# Can every tromino (including those with gaps) tile the plane?

I've generalized trominos to include "gaps", i.e. they are formed by removing all but $3$ squares from an $n$-omino where $n$ is finite.

The generalized trominos pictured above can tile the plane using only translation.

The generalized trominos pictured above can tile the plane using only translation and reflection.



Can all generalized trominos tile the plane employing translation, rotation, and reflection?

More interestingly, can all generalized $k$-polyhypercubes tile $k-1$ dimensional space?

• Try XX X, and related, for a series that does not tile the plane without rotation. – The Masked Avenger Mar 20 '15 at 21:33
• Note that this is problem is invariant under the action of $GL_n(\mathbb{Z})$, so if it holds for one linearly independent omino, it holds for all of them. But classifying exactly when a non-independent omino tiles seems nontrivial. – Eric Wofsey Mar 20 '15 at 21:41
• I'll edit the question to remove the "only using translation" criterion. – Bob Spaghetti Mar 20 '15 at 21:43
• If two of the three points (=unit squares) have a common coordinate, it is easy: Suppose wlog the squares are at (0,0), (0,a) and (x,y). Fill with (a-1) horizontal translates. So we have covered [0,2a-1] x {0} and [x,x+a-1] x {y}. Add the rotation of this whole thing around the point (x/2-1/2, y/2) - and we have covered two parallel segments (i.e. strips of width 1) of equal length, which tile the whole plane. So "only" the general case with all coordinates different remains. – Wolfgang Mar 21 '15 at 10:25
• I don't have access to Newman, D. J., Tesselation of integers, J. Number Theory 9 (1977), 107–111, but I believe it solves the problem of tiling the integers using only translation, if the tile has a prime power number of elements (e.g., $3$). Type "tiling the integers" into Google for other work on the topic. – Gerry Myerson Mar 21 '15 at 21:59

Each four-celled animal tiles the plane!

http://www.sciencedirect.com/science/article/pii/0097316585901050

The corresponding result for 1D three-celled "animals" holds as well. This was a recent problem in the German Math Olympiad (Problem 531046).

Together with Wolfgang's comment this solves the original problem, since if all coordinates of the three squares are different, then one starts by horizontal translation of the "animal" by all elements of $\mathbb Z$, and then uses the 1D result.

• Thanks for the information, however not everyone (including myself) is going to have access to that article. Using my original terminology, I wonder now if all generalized $k$-polyhypercubes can tile $n$ dimensional space where $n \geq k - 2$. – Bob Spaghetti Mar 22 '15 at 0:06
• @IanFarrell: I placed the Coppersmith paper, "Each four-celled animal tiles the plane," at this PDF download link. – Joseph O'Rourke Mar 22 '15 at 0:45
• mathoverflow.net/questions/49915/… has related information. See in particular the most recent comments. – user69547 Mar 22 '15 at 7:50

Imagining the plane as a checkerboard, there are two exhaustive cases for generalized trominos:

(1) All monominos are the same color. (2) Exactly two monominos are the same color.$\\\ \\\$

Case (1) can split into two exhaustive sub-cases.

(1.1) Two pairs of monominos are an equal distance apart either horizontally and/or vertically.

• These generalized trominos will always tile the plane using only translation.

1. In the direction of the equal gaps, copy translates $n$ times where $n$ is the size of the gap.
2. Copy the result and join them together in the opposite direction of the equal gaps an infinite number of times.
3. Copy the result and join them together in the direction of the equal gaps an infinite number of times.

Step 1 and the first six iterations of step 2 are depicted above.

(1.2) Three pairs of monominos are an unequal distance apart both horizontally and vertically.

• user69547's answer demonstrates these generalized trominos can tile the plane employing both translation and reflection.$\\\ \\\$

Case (2) can split into three exhaustive sub-cases.

(2.1) No pairs of monominos are collinear.

• I believe these generalized trominos can tile the plane using only translation based on data.

(2.2) One or two distinct pairs of monominos are collinear.

• These generalized trominos can always tile the plane using only translation by recursively generating infinite parallel lines at an equal distance from one another.

1. Copy translates horizontally to the right until doing so will cause a collision.
2. Copy the result and join the left-side of the bottom line to the right-side of the top line an infinite number of times.
3. Copy the result and join the infinite number of line segments together horizontally an infinite number of times.
4. Copy translates of all infinite horizontal lines vertically until no gaps remain.

Step 1 and the first iteration of step 2 are depicted above.

(2.3) Three distinct pairs of monominos are collinear.

• user69547's answer demonstrates these generalized trominos can tile the plane employing both translation and reflection.
• Do not vandalise your post – user88558 Jan 18 '17 at 1:47
• Bob, if you wish to delete your own post, you can do so. meta.stackexchange.com/questions/5221/… Or someone can do that for you. But Yvette is right: mutilating a post is considered a site violation. – Todd Trimble Jan 18 '17 at 3:19