16 Minimal Convex Imaginary Cubes

Woodworks by Hiroshi Nakagawa



A convex imaginary cube is obtained by truncating a cube (possibly infinitely many times). A minimal convex imaginary cube is a convex imaginary cube such that if it is further truncated then it no longer has the same three square projections. One can show that a minimal convex imaginary cube is a convex polygon. One can also show that the set of minimal convex imaginary cubes is divided into 16 equivalence classes. The list of representatives of 16 equivalence classes is here . For the details, see Bridges 2010 paper. Here is all the representative minimal convex imaginary cubes, made of wood.

It is difficult to imagine that these polyhedra have this property if they are looked at solely, but once each of them is put in an acrylic resin box with one side open, one can easily see that it is an imaginary cube just by looking at it from the faces of the box. It is a good mathematical puzzle how to place imaginary cubes in a box.





The 16 imaginary cubes are minimal convex ones. If there is an imaginary cube, its convex hull is also an imaginary cube. Therefore, we are interested in convex ones. If there is a convex imaginary cube, any object which containes this and is contained in the surrounding cube is also an imaginary cube. Therefore, we are interested in minimal convex ones for a fixed surrounding cube. One can easily show that minimal convex imaginary cube is a polyhedron. Therefore, in all of the 6 square appearances, the square is divided into polygons. We consider two divisions of a square into polygons are equivalent if both consists of same kinds of polygons (like triangle, tetragon,..) connected in the same way through edges. We consider two minimal convex imaginary cubes equivalent if they have the same kind of divisions in all the six appearances. With this equivalence, we can show that there are 16 minimal convex imaginary cubes in this picture.



Paper models of all of them were made with the help of Kei Terayama in 2007.

Paper models of this polyhedra available here It is only permitted for personal and educational uses.