Name this periodic tiling - MathOverflow

Name this periodic tiling

2011-10-12T20:16:00Z

Hello MO,

I've been working on a problem I'm working on in ergodic theory (finding Alpern lemmas for measure-preserving $\mathbb R^d$ actions) and have found some neat tilings, that I presume were previously known.

They are periodic tilings of $\mathbb R^d$ by a single prototile consisting of any box with a smaller box removed from a corner. The two-dimensional case is illustrated in the figure below:

There exist versions of this in arbitrary dimensions, irrespective of the size of the boxes. Furthermore, there appear to be many essentially different tilings using the same prototile.

Does anybody recognize the $d$-dimensional version, or have a name for it?

Thanks

Answer by Ryan Budney for Name this periodic tiling

2011-10-12T21:55:33Z

Hi Anthony,

I maybe should walk down the hall... but this is easier.

Dual to the $d$-cubical tiling of $\mathbb R^d$ would be the "cross-polytope" tiling. This is the tiling made up of the duals -- the vertices are the centres of the $d$-cubes, the edges of the cross-polytope are the faces of the $d$-cubes, and so on.

To me it looks like you get your tiling from the cross-polytope tiling, by simply scaling up each tile appropriately -- scaling each tile at one of its vertices, and doing the scaling symmetrically with respect to the translation symmetry of the tiling. So as you scale, part of the tile vanishes (from a growing tile eating it up) and part gets created (via scaling).

Your picture appears to be consistent with something like that. Or rather than the cross-polytope tiling, it could the the same idea but with the cubical tiling.

edit:

Take this procedure for generating a tiling of $\mathbb R^n$. Let $M$ be an $n \times n$ invertible matrix with real entries. Let $\vec v \in \mathbb Z^n$. In the lexicographical order on $\mathbb Z^n$ we can lay down a "tile" being $[0,1]^n + M\vec v$, where whenever we place a tile, it overwrites any old tile that it may be placed on top of. Provided the norm of the matrix is small enough, this procedure writes over the entire plane. It produces a tiling a fair bit more general than what you're talking about. I'd call it perhaps a linear overlapping translation of the cubical tiling. But I don't know if there's standard names for such thing. Perhaps "lizard scales" ?

The tiling in your picture looks like something like this, with a 2x2 matrix with entries (left to right then top to bottom) $2/3, -1/3, 1/3, 3/4$. I seem to have forgotten how to typeset a $2\times 2$ matrix in mathjax or whatever is powering this website nowadays...

Does that make sense?

Answer by Kevin O'Bryant for Name this periodic tiling

2011-10-13T02:03:00Z

Kolountzakis worked with some tilings of this sort in his paper "Lattice tilings by cubes: whole, notched and extended", Electr. J. Combinatorics 5 (1998), 1, R14.