Multidimensional Impossible Polycubes

Koji Miyazaki
Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (2013)
Pages 79–86 Regular Papers


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 2- or 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.