Test Your Intuition (34): Tiling high dimensional spaces with two-dimensional tiles.

Combinatorics and more 2018-04-15

A tile $T$ is a finite subset of $\mathbb Z^d$ . We can ask if $\mathbb Z^d$ can or cannot be partitioned into copies of $T$ . If $\mathbb Z^d$ can be partitioned into copies of $T$ we say that $T$ tiles $\mathbb Z^d$ .

Here is a simpe example. Let $T$ consists of 24 points of the 5 by 5 planar grid minus the center point. $T$ cannot tile $\mathbb Z^2$ .

Test your intuition: Does $T$ tiles $\mathbb Z^d$ for some $d>2$ ?

If you prefer you can think about the simpler case of $T_0$ consisting of eight points: the 3 by 3 grid minus the center.