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.
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Can all generalized trominos tile the plane employing translation, rotation, and reflection?
More interestingly, can all generalized $k$-polyhypercubes tile $k-1$ dimensional space?
 A: 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.
A: 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.


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


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*In the direction of the equal gaps, copy translates $n$ times where $n$ is the size of the gap.

*Copy the result and join them together in the opposite direction of the equal gaps an infinite number of times.

*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.


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*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.


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*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.


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


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*Copy translates horizontally to the right until doing so will cause a collision.

*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.

*Copy the result and join the infinite number of line segments together horizontally an infinite number of times.

*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.


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

