Testing *My* Intuition (34): Tiling High Dimension with an Arbitrary Low-Dimensional Tile.

Test your intuition 34 asked the following:

A tile is a finite subset of . We can ask if can or cannot be partitioned into copies of . If   can be partitioned into copies of we say that tiles .

Here is a simpe example. Let consists of 24 points of the 5 by 5 planar grid minus the center point. cannot tile .

Test your intuition: Does tiles for some ?

We had a poll and 58% of voters said YES. The answer is

YES!

As a matter of fact Adam Chalcraft have made the beautiful conjecture that every tile in tiles for some large .  This conjecture was proved by Vytautas Gruslys, Imre Leader,  and Ta Sheng Tan in their remarkable paper Tiling with arbitrary tiles.

Theorem (Vytautas Gruslys, Imre Leader,  and Ta Sheng Tan): Let   be a tile. Then tiles for some .

But wait, what about our tile T? After seeing the abstract of Imre Leader’s lecture, looking briefly at the paper which contained the 5 by 5 minus the middle example, I posted the question on my blog.  But then driving to Jerusalem I suddenly was sure that there is no way in the world the hole in T can be filled up by another tile of the same shape.  T is simply too fat, I thought. I must have missed something –  some extra condition or subtelty that I overlooked. It turned out that my intuition was wrong already after I saw the right answer. (This does happen from time to time.)

When I had a chance I looked again at the paper, and saw a beautiful picture explaining how the hole can be filled in four dimension. (BTW, I don’t know what is the minimum dimension that T can tile.)