Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture (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.

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