertex of both of the piece. It means that the collection of the pieces and their faces, edges, and vertices form a polytopal complex [1], and we call it the polytopal complex condition. Another one is that the union of the pieces forms a star-polyhedron with respect to the center point of the puzzle box. Here, a polyhedron \(Q\) is a star-polyhedron with respect to a point \(p\) if every line segment connecting \(p\) and a point of \(Q\) is contained in \(Q\). We call it the star-polyhedron condition. Under these conditions, this puzzle has a unique solution up to the rotation of the cube. We analyze this solution based on geometrical properties of H and T in Section 4.