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

Posted on April 15, 2018 by Gil Kalai

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. Sándor Kolumbán says:
    April 15, 2018 at 6:27 pm

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

    Reply
  3. Jineon Baek says:
    April 17, 2018 at 6:29 pm

    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…

    Reply
  4. bjonas says:
    April 22, 2018 at 8:37 pm

    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/

    Reply

  5. Pingback: Testing *My* Intuition (34): Tiling High Dimension with an Arbitrary Low-Dimensional Tile. | Combinatorics and more

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