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

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.

I forgot to add polls…

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

1. I can see T_0 tiling Z^3. My intuition says T cannot do it. We’ll see 🙂

• Johnson says:

I’d like to have some hints on how T_0 can tile Z^3.

2. Gil Kalai says:

Polls added!

3. Jineon Baek says:

I really want to do a ‘weighted vote’, like, 70% yes and 30% no. Think I will wait and observe the problem till the intuition shapes into something…

4. bjonas says:

This isn’t for my intuition, because I remember this question from MathOverflow. It’s a really good one. SPOILER: https://mathoverflow.net/q/49915/