## 2 Minimal Convex Imaginary Cubes  
We start with defining a (minimal convex) imaginary cube of a cube \(C\). **Definition 1**. : _For a cube \(C\), we say that an object is an imaginary cube of \(C\) if it has square projections in three orthogonal directions just as \(C\) has. We say that an object is a minimal convex imaginary cube of \(C\) if it is minimal among convex imaginary cubes of \(C\) with respect to the inclusion order. We say that an object is a (minimal convex) imaginary cube if it is a (minimal convex) imaginary cube of some cube._  
Note that a minimal convex imaginary cube is not minimal among convex imaginary cubes with respect to the inclusion order because it contains an arbitrary small cube which is an imaginary cube. Therefore, we first fix a cube \(C\) and define a minimal convex imaginary cube of \(C\) in defining a minimal convex imaginary cube. Since an object is convex if it is the intersection of half-spaces where a half-space is one side of a plane, an object is a convex imaginary cube of \(C\) if it is obtained by cutting off \(C\) with planes so that it has the same square projection images as \(C\). A convex imaginary cube of \(C\) is minimal if any more cut with a plane will destroy one of the three square projection images. We fix a cube \(C\) and study the problem of characterizing minimal convex imaginary cubes of \(C\). It is immediate that a convex object is an imaginary cube of \(C\) if and only if it is in \(C\) and has intersections with all the twelve \(C\)-edges. Therefore, a minimal convex imaginary cube of \(C\) is the convex hull of its intersection with the \(C\)-edges, which is a polyhedron. Thus, 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. Since a minimal convex imaginary cube is a polyhedron, we only need to specify the set of vertices to identify it. Since each vertex is on a \(C\)-edge, vertices of a minimal convex imaginary cube of \(C\) are divided into two categories; v-vertices which are also vertices of \(C\), and e-vertices which are not vertices of \(C\). If an e-vertex and another vertex of a minimal convex imaginary cube are on the same \(C\)-edge, then we can remove the e-vertex to have a smaller convex imaginary cube of \(C\) and contradict to the minimality condition. Therefore, there is at most one e-vertex on each \(C\)-edge, and if there is a v-vertex on a \(C\)-edge, then there is no e-vertex on that edge. In this way, each minimal convex imaginary cube \(P\) of \(C\) determines a subset \(S\) of vertices of \(C\) which is the set of v-vertices of \(P\), and \(P\) has one e-vertex on each \(C\)-edge both of whose endpoints are not in \(S\). Thus, we define equivalence of minimal convex imaginary cubes as follows. **Definition 2**. :
1. _Two minimal convex imaginary cubes of_ \(C\) _are equivalent if they have the same set of v-vertices._
2. _Two minimal convex imaginary cubes of_ \(C\) _are rotationally (or reflectively) equivalent if they are equivalent minimal convex imaginary cubes of_ \(C\) _modulo rotations (or reflections) of_ \(C\)_._  
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 \(C\)-edge both of whose endpoints are not in \(S\). However, not all the imaginary cubes obtained in this way are minimal. For a vertex \(a\) of \(C\), we denote by \(\mathrm{star}_{a}\) the set of vertices which is composed of \(a\) and all the three adjacent \(C\)-vertices of \(a\), and call it a star. If \(S\) contains a star \(\mathrm{star}_{a}\), then an imaginary cubeobtained in this way is not minimal because we have a smaller convex imaginary cube by removing \(a\) from the set of vertices.