A triangle $T(n)$ is a polyomino with columns on the same base with lengths $1, 2, 3, \cdots, n$.
In this slide deck (original PPT, images may be corrupt), Friedman looks at tilings of these triangles by exactly two tiles. I'm wondering what single tiles can tile triangles. So far, I have been only able to find tilings by smaller right trominoes (and monominoes, of course), such as this one:
In the deck, Friedman identifies some properties of tiling sets that can tile triangles, which limits the possibilities if the tileset has only one tile:
- Tiles must have the diagonal property; i.e. the ability to cover more diagonal cells (red, marked with a dot) than those below them (green, marked with a square):
- Tiles must have the bottom property; i.e. the ability to cover more bottom cells (red, marked with a dot) than those above them (green, marked with a square):
This already limits possible tiles to only triangles for small polyominoes. The T-tetromino and the polyomino below have both these properties:
(Clearly, it cannot tile a triangle. Other examples are similar if the tiles are small... but its not clear what may happen if the tiles are much bigger.)
If the rectangular hull is not square, we can place a tile so that its hull is vertical or horizontal. The tile in the top corner must be vertical, and the one in the bottom must be horizontal. Therefore there must be at least one transition between horizontal and vertical. This is useful to eliminate certain infinite patterns. For example, the T-tetromino can fit in the top corner in only one way. The next open diagonal cell and its neighbors can only be tiled with vertical tetrominoes; and so all diagonal cells can only be tiled with vertical dominoes. But we eventually need a horizontal tile; since this is impossible there is no tiling of triangles by the T-tetromino.
This is called the "orientation theorem" in the deck.
A region/tile is called balanced if it has the same number of black and white squares under the checkerboard coloring. Triangles are never balanced, so they cannot be tiled by a tile that is balanced.
All these above bring us quite far. My question is:
Are there any other shapes that can tile triangles?
(An earlier version of this question said the only tilings I could find were by other triangles. Those look feasible, but I have not in fact be able to find one of those either. Note that none of the properties mentioned above can be used to rule out triangular tiles.)