Aperiodic cube tilings in all dimensions $n\geq3$

geometryplane-geometrytessellationstiling

In a paper the following statement is made (link):

"We note that in high dimensions there are many "exotic" cube tilings. There are aperiodic cube tilings in all dimensions $n\geq3$, while in dimensions $n\geq10$ there are cube tilings in which no two cubes share a common $(n-1)$-face (Lagarias and Shor [8])"

The last part is not relevant for the question, and the reference only verifies that part. The authors do not provide any reference for the claim "There are aperiodic cube tilings in all dimensions $n\geq 3$ ".

I have done some extensive searching and I cannot find any examples of this. There are many who constructs aperiodic tiles from the unit cube/n-cube/n-dimensional hypercube, but I have not found an example which uses an unmodified(!) version. In this context, especially considering the latter part of the sentence (which only is for unmodified cubes) "cube tilings" here must be of the unmodified type. Examples of modified ones are ones such as Ammann tiles, or the L tile by Chaim Goodman-Strauss.

Given that the unit cube is so important in mathematics, why is there no mention of such a tiling on even the wikipedia list of aperiodic tiles? And given that they have not referenced this, is this common knowledge? Or is it totally wrong since I can find no books/papers/drawings/evidence of this.

Best Answer

I cannot provide a reference and I am not sure whether I read about it somewhere, but the following construction comes to mind.

Start with the usual regular tessellation of the 3-dimensional space by cubes.

Now,

  1. every column of cubes of the form $(x,y,z)$ with fixed even $x,y$ and $z$ running over all integers is shifted in the vertical direction; we require the shifts to be different and aperiodic over $x,y$. (For example, we can shift the column $(x,y,z)$ by $0.1$ if the total number of 1 bits in the binary expansion for $|x|$ and $|y|$ is odd or by $0.2$ otherwise, in the Thue-Morse spirit.)
  2. Every row of cubes of the form $(x,y,z)$ with fixed odd $x$, fixed even $z$, and $y$ running over all integers is shifted in the $y$ direction; again, the shifts are different and aperiodic over $x,z$.
  3. Every row of cubes of the form $(x,y,z)$ with fixed odd $y$, fixed odd $z$, and $x$ running over all integers is shifted in the $x$ direction; the shifts are different and aperiodic over $y,z$.

All these rows and columns can be shifted independently. The resulting cubic tessellation is aperiodic in all direction. This apparently generalizes to higher dimensions but doesn't work in the plane.

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