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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Col. George Sicherman [ HOME | MAIL ]