Multidimensional Impossible Polycubes

We first derive a 3-dimensional impossible polycube by forcibly deforming the projection of a 3-dimensional polycube. This procedure is extended into n(≥4)-space to construct n-dimensional impossible polycubes represented in 2or 3-space. They are useful as fundamental grid patterns for imaging various n-dimensional impossible figures in our 3-space. On 2-space, especially, each pattern can be composed of [n/2] kinds of rhombi grouped into n congruent periodic portions which spirally fill a semi-regular 2n-gon. The same [n/2] kinds of rhombi compose a radial quasi-periodic pattern in a regular 2n-gon which is derived from the projection of an n-dimensional polycube. Basic n-dimensional figures in this paper An n-dimensional impossible polycube (an impossible n-polycube) in this paper is composed of mainly two kinds of n-dimensional regular polytopes. One is the n-dimensional cube (n-cube) as the 2n-cell composed of 2n numbers of (n-1)-cubes, while the other is the n-dimensional regular tetrahedron (n-tetrahedron) as the (n+1)-cell or n-dimensional simplex composed of n+1 numbers of (n-1)-tetrahedra. Several n-cubes may construct an n-polycube which is a portion of space-filling agglomeration of n-cubes. On the contrary, 3-dimensional regular tetrahedra (3-tetrahedra) can fill 3-space with 3-dimensional regular octahedra, each being the dual of a 3-cube. Buckminster Fuller’s octet-truss or the polyoctet in this paper is a portion of this triangular 3-dimensional space-filling agglomeration (Figure 1). They are usually represented by orthogonal isometric projections into 2or 3-space in this paper. Figure 1 : Left three columns, solid models with each line pattern of n-cubes embedded in regular 2n-gons (top row) and that of n-tetrahedra embedded in regular (n+1)-gons (bottom row). From left to right, n=3, 4, and 5. Right end column, line patterns of a 3-polycube (top) and a polyoctet (bottom). CG: M. Ishii. Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture