Multidimensional Impossible Polycubes

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Koji Miyazaki: Multidimensional Impossible Polycubes. In: Bridges 2013. Pages 79–86



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.

Extended Abstract


Used References

[1] K. Miyazaki, “Impossible Polycube – four-dimensional version”, Experience-centered Approach and Visuality In The Education of Mathematics and Physics, S. Jablan, et al. ed., the Kaposvar University (2012), pp.180-182.

[2] S. Kim, “An Impossible Four-Dimensional Illusion”, Hypergraphics: Visualizing Complex Relation- ships in Art, Science and Technology, D. Brisson ed., Westview Press (1978), pp.187-239.

[3] B.Grünbaum, G.C.Shephard, “Tilings and Patterns”, W.H.Freeman (1987), pp.556-557.


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