In 2000, Livio started to study tilings by polydoms. See this page of Livio's at Giovanni Resta's site iread.it.
It has Dom triangles aligned to three different square grids.
In other words, it has two of what Livio calls jumps of the grid.
Of the 13 didoms, the Kite is unique in that it does not conform to a square grid. So any tiling that uses Kites will have grid jumps. The Brick is simply a polyomino with 2 cells, a domino. In Livio's three-grid construction above, the pieces can be joined in pairs to form 24 Kites and 26 Bricks.
After showing a 10×25 rectangle tiled with 52 Kites and 73 Bricks, Livio remarks:
It's possible to cover a 10×10 square with 50 bricks, it's trivial. Also to insert 6 kites and 44 bricks it's easy. 28 kites and 22 bricks are less easy. Do you want to try? Is 28 the maximum number of kites?
As Livio observes, any rectangle with integer sides and even area can be tiled with Bricks alone. We may adapt Livio's challenge to any such rectangle: how many Kites can we use in tiling it with Kites and Bricks?
The smallest rectangle that can be tiled using some Kites is 4×5. It has 6 Kites and 4 Bricks:
Here are the greatest known numbers of Kites for tiling various rectangles with Kites and Bricks. How many can you find?
Height and Width | Kites | Bricks | Kite Percentage | Solution |
---|---|---|---|---|
6×6 | 8 | 10 | 44.444 | CLICK |
6×7 | 10 | 11 | 47.619 | CLICK |
6×9 | 12 | 15 | 44.444 | CLICK |
8×9 | 20 | 16 | 55.555 | CLICK |
7×14 | 28 | 21 | 57.143 | CLICK |
10×10 | 28 | 22 | 56.000 | CLICK |
9×16 | 44 | 28 | 61.111 | CLICK |
12×14 | 52 | 32 | 61.905 | CLICK |
12×15 | 58 | 32 | 64.444 | CLICK |
These solutions with maximum Kites are not necessarily unique.
Last revised 2024-01-29.