# Didom Kites and Bricks

## Introduction

A *polydom* (or *polyDom*) is a plane polygon
formed by joining diagonal halves of a domino (1×2 right
triangles) at their short legs, long legs, half long legs,
and hypotenuses.
The term *polyDom*
was proposed by Andrew Clarke
and chosen in an election organized by Livio Zucca.
In 2000, Livio
started to study tilings by polydoms.
See this page
of Livio's at
Giovanni Resta's site
**iread.it**.

## Polydom Tilings

Livio and his contributors show a variety of plane shapes
tiled with monodoms or polydoms.
Here is one of Livio's constructions:

It has Dom triangles aligned to three different square grids.
In other words, it has two of what Livio calls jumps of the grid.

## Tiling Rectangles

About 2/3 of the way down Livio's page,
Livio considers tiling rectangles with copies of
two didoms, the *Kite* and the *Brick*:

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.*

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Polyform Curiosities

Col. George Sicherman
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