## 2 Minimal Convex Imaginary Cubes
On the other hand, if \(S\) does not contain a star, then an imaginary cube obtained in this way is a minimal convex imaginary cube of \(C\), because if we remove a vertex then there exists a \(C\)-edge without a vertex. **Theorem 3**. : _There is an one-to-one correspondence between equivalence classes of minimal convex imaginary cubes of \(C\) and subsets of \(C\)-vertices without a star._  
Through the enumeration of such subsets, we have the following. **Corollary 4**. :
1. _There are 183 equivalence classes of minimal convex imaginary cubes of a given cube_ \(C\)_._
2. _There are 16 equivalence classes modulo rotational equivalence._
3. _There are 15 equivalence classes modulo reflective equivalence._  
The cube-vertices column of Table 1 lists all the 16 subsets of \(C\)-vertices without stars modulo rotational equivalence. Note that No.10(L) and No.10(R) form a pair of mirror images and all the other ones have mirror symmetry. Though some imaginary cubes are minimal convex imaginary cubes of two different cubes, such an imaginary cube is in the same equivalence class (No. 5) for both of the cubes, as we will explain in the next section. Therefore, we can say that there are 16 (or 15) equivalence classes of minimal convex imaginary cubes modulo rotational (or reflective) equivalence. As in Table 1, we define a representative imaginary cube of each equivalence class by taking middle points of cube-edges as e-vertices. The two imaginary cubes H and T appear as the representative imaginary cube of No. 5 and No. 8, respectively. This choice of the representative is natural in that the rotation group (or the full symmetry group) of such an imaginary cube is bigger or equal to that of the corresponding cube with the v-vertices colored. Note that they are not equal for H as we will explain in the next section in detail. Note that No.10(L) and No.10(R) have rotational symmetry though they do not have mirror symmetry. The nets of the 16 representatives of minimal convex imaginary cubes are available from the author's homepage [2]. Figure 5 shows wooden models of all the 16 representatives. A sculpture which uses all of these wooden pieces is presented in [3]. **Figure 5**. **(a)** Wooden imaginary cubes; **(b)** The same as **(a)** from a different inclination. Woodworks by [PERSON]. \begin{table}
\begin{tabular}{|c c c|c c c|} \hline
**Number** & **Cube-** & **Imaginary** & **Number** & **Cube-** & **Imaginary** \\
**\#vertices** & **Cube-** & **cube** & **(\#faces,** **\#vertices**) & **vertices** & **vertices** & **cube** \\ \hline \hline
1 & & & & 9 & \\ (14,12) & & & & (10,8) & \\ Cubocta- & & & & Quadric & \\ hedron & \(\{\}\) & & antiprismoid & \(\{1,2,3,4\}\) & \\ \hline
2 & & & & 10(L) & \\ (13,10) & & & (10,7) & \\ \(\{1\}\) & & & & \(\{1,2,3,6\}\) & \\ \hline
3 & & & & 10(R) & \\ (12,9) & & & (10,7) & \\ \(\{1,2\}\) & & & & \(\{1,2,3,7\}\) & \\ \hline
4 & & & & 11 & \\ (11,8) & & & (8,6) & \\ \(\{1,4\}\) & & & & \(\{1,2,3,8\}\) & \\ \hline
5 & & & & & \\ (12,8) & & & & & \\ Hexagonal & & & & & \\ bipyramid & \(\{1,8\}\) & & & & \\ \hline
6 & & & & 13 & \\ (11,8) & & & & Regular & \\  & \(\{1,2,3\}\) & & & tetrahedron & \\ \hline
7 & & & & 14 & \\ (10,7) & & & & (8.6) & \\  & \(\{1,2,7\}\) & & & & \\ \hline
8 & & & & 15 & \\ Triangular & & & & & \\ antiprismoid & \(\{1,4,6\}\) & & & \\ \hline \end{tabular}
\end{table} 
Table 1: The 16 (or 15) representatives of minimal convex imaginary cubes.